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Chen3, 4 +1Key Laboratory for Matter Microstructure and Function of Hunan Province, +Key Laboratory of Low-Dimensional Quantum Structures and Quantum Control of Ministry of Education, +School of Physics and Electronics, Hunan Normal University, Changsha 410081, China +2Key Laboratory for Micro-Nano Physics and Technology of Hunan Province, +College of Materials Science and Engineering, Hunan University, Changsha 410082, China +3Shenyang National Laboratory for Materials Science, Institute of Metal Research, +Chinese Academy of Sciences, 110016 Shenyang, People’s Republic of China. +4School of Materials Science and Engineering, University of Science and +Technology of China, 110016 Shenyang, People’s Republic of China. +(Dated: January 3, 2023) +Ferroelectric control of two-dimensional magnetism is promising in fabricating electronic devices +with high speed and low energy consumption. The newly discovered layered MnBi2Te4(Bi2Te3)n +and their Sb counterparts exhibit A-type antiferromagnetism with intriguing topological proper- +ties. Here, we propose to obtain tunable magnetic multistates in their thin films by ferroelectrically +manipulating the interlayer magnetic couplings (IMCs) based on the Heisenberg model and first- +principles calculations. Our strategy relies on that interfacing the thin films with appropriate ferro- +electric materials can switch on/off an interlayer hopping channel between Mn-eg orbitals as the po- +larizations reversed, thus resulting in a switchable interlayer antiferromagnetism-to-ferromagnetism +transition. On the other hand, the interface effect leads to asymmetric energy barrier heights for +the two polarization states. These properties allow us to build ferroelectrically switchable triple +and quadruple magnetic states with multiple Chern numbers in thin films. Our study reveals that +ferroelectrically switchable magnetic and topological multistates in MnBi2Te4 family can be ob- +tained by rational design for multifunctional electronic devices, which can also be applied to other +two-dimensional magnetic materials. +I. +INTRODUCTION +Two-dimensional (2D) magnetic materials provide an +ideal platform to explore novel magnetic and electronic +properties1–7. The delicate interlayer exchange couplings +in these systems enable a variety of methods of manipu- +lating their magnetism. For instance, recent studies re- +vealed that the A-type antiferromagnetic (AFM) CrI3 +bilayer could be tuned into ferromagnetic (FM) by ex- +ternal electric field8,9, electrostatic doping10,11, and in- +terface engineering12–17. Twisting the bilayer may yield +non-collinear magnetism18–20. +Recently, +the +MnBi2Te4 +family, +i.e., +MnBi2Te4(Bi2Te3)n and MnSb2Te4(Bi2Te3)n, which are +hereafter referred to as MAT, have much great attention +due to the coexistence and interesting interplay of +intrinsic magnetism and band topology in them21–37. +This series of materials also have a layered van der +Waals (vdW) structure with an A-type AFM structure +as revealed by experiments38–41, which preserves the +combination of the time-reversal and a half lattice +translation symmetry. As a result, many of them show +nontrivial topological properties such as topological +insulators21 and axion insulators22. +Moreover, the +systems can be turned into Weyl semimetals as the +interlayer couplings become ferromagnetic22,23,36. +The A-type AFM coupling in MAT yields unusual +even-odd layer-number dependent magnetism for their +thin films35,42,43. The even-number (even-N) systems are +expected to have no net magnetization. Whereas those +with odd-number (odd-N) layers have uncompensated +magnetization. This difference can lead to distinct topo- +logical properties for them. For instance, the thin films +of MnBi2Te4 with odd-N septuple layers are quantum +anomalous Hall insulators. However, those with even-N +layers have a zero Chern number42. Instead, their topo- +logical properties can be characterized by the so-called +pseudospin Chern number44. Due to the weak interlayer +interaction, small magnetic fields could induce spin-flip +transitions, giving rise to an AFM-to-FM transition in +the IMCs35,40. +Chemical dopings45,46 and antisite de- +fects47–51 can also be used to manipulate the IMCs in +these systems, although they may complicate the nature +of the surface states. +First-principles calculations sug- +gest that the AFM double-septuple MnBi2Te4 could be +driven into a Chern insulator with a high Chern number +under electric fields52. +In this work, we propose to ferroelectrically tune the +IMCs in MnBi2Te4 thin films for magnetic multistates by +interface, which is desired for memory devices with high +density storage, high speed, and low power consump- +tion. We reveal that hole doping can lead to an interlayer +AFM-to-FM transition in MAT bilayers based on the un- +derstanding of the IMCs using the Heisenberg model. We +provide a guideline for designing ferroelectric substrates +that may induce transitions in the interlayer exchange +couplings, i.e., polarization dependent IMCs, as demon- +strated by our first-principles calculations. Moreover, we +arXiv:2301.00515v1 [cond-mat.mtrl-sci] 2 Jan 2023 + +2 +find that the interface effect results in symmetry breaking +in the two polarization states of the FE substrate. This +asymmetry allows us to design switchable magnetic mul- +tistates in sandwich structures made of MnBi2Te4 mul- +tilayers and 2D ferroelectric (FE) materials, which also +exhibit distinct electronic and topological properties. +We begin by presenting the concept of FE tuning of +IMCs in MAT bilayers, which is shown in Fig. 1. In these +systems, each Mn atom is coordinated with six chalcogen +atoms, which form a distorted octahedron. The Mn-3d +orbitals are split into triply degenerate t2g states and the +doubly degenerate eg states due to the octahedral ligand +field. The states are further split due to the magnetic +exchange interaction between the Mn atoms. As a re- +sult, the majority states of the t2g and eg orbitals are +fully occupied by the five d electrons of the Mn2+ ions +(see Fig. 1), resulting in a high spin state for the Mn2+ +ions. +Like the 2D magnetic bilayers reported by Refs +31 and 53, this type of occupation favors AFM IMCs +between the Mn+2 ions, which are mediated by the p- +orbitals of the nonmetallic atoms (denoted by {p . . . p}). +Whereas FM IMCs are energetically unfavorable because +the e↑ +g − {p . . . p} − e↑ +g hopping between the Mn-d or- +bitals of adjacent layers is prohibited31,53,54. +For our +systems, reducing the occupation of the d orbitals makes +the e↑ +g − {p . . . p} − e↑ +g hopping channel energetically fa- +vorable, thus enhancing the stability of the FM IMCs. +Indeed, our DFT calculations indicate that all the MAT +bilayers undergo the AFM-to-FM transition by small hole +dopings (see Fig. 1c and Fig. S1), which is also expected +for their multilayers. +The IMCs in MAT-2L can be understood using the +following spin Hamiltonian. +H = +� +i,j +Jt +∥Si · Sj + +� +m,n +Jb +∥Sm · Sn + +� +i,m +J⊥Si · Sm, (1) +where J∥ and J⊥ denote the intra- and interlayer ex- +change interactions between the Mn ions, respectively. +The intralayer ones are denoted by Jt +∥ for the top layer +and Jb +∥ for the bottom layer, for which only the first +nearest-neighbor interactions are taken into account. Jt +∥ +are equal to Jb +∥ for the freestanding MAT-2L. Whereas +for the interlayer ones, the second nearest neighbors are +included. +For the bilayers without doping, we obtain +positive J1st +⊥ +and negative J2nd +⊥ +(see Fig. 1d, Fig. S2 and +Table S1). +Note that the magnitude of J1st +⊥ +is larger +than that of J2nd +⊥ +. So, the sum of J1st +⊥ +and J2nd +⊥ +, i.e., +¯ +J⊥ = J1st +⊥ ++ J2nd +⊥ +, is positive, which gives rise to AFM +IMCs. Introducing hole doping suppresses J1st +⊥ +while en- +hances J2nd +⊥ +. As a result, ¯ +J⊥ decreases with increasing +of the hole doping and eventually changes its sign across +the AFM-to-FM transition (see Fig. 1d). +The hole doping over the MAT bilayers can be achieved +via interfacial charge transfer which requires suitable +band alignments between them and the substrates. +When their bands are in the type-I or type-II align- +ment, interfacial charge transfer can be negligible. +In +(b) +MAT-2L(FM)/P↓ +Mn1 +t2g +eg +EF +t2g +eg +P↓ +e +CBM +VBM +Mn2 +t2g +eg +t2g +eg +MAT-2L(AFM)/P↑ +(a) +Mn1 +t2g +eg +EF +t2g +eg +Mn2 +t2g +eg +t2g +eg +e +P↑ +CBM +VBM +(c) +0.4 +0 +-0.4 +0.2 +-0.2 +-0.6 +∆EFM-AFM (meV) +n (hole/Mn pair) +0 +0.02 +0.04 +0.06 +AFM +FM +MnBi2Te4-2L +MnSb2Te4-2L +(d) +n (hole/Mn pair) +0 +0.02 +0.04 +0.06 +AFM +FM +MnSb2Te4-2L +0.08 +0 +-0.08 +0.04 +-0.04 +-0.12 +n (hole/Mn pair) +0 +0.02 +0.04 +0.06 +AFM +FM +J⊥ (meV) +MnBi2Te4-2L +1st +J⊥ +2nd +J⊥ +J⊥ +FIG. 1. Interface engineering of the IMCs in MAT bilayers. +(a) Spin states of Mn ions for the AFM interlayer coupling. In +the presence of a substrate that has a type-I or type-II band +alignment with the bilayer, the IMCs remain AFM. (b) The +IMCs becomes FM when there is a type-III band alignment +between them so that the CBM of the substrate lower than +the VBM of the MAT bilayer. In (b), the white circles denote +hole dopings to the Mn-eg orbital. In (a, b), arrows denote +spins. The dark and light red curves with an arrow denote +hopping channels. +The one marked by a cross means that +electron hoppings are prohibited. We use a FE material as +the substrate, whose polarizations are labeled by P. P ↑ and +P ↓ represent the up and down polarizations, respectively. (c) +Energy difference between the FM and AFM states as a func- +tion of hole doping for freestanding MnBi2Te4 and MnSb2Te4 +bilayers. ∆E = EF M − EAF M. (d) Doping dependence of J⊥ +for the two systems. +¯ +J⊥ = J1st +⊥ ++ J2nd +⊥ +. +these cases, the IMCs are most likely to be AFM. In +contrast, electrons will be transferred from the MAT bi- +layer to the substrate when they are in the type-III band +alignment that the valence band maximum (VBM) of +the MAT bilayer are higher than the conduction band +minimum (CBM) of the substrate.Namely, hole doping +is introduced to the MAT bilayer, which is desired for +the AFM-to-FM transition. Ferroelectrically switchable +IMCs may be achieved if a 2D FE materials serves as the +substrate so that reversing its polarizations gives rise to +a switching of the band alignment from type-III to type-I +(II) or vice versa (see Fig. 1). However, one can expect +that the transferred electrons mainly come from the in- +terfacial MAT layer because of the vdW-type interlayer +bonding.Therefore, the spin-flipping most likely happen +to the interfacial MAT layer rather than those further +away from the substrate. +We now come to first-principles calculations of the het- +erostructures of MAT thin films and 2D FE materials, +which were performed using the Vienna Ab initio Sim- +ulation Package55. We choose In2Se3 monolayer as the +substrate, which has been experimentally proved since +its prediction in 201756–58. +Their heterostructures are +built by slightly adjusting the lattice of In2Se3 (The lat- +tice mismatch between them is small). The pseudopoten- +tials were constructed by the projector augmented wave +method59,60. +An 11 × 11 × 1 and 21 × 21 × 1 Γ- + +3 +centered k-mesh were used to sample the 2D Brillouin +zone for structural relaxation and electronic structure +calculations, respectively. The plane-wave energy cutoff +is set to 400 eV for all calculations. +A 20 ˚A vacuum +region was used between adjacent plates to avoid the +interaction between neighboring periodic images. +Van +der Waals (vdW) dispersion forces between the adsorbate +and the substrate were accounted for through the DFT- +D361. Different vdW methods/functionals such as DFT- +D2 and optPBE-vdW were also used for comparison62–64. +The systems were fully relaxed until the residual force on +each atom is less than 0.01 eV/˚A. The DFT+U method65 +is used to treat electron correlations due to the partially +filled d-orbital of Mn for which a value of 5.34 eV is +used21. Our results on the structural properties, mag- +netism, and band structures of the free-standing MAT +films are consistent with previous studies21,32,36. +The +kinetic pathways of transitions between different polar- +ization states are calculated using the climbing image +nudge elastic band (CI-NEB) method66,67. The topolog- +ical properties calculations were done using the WAN- +NIER9068 and WannierTools package69. +We have performed careful calculations over a number +of stacking configurations (see Fig. S3, and Table S2). +The lowest energy configuration is shown in Fig. 2a, in +which the interfacial Se and Te are in the hollow sites. +The stacking order is the same as the one for MnBi2Te4 +monolayer on In2Se370, which shows up for all MAT bi- +layers on In2Se3. Table I summarizes the stability of the +two magnetic states for MAT bilayers on In2Se3 mono- +layer in different polarization states. One can see that for +all the MAT bilayers the IMCs remain AFM when the +polarization is pointing toward the interface, but become +FM as the polarization is reversed. The trend is indepen- +dent of the vdW functionals/methods (Table S3). Below +we focus on the electronic structure of MnBi2Te4 bilayer +on In2Se3 monolayer, i.e., MnBi2Te4-2L/In2Se3, which +are shown in Figs. 2b-d. Those for all other MAT systems +are shown in Figs.S4-S6 since they show pretty much the +same trend as MnBi2Te4-2L/In2Se3. The Fermi level is +located in the band gap for the AFM state. Whereas for +the FM state, the conduction band of In2Se3 is shifted +down into the valence band of MnBi2Te4-2L such that +the Fermi level is crossing the valence band of the latter. +This feature favors interfacial charge transfer. Fig. 2d de- +picts the differential charge density for the two magnetic +states, which indicates that there is almost negligible in- +terfacial charge transfer between the MnBi2Te4-2L and +In2Se3 for the AFM state. In contrast, the charge trans- +fer is much more significant for the FM state than that +for the AFM state. +A close inspection finds that the +charge density on the Mn atoms in the interfacial layer +becomes positive. This confirms the picture of hole dop- +ing over this layer and opens up the e↑ +g − {p . . . p} − e↑ +g +hopping chanel. +Consequently, the FM state becomes +energetically favorable for this type of band structure. +Thus, the FE In2Se3 monolayer fits the criterion for a +substrate that gives switchable band alignments between +MBT-2L(FM)/IS(↓) +0.00 +0.04 +-0.04 +∆ρ (eÅ) +MBT-2L(AFM)/IS(↑) +0.00 +0.04 +-0.04 +∆ρ (eÅ) +0 +10 +20 +30 +40 +50 +Z(Å) +P +d +P +d +(a) +a +c +(d) +Energy (eV) +1 +0 +-1 +Μ +Γ +Κ +Μ +Γ +��� +��� +��� +� +(c) +(b) +MnBi2Te4-2L(AFM)/In2Se3(↑) +Μ +Γ +Κ +Μ +In2Se3 +MnBi2Te4-2L +� +M +K +K' +Energy (eV) +1 +0 +-1 +MnBi2Te4-2L(FM)/In2Se3(↓) +Mn +Bi +Te +In +Se +a +b +FIG. 2. +Ferroelectric control of AFM-to-FM transition in +MnBi2Te4 bilayers. (a) Geometric structures of MnBi2Te4- +2L/In2Se3 heterostructures. Left panel shows the top view +of the lowest energy configuration. +Middle and right pan- +els show the side view of the structures with different po- +larizations. +The thin purple arrows denote spins of the +Mn ions. While the thick blue arrows denote polarizations +of the FE substrate. +(b, c) Band structures for the two +states in (a), respectively, i.e., MnBi2Te4-2L(AFM)/In2Se3(↑) +and MnBi2Te4-2L(FM)/In2Se3(↓).(d) Planar-averaged differ- +ential charge density ∆ρ(z) for the two states shown in (b) +and (c). +The insets show the density contour at 0.00015 +e/˚A3. Here, abbreviations (MBT-2L(AFM)/IS(↑) and MBT- +2L(FM)/IS(↓)) are used by incorporating the IMCs of the +MnBi2Te4-2L and the polarization states of In2Se3 for sim- +plicity. +type-II and type-III with MnBi2Te4-2L. Moreover, the +trend that the charge transfer mainly happened to the +interfacial layer also suggests that the spin-flipping ac- +companied by the AFM-FM transition takes place to the +interfacial MnBi2Te4 layer. For the trilayers and quad- +layers, our calculations find the same trend in the spin- +flipping as the bilayers (Figs. S7 and S8). +On the other hand, the interface has a significant im- +pact on the polarization states of the FE In2Se3 mono- +layer by introducing a coupling between its polarizations +and the local dipoles of MAT. This coupling breaks the +symmetry of the two polarization states, that is, it gives +rise to asymmetric barrier heights for the two polariza- +tion states. Fig. 3a shows that the state with the po- +larizations pointing toward the MnBi2Te4 bilayer has a +barrier height of about 152 meV (∆GT ), which is about +68 meV lower than the one with polarizations pointing +away from the interface (∆GA). Therefore, the critical +electric fields needed to flip the polarizations for the for- +mer (ET ) is smaller than that for the latter (EA), i.e., +ET < EA. +The asymmetric barrier heights along with the unique +polarization-dependent IMCs allow designing ferroelec- + +4 +FE1 +FE1 +FE2 +FE2 +MBT/P↑ +MBT/P↓ +(c) +0 +200 +100 +300 +Energy (meV) +400 +Q2 +Q3 +Q4 +Q1 +Q1 +(e) +153 +220 +307 +220 +153 +243 +E↓ +E↑ +E↓ +E↑ +E↑ +E↓ +P +P +(a) +0 +200 +100 +250 +Energy (meV) +50 +150 +MBT-2L/IS(↓) +MBT-2L/IS(↑) +E↓ > E2 +A +Q1 +P +P +Q2 +P +P +E1 +T < E↓ < E2 +A +E↑ > E1 +A +Q4 +P +P +Q3 +P +P +E↑ > E2 +T +E1 +A < E↑ < E2 +T +E↓ > E1 +T +(d) +ET < E↓ < EA +(b) +T1 +T2 +T3 +E↑ > E A +E↓ > E A +ET < E↑ < EA +P +P +P +P +P +P +ΔGT +(ET) +ΔGA +(EA) +ΔG1 +T +(E1 +T) +ΔG1 +A +(E1 +A) +ΔG2 +T +(E2 +T) +ΔG2 +A +(E2 +A) +FIG. 3. Magnetic multistates in MnBi2Te4 thin films. (a) Kinetic pathway of the FE phase transforming in MnBi2Te4-2L/In2Se3 +(abbreviated as MBT-2L/IS). Interface effects lead to asymmetric barrier heights for the two polarization states, which are +labelled as ∆GT and ∆GA as the polarization point toward and away from the interface, respectively. Correspondingly, the +critical electric fields are labelled as ET and EA, respectively. (b) Triple magnetic states in In2Se3/MnBi2Te4-3L/In2Se3 and +schematic FE transforming by controlling the external electric field. E↑ (E↓) represents the external electric fields along the z +(-z) axis. (c) Requirement of energy barriers of the two different FE layers for quadruple magnetic states in sandwich structure +FE1/MBT-4L/FE2, ∆GT +1 < ∆GA +1 < ∆GT +2 < ∆GA +2 . ∆GT +1 and ∆GA +1 are for one FE layer (FE1), which is colored in blue. +Critical electric fields needed to overcome these barriers are denoted as ET +1 and EA +1 ,respectively. +∆GT +2 and ∆GA +2 are for +the other layer (FE2) colored in red, for which the critical fields are ET +2 and EA +2 , respectively. (d) Schematic illustration of +quadruple magnetic states in FE1/MnBi2Te4-4L/FE2 and transforming between the states under electric fields. (e) Kinetic +pathways of the quadruple states in In2SSe2/MBT-4L/In2Se3 during FE transforming. The convention of labeling spins of the +Mn+2 ions and the polarizations of In2Se3 is the same as in Fig. 2. +trically switchable magnetic multistates for MAT multi- +layers. We illustrate the concept in MnBi2Te4 trilayers +and quadlayers, i.e., MnBi2Te4-3L and MnBi2Te4-4L. We +first sandwich MnBi2Te4-3L in between two In2Se3 layers +(Fig. 3b). Suppose that both the top and bottom In2Se3 +layers have up polarizations, which can be achieved by +applying external electric fields anyway. +According to +the polarization dependent IMCs discussed above, spins +in the MnBi2Te4 layer next to the top In2Se3 layer will be +flipped so that it will beferromagnetically coupled with +the underneath MnBi2Te4 layer. We label this magnetic +state as T1. Then one can apply an electrical field E↓ +antiparallel to the z axis that is larger than the critical +field overcoming ∆GT but smaller than the critical field +required to overcome ∆GA, i.e., ET < E↓ < EA. As a re- +sult, the polarizations in the bottom layer will be reversed +while those in the top layer will remain unchanged. Then, +the magnetization of the bottom MnBi2Te4 layer will be +flipped to be ferromagnetically coupled with the adjacent +MnBi2Te4 layer, i.e., T2 in Fig. 3b. Further increasing +the electric field such that E↓ > EA will also drive the +polarizations of the top In2Se3 layer to be flipped. Cor- +respondingly, the magnetizations of the top MnBi2Te4 +layer will be flipped, for which the magnetic state is la- +belled as T3. Now, an electric field along the z axis, i.e., +E↑, will first force the polarization of the bottom In2Se3 +to be reversed when ET < E↑ < EA. As a result, the +system will flow into T2. +Futher enhancing E↑ to the +level that E↑ > EA will drive the system back into T1. +So the whole system have triple magnetic states, which +can be ferroelectrically controlled. Likewise, sandwiching +thicker films than triple layers by the same FE layers also +gives rise to triple magnetic states. +More magnetic states can be obtained by sandwiching +the MAT thin films in between two different FE layers +with a special combination of the barrier heights. Such a +combination requires that the highest barrier for one FE +monolayer should be lower than the lowest barrier for the +other FE layer. We depict the barrier heights for the two +different FE layers in Fig3c, ∆GT +1 and ∆GA +1 are for one + +5 +TABLE I. Energy difference between the interlayer FM and +AFM states for freestanding MAT bilayers and their bilayers +supported by In2Se3 monolayer in different polarization states +(denoted by arrows). ∆E = EF M − EAF M, EF M (EAF M) +represents the total energy of the FM (AFM) state. +Systems +∆E [meV/(Mn pair)] +IMCs +MnBi2Te4-2L +0.21 +AFM +MnBi2Te4-2L/In2Se3(↑) +0.22 +AFM +MnBi2Te4-2L/In2Se3(↓) +-0.16 +FM +MnSb2Te4-2L +0.39 +AFM +MnSb2Te4-2L/In2Se3(↑) +0.36 +AFM +MnSb2Te4-2L/In2Se3(↓) +-0.40 +FM +MnBi4Te7-2L +0.03 +AFM +MnBi4Te7-2L/In2Se3(↑) +0.03 +AFM +MnBi4Te7-2L/In2Se3(↓) +-0.01 +FM +MnSb4Te7-2L +0.06 +AFM +MnSb4Te7-2L/In2Se3(↑) +0.09 +AFM +MnSb4Te7-2L/In2Se3(↓) +-0.04 +FM +FE layer (FE1), to which the corresponding critical elec- +tric fields are ET +1 and EA +1 , respectively. Whereas ∆GT +2 +and ∆GA +2 are for the other layer colored in red (FE2), +for which the critical fields are ET +2 and EA +2 , respectively. +In the case that ∆GT +1 < ∆GA +1 < ∆GT +2 < ∆GA +2 , i.e., +ET +1 < EA +1 < ET +2 < EA +2 , a layer-by-layer flipping mech- +anism for the FE contacts can be achieved by properly +controlling the electric field. As a result, one can have +quadruple magnetic states based on the polarization- +dependent IMCs in MAT heterostructures (Fig. 3d). Our +calculations find that the barrier heights of In2Se3 and +In2SSe2 monolayers fit the above requirement for the +quadruple magnetic states. Specifically, we obtain 245 +meV (∆GT +2 ) and 308 meV (∆GA +2 ) for In2SSe2 with po- +larizations pointing toward and away from the MnBi2Te4 +layer (see Figs. +S9 and S10), respectively, which are +larger than those of In2Se3 (see Fig. 3a, 152 meV for +∆GT +1 and 220 meV for ∆GA +1 ). In Fig. 3e, we show the +kinetic pathway of transforming the polarization states, +which suggests that the quadruple states are ferroelectri- +cally switchable. +The ferroelectrically tunable magnetic multistates give +rise to a variety of distinct topological properties the +MAT thin films. We perform calculations of the Chern +number (C) for the MnBi2Te4 multilayers with the mag- +netic states show in Figs. 2 and 3. For the bilayer sys- +tems, the topological properties of MnBi2Te4 remain un- +changed upon interfacing, i.e., C = 1 for the FM state +and C = 0 for the AFM state, which is also supported +by the results of edge states (Figs. S11 and S12). For +MnBi2Te4-3L, we have C = 0, -1, and 0 for T1, T2, and +T3, respectively. +Whereas for MnBi2Te4-4L, there are +three Chern numbers for the quadruple states, i.e., 1, +0, and -1. Table II summarizes the Chern numbers for +the studied MnBi2Te4 thin films. We expect that such a +TABLE II. Chern number (C) for MnBi2Te4 multilayers with +different magnetic states. Arrows denote the magnetizations +on Mn ions. +Systems +IMCs +C +MnBi2Te4-2L +M1 (↑↓) +0 +M2 (↓↓) +-1 +MnBi2Te4-3L +T1 (↓↓↑) +0 +T2 (↓↓↓) +-1 +T3 (↑↓↓) +0 +MnBi2Te4-4L +Q1 (↑↑↓↑) +1 +Q2 (↑↑↓↓) +0 +Q3 (↓↑↓↓) +-1 +Q4 (↓↑↓↑) +0 +ferroelectrically tunable multiplet for the Chern number +may be seen in other MAT multilayers. +In conclusion, we have proposed to ferroelectrically +tune the magnetism of MAT thin films using model and +first-principles calculations. The scheme is based on the +fact that the IMCs are strongly dependent on the oc- +cupation of d-orbitals of the Mn2+ ions. The variation +in the occupation can be controlled by interfacing the +films with a FE layer with appropriate band alignments. +We have demonstrated the concept in MAT/In2Se3 het- +erostructures by performing first-principles calculations. +We find that the interfacing effect mainly has an im- +pact on the interfacial MAT layer. Specifically, there is +spin-flipping in the interfacial layer when polarizations of +the In2Se3 are reversed, which results in ferroelectrically +switchable IMCs and an AFM-to-FM transition. On the +other hand, the interfacing effect leads to asymmetric en- +ergy barrier heights, which means that different electric +fields are needed to switch the polarizations for the two +states. We further show that this physics can be used to +build magnetic multistates in their sandwich structures. +Our calculations suggest that triple and quadruple mag- +netic states with tunable Chern number can be obtained +for MnBi2Te4 thin films by sandwiching them in between +appropriate FE layers. Our results will not only attract +experimental interest in FE control of the magnetism and +topological properties of MAT thin films, but also inspire +designing novel magnetism in other 2D materials. +Acknowledgments +We thank Haijun Zhang and Zhixin Guo for useful dis- +cussions. 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Rev. +B 101, 184426 (2020). + diff --git a/3NAyT4oBgHgl3EQfo_gV/content/tmp_files/load_file.txt b/3NAyT4oBgHgl3EQfo_gV/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ce3b608a4f4ae43864d71115512ca9e91b27b1c5 --- /dev/null +++ b/3NAyT4oBgHgl3EQfo_gV/content/tmp_files/load_file.txt @@ -0,0 +1,1353 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf,len=1352 +page_content='Ferroelectrically tunable magnetic and topological multistates in thin films of MnBi2Te4 family Guoliang Yu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='1 Chuhan Tang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='1 Zhiqiang Tian,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='1 Ziming Zhu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='1 Anlian Pan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='2 Mingxing Chen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' ∗ and Xing-Qiu Chen3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' 4 1Key Laboratory for Matter Microstructure and Function of Hunan Province,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Key Laboratory of Low-Dimensional Quantum Structures and Quantum Control of Ministry of Education,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' School of Physics and Electronics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Hunan Normal University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Changsha 410081,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' China 2Key Laboratory for Micro-Nano Physics and Technology of Hunan Province,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' College of Materials Science and Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Hunan University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Changsha 410082,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' China 3Shenyang National Laboratory for Materials Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Institute of Metal Research,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Chinese Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' 110016 Shenyang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' People’s Republic of China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' 4School of Materials Science and Engineering, University of Science and Technology of China, 110016 Shenyang, People’s Republic of China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' (Dated: January 3, 2023) Ferroelectric control of two-dimensional magnetism is promising in fabricating electronic devices with high speed and low energy consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' The newly discovered layered MnBi2Te4(Bi2Te3)n and their Sb counterparts exhibit A-type antiferromagnetism with intriguing topological proper- ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Here, we propose to obtain tunable magnetic multistates in their thin films by ferroelectrically manipulating the interlayer magnetic couplings (IMCs) based on the Heisenberg model and first- principles calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Our strategy relies on that interfacing the thin films with appropriate ferro- electric materials can switch on/off an interlayer hopping channel between Mn-eg orbitals as the po- larizations reversed, thus resulting in a switchable interlayer antiferromagnetism-to-ferromagnetism transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' On the other hand, the interface effect leads to asymmetric energy barrier heights for the two polarization states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' These properties allow us to build ferroelectrically switchable triple and quadruple magnetic states with multiple Chern numbers in thin films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Our study reveals that ferroelectrically switchable magnetic and topological multistates in MnBi2Te4 family can be ob- tained by rational design for multifunctional electronic devices, which can also be applied to other two-dimensional magnetic materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' INTRODUCTION Two-dimensional (2D) magnetic materials provide an ideal platform to explore novel magnetic and electronic properties1–7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' The delicate interlayer exchange couplings in these systems enable a variety of methods of manipu- lating their magnetism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' For instance, recent studies re- vealed that the A-type antiferromagnetic (AFM) CrI3 bilayer could be tuned into ferromagnetic (FM) by ex- ternal electric field8,9, electrostatic doping10,11, and in- terface engineering12–17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Twisting the bilayer may yield non-collinear magnetism18–20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Recently, the MnBi2Te4 family, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=', MnBi2Te4(Bi2Te3)n and MnSb2Te4(Bi2Te3)n, which are hereafter referred to as MAT, have much great attention due to the coexistence and interesting interplay of intrinsic magnetism and band topology in them21–37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' This series of materials also have a layered van der Waals (vdW) structure with an A-type AFM structure as revealed by experiments38–41, which preserves the combination of the time-reversal and a half lattice translation symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' As a result, many of them show nontrivial topological properties such as topological insulators21 and axion insulators22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Moreover, the systems can be turned into Weyl semimetals as the interlayer couplings become ferromagnetic22,23,36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' The A-type AFM coupling in MAT yields unusual even-odd layer-number dependent magnetism for their thin films35,42,43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' The even-number (even-N) systems are expected to have no net magnetization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Whereas those with odd-number (odd-N) layers have uncompensated magnetization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' This difference can lead to distinct topo- logical properties for them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' For instance, the thin films of MnBi2Te4 with odd-N septuple layers are quantum anomalous Hall insulators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' However, those with even-N layers have a zero Chern number42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Instead, their topo- logical properties can be characterized by the so-called pseudospin Chern number44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Due to the weak interlayer interaction, small magnetic fields could induce spin-flip transitions, giving rise to an AFM-to-FM transition in the IMCs35,40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Chemical dopings45,46 and antisite de- fects47–51 can also be used to manipulate the IMCs in these systems, although they may complicate the nature of the surface states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' First-principles calculations sug- gest that the AFM double-septuple MnBi2Te4 could be driven into a Chern insulator with a high Chern number under electric fields52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' In this work, we propose to ferroelectrically tune the IMCs in MnBi2Te4 thin films for magnetic multistates by interface, which is desired for memory devices with high density storage, high speed, and low power consump- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' We reveal that hole doping can lead to an interlayer AFM-to-FM transition in MAT bilayers based on the un- derstanding of the IMCs using the Heisenberg model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' We provide a guideline for designing ferroelectric substrates that may induce transitions in the interlayer exchange couplings, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=', polarization dependent IMCs, as demon- strated by our first-principles calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Moreover, we arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='00515v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='mtrl-sci] 2 Jan 2023 2 find that the interface effect results in symmetry breaking in the two polarization states of the FE substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' This asymmetry allows us to design switchable magnetic mul- tistates in sandwich structures made of MnBi2Te4 mul- tilayers and 2D ferroelectric (FE) materials, which also exhibit distinct electronic and topological properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' We begin by presenting the concept of FE tuning of IMCs in MAT bilayers, which is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' In these systems, each Mn atom is coordinated with six chalcogen atoms, which form a distorted octahedron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' The Mn-3d orbitals are split into triply degenerate t2g states and the doubly degenerate eg states due to the octahedral ligand field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' The states are further split due to the magnetic exchange interaction between the Mn atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' As a re- sult, the majority states of the t2g and eg orbitals are fully occupied by the five d electrons of the Mn2+ ions (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' 1), resulting in a high spin state for the Mn2+ ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Like the 2D magnetic bilayers reported by Refs 31 and 53, this type of occupation favors AFM IMCs between the Mn+2 ions, which are mediated by the p- orbitals of the nonmetallic atoms (denoted by {p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' p}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Whereas FM IMCs are energetically unfavorable because the e↑ g − {p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' p} − e↑ g hopping between the Mn-d or- bitals of adjacent layers is prohibited31,53,54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' For our systems, reducing the occupation of the d orbitals makes the e↑ g − {p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' p} − e↑ g hopping channel energetically fa- vorable, thus enhancing the stability of the FM IMCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Indeed, our DFT calculations indicate that all the MAT bilayers undergo the AFM-to-FM transition by small hole dopings (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' 1c and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' S1), which is also expected for their multilayers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' The IMCs in MAT-2L can be understood using the following spin Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' H = � i,j Jt ∥Si · Sj + � m,n Jb ∥Sm · Sn + � i,m J⊥Si · Sm, (1) where J∥ and J⊥ denote the intra- and interlayer ex- change interactions between the Mn ions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' The intralayer ones are denoted by Jt ∥ for the top layer and Jb ∥ for the bottom layer, for which only the first nearest-neighbor interactions are taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Jt ∥ are equal to Jb ∥ for the freestanding MAT-2L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Whereas for the interlayer ones, the second nearest neighbors are included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' For the bilayers without doping, we obtain positive J1st ⊥ and negative J2nd ⊥ (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' 1d, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' S2 and Table S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Note that the magnitude of J1st ⊥ is larger than that of J2nd ⊥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' So, the sum of J1st ⊥ and J2nd ⊥ , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=', ¯ J⊥ = J1st ⊥ + J2nd ⊥ , is positive, which gives rise to AFM IMCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Introducing hole doping suppresses J1st ⊥ while en- hances J2nd ⊥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' As a result, ¯ J⊥ decreases with increasing of the hole doping and eventually changes its sign across the AFM-to-FM transition (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' 1d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' The hole doping over the MAT bilayers can be achieved via interfacial charge transfer which requires suitable band alignments between them and the substrates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' When their bands are in the type-I or type-II align- ment, interfacial charge transfer can be negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' In (b) MAT-2L(FM)/P↓ Mn1 t2g eg EF t2g eg P↓ e CBM VBM Mn2 t2g eg t2g eg MAT-2L(AFM)/P↑ (a) Mn1 t2g eg EF t2g eg Mn2 t2g eg t2g eg e P↑ CBM VBM (c) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='4 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='6 ∆EFM-AFM (meV) n (hole/Mn pair) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='06 AFM FM MnBi2Te4-2L MnSb2Te4-2L (d) n (hole/Mn pair) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='06 AFM FM MnSb2Te4-2L 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='08 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='12 n (hole/Mn pair) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='06 AFM FM J⊥ (meV) MnBi2Te4-2L 1st J⊥ 2nd J⊥ J⊥ FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Interface engineering of the IMCs in MAT bilayers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' (a) Spin states of Mn ions for the AFM interlayer coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' In the presence of a substrate that has a type-I or type-II band alignment with the bilayer, the IMCs remain AFM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' (b) The IMCs becomes FM when there is a type-III band alignment between them so that the CBM of the substrate lower than the VBM of the MAT bilayer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' In (b), the white circles denote hole dopings to the Mn-eg orbital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' In (a, b), arrows denote spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' The dark and light red curves with an arrow denote hopping channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' The one marked by a cross means that electron hoppings are prohibited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' We use a FE material as the substrate, whose polarizations are labeled by P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' P ↑ and P ↓ represent the up and down polarizations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' (c) Energy difference between the FM and AFM states as a func- tion of hole doping for freestanding MnBi2Te4 and MnSb2Te4 bilayers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' ∆E = EF M − EAF M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' (d) Doping dependence of J⊥ for the two systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' ¯ J⊥ = J1st ⊥ + J2nd ⊥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' these cases, the IMCs are most likely to be AFM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' In contrast, electrons will be transferred from the MAT bi- layer to the substrate when they are in the type-III band alignment that the valence band maximum (VBM) of the MAT bilayer are higher than the conduction band minimum (CBM) of the substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='Namely, hole doping is introduced to the MAT bilayer, which is desired for the AFM-to-FM transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Ferroelectrically switchable IMCs may be achieved if a 2D FE materials serves as the substrate so that reversing its polarizations gives rise to a switching of the band alignment from type-III to type-I (II) or vice versa (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' However, one can expect that the transferred electrons mainly come from the in- terfacial MAT layer because of the vdW-type interlayer bonding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='Therefore, the spin-flipping most likely happen to the interfacial MAT layer rather than those further away from the substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' We now come to first-principles calculations of the het- erostructures of MAT thin films and 2D FE materials, which were performed using the Vienna Ab initio Sim- ulation Package55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' We choose In2Se3 monolayer as the substrate, which has been experimentally proved since its prediction in 201756–58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Their heterostructures are built by slightly adjusting the lattice of In2Se3 (The lat- tice mismatch between them is small).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' The pseudopoten- tials were constructed by the projector augmented wave method59,60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' An 11 × 11 × 1 and 21 × 21 × 1 Γ- 3 centered k-mesh were used to sample the 2D Brillouin zone for structural relaxation and electronic structure calculations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' The plane-wave energy cutoff is set to 400 eV for all calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' A 20 ˚A vacuum region was used between adjacent plates to avoid the interaction between neighboring periodic images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Van der Waals (vdW) dispersion forces between the adsorbate and the substrate were accounted for through the DFT- D361.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Different vdW methods/functionals such as DFT- D2 and optPBE-vdW were also used for comparison62–64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' The systems were fully relaxed until the residual force on each atom is less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='01 eV/˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' The DFT+U method65 is used to treat electron correlations due to the partially filled d-orbital of Mn for which a value of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='34 eV is used21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Our results on the structural properties, mag- netism, and band structures of the free-standing MAT films are consistent with previous studies21,32,36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' The kinetic pathways of transitions between different polar- ization states are calculated using the climbing image nudge elastic band (CI-NEB) method66,67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' The topolog- ical properties calculations were done using the WAN- NIER9068 and WannierTools package69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' We have performed careful calculations over a number of stacking configurations (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' S3, and Table S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' The lowest energy configuration is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' 2a, in which the interfacial Se and Te are in the hollow sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' The stacking order is the same as the one for MnBi2Te4 monolayer on In2Se370, which shows up for all MAT bi- layers on In2Se3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Table I summarizes the stability of the two magnetic states for MAT bilayers on In2Se3 mono- layer in different polarization states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' One can see that for all the MAT bilayers the IMCs remain AFM when the polarization is pointing toward the interface, but become FM as the polarization is reversed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' The trend is indepen- dent of the vdW functionals/methods (Table S3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Below we focus on the electronic structure of MnBi2Te4 bilayer on In2Se3 monolayer, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=', MnBi2Te4-2L/In2Se3, which are shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' 2b-d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Those for all other MAT systems are shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='S4-S6 since they show pretty much the same trend as MnBi2Te4-2L/In2Se3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' The Fermi level is located in the band gap for the AFM state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Whereas for the FM state, the conduction band of In2Se3 is shifted down into the valence band of MnBi2Te4-2L such that the Fermi level is crossing the valence band of the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' This feature favors interfacial charge transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' 2d de- picts the differential charge density for the two magnetic states, which indicates that there is almost negligible in- terfacial charge transfer between the MnBi2Te4-2L and In2Se3 for the AFM state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' In contrast, the charge trans- fer is much more significant for the FM state than that for the AFM state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' A close inspection finds that the charge density on the Mn atoms in the interfacial layer becomes positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' This confirms the picture of hole dop- ing over this layer and opens up the e↑ g − {p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' p} − e↑ g hopping chanel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Consequently, the FM state becomes energetically favorable for this type of band structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Thus, the FE In2Se3 monolayer fits the criterion for a substrate that gives switchable band alignments between MBT-2L(FM)/IS(↓) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='04 ∆ρ (eÅ) MBT-2L(AFM)/IS(↑) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content="04 ∆ρ (eÅ) 0 10 20 30 40 50 Z(Å) P d P d (a) a c (d) Energy (eV) 1 0 1 Μ Γ Κ Μ Γ ��� ��� ��� � (c) (b) MnBi2Te4-2L(AFM)/In2Se3(↑) Μ Γ Κ Μ In2Se3 MnBi2Te4-2L � M K K' Energy (eV) 1 0 1 MnBi2Te4-2L(FM)/In2Se3(↓) Mn Bi Te In Se a b FIG." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Ferroelectric control of AFM-to-FM transition in MnBi2Te4 bilayers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' (a) Geometric structures of MnBi2Te4- 2L/In2Se3 heterostructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Left panel shows the top view of the lowest energy configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Middle and right pan- els show the side view of the structures with different po- larizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' The thin purple arrows denote spins of the Mn ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' While the thick blue arrows denote polarizations of the FE substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' (b, c) Band structures for the two states in (a), respectively, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=', MnBi2Te4-2L(AFM)/In2Se3(↑) and MnBi2Te4-2L(FM)/In2Se3(↓).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' (d) Planar-averaged differ- ential charge density ∆ρ(z) for the two states shown in (b) and (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' The insets show the density contour at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='00015 e/˚A3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Here, abbreviations (MBT-2L(AFM)/IS(↑) and MBT- 2L(FM)/IS(↓)) are used by incorporating the IMCs of the MnBi2Te4-2L and the polarization states of In2Se3 for sim- plicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' type-II and type-III with MnBi2Te4-2L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Moreover, the trend that the charge transfer mainly happened to the interfacial layer also suggests that the spin-flipping ac- companied by the AFM-FM transition takes place to the interfacial MnBi2Te4 layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' For the trilayers and quad- layers, our calculations find the same trend in the spin- flipping as the bilayers (Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' S7 and S8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' On the other hand, the interface has a significant im- pact on the polarization states of the FE In2Se3 mono- layer by introducing a coupling between its polarizations and the local dipoles of MAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' This coupling breaks the symmetry of the two polarization states, that is, it gives rise to asymmetric barrier heights for the two polariza- tion states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' 3a shows that the state with the po- larizations pointing toward the MnBi2Te4 bilayer has a barrier height of about 152 meV (∆GT ), which is about 68 meV lower than the one with polarizations pointing away from the interface (∆GA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Therefore, the critical electric fields needed to flip the polarizations for the for- mer (ET ) is smaller than that for the latter (EA), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=', ET < EA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='The asymmetric barrier heights along with the unique ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='polarization-dependent IMCs allow designing ferroelec- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='FE1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='FE1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='FE2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='FE2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='MBT/P↑ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='MBT/P↓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='(c) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='Energy (meV) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='Q2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='Q3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='Q4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='Q1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='Q1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='(e) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='153 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='220 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='307 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='220 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='153 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='243 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='E↓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='E↑ ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='250 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='Energy (meV) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='150 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='E↑ > E2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='E1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='A < E↑ < E2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='E↓ > E1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='(d) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='ET < E↓ < EA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='(b) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='T1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='T2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='T3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='E↑ > E A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='E↓ > E A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='ET < E↑ < EA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='P ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='P ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='P ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='P ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='P ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='P ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='ΔGT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='(ET) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='ΔGA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='(EA) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='ΔG1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='(E1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='T) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='ΔG1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='(E1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='A) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='ΔG2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='(E2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='T) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='ΔG2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='(E2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='A) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Magnetic multistates in MnBi2Te4 thin films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' (a) Kinetic pathway of the FE phase transforming in MnBi2Te4-2L/In2Se3 (abbreviated as MBT-2L/IS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Interface effects lead to asymmetric barrier heights for the two polarization states, which are labelled as ∆GT and ∆GA as the polarization point toward and away from the interface, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Correspondingly, the critical electric fields are labelled as ET and EA, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' (b) Triple magnetic states in In2Se3/MnBi2Te4-3L/In2Se3 and schematic FE transforming by controlling the external electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' E↑ (E↓) represents the external electric fields along the z (-z) axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' (c) Requirement of energy barriers of the two different FE layers for quadruple magnetic states in sandwich structure FE1/MBT-4L/FE2, ∆GT 1 < ∆GA 1 < ∆GT 2 < ∆GA 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' ∆GT 1 and ∆GA 1 are for one FE layer (FE1), which is colored in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Critical electric fields needed to overcome these barriers are denoted as ET 1 and EA 1 ,respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' ∆GT 2 and ∆GA 2 are for the other layer (FE2) colored in red, for which the critical fields are ET 2 and EA 2 , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' (d) Schematic illustration of quadruple magnetic states in FE1/MnBi2Te4-4L/FE2 and transforming between the states under electric fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' (e) Kinetic pathways of the quadruple states in In2SSe2/MBT-4L/In2Se3 during FE transforming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' The convention of labeling spins of the Mn+2 ions and the polarizations of In2Se3 is the same as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' trically switchable magnetic multistates for MAT multi- layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' We illustrate the concept in MnBi2Te4 trilayers and quadlayers, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=', MnBi2Te4-3L and MnBi2Te4-4L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' We first sandwich MnBi2Te4-3L in between two In2Se3 layers (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' 3b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Suppose that both the top and bottom In2Se3 layers have up polarizations, which can be achieved by applying external electric fields anyway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' According to the polarization dependent IMCs discussed above, spins in the MnBi2Te4 layer next to the top In2Se3 layer will be flipped so that it will beferromagnetically coupled with the underneath MnBi2Te4 layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' We label this magnetic state as T1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Then one can apply an electrical field E↓ antiparallel to the z axis that is larger than the critical field overcoming ∆GT but smaller than the critical field required to overcome ∆GA, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=', ET < E↓ < EA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' As a re- sult, the polarizations in the bottom layer will be reversed while those in the top layer will remain unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Then, the magnetization of the bottom MnBi2Te4 layer will be flipped to be ferromagnetically coupled with the adjacent MnBi2Te4 layer, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=', T2 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' 3b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Further increasing the electric field such that E↓ > EA will also drive the polarizations of the top In2Se3 layer to be flipped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Cor- respondingly, the magnetizations of the top MnBi2Te4 layer will be flipped, for which the magnetic state is la- belled as T3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Now, an electric field along the z axis, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=', E↑, will first force the polarization of the bottom In2Se3 to be reversed when ET < E↑ < EA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' As a result, the system will flow into T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Futher enhancing E↑ to the level that E↑ > EA will drive the system back into T1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' So the whole system have triple magnetic states, which can be ferroelectrically controlled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Likewise, sandwiching thicker films than triple layers by the same FE layers also gives rise to triple magnetic states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' More magnetic states can be obtained by sandwiching the MAT thin films in between two different FE layers with a special combination of the barrier heights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Such a combination requires that the highest barrier for one FE monolayer should be lower than the lowest barrier for the other FE layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' We depict the barrier heights for the two different FE layers in Fig3c, ∆GT 1 and ∆GA 1 are for one 5 TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Energy difference between the interlayer FM and AFM states for freestanding MAT bilayers and their bilayers supported by In2Se3 monolayer in different polarization states (denoted by arrows).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' ∆E = EF M − EAF M, EF M (EAF M) represents the total energy of the FM (AFM) state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Systems ∆E [meV/(Mn pair)] IMCs MnBi2Te4-2L 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='21 AFM MnBi2Te4-2L/In2Se3(↑) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='22 AFM MnBi2Te4-2L/In2Se3(↓) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='16 FM MnSb2Te4-2L 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='39 AFM MnSb2Te4-2L/In2Se3(↑) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='36 AFM MnSb2Te4-2L/In2Se3(↓) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='40 FM MnBi4Te7-2L 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='03 AFM MnBi4Te7-2L/In2Se3(↑) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='03 AFM MnBi4Te7-2L/In2Se3(↓) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='01 FM MnSb4Te7-2L 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='06 AFM MnSb4Te7-2L/In2Se3(↑) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='09 AFM MnSb4Te7-2L/In2Se3(↓) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='04 FM FE layer (FE1), to which the corresponding critical elec- tric fields are ET 1 and EA 1 , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Whereas ∆GT 2 and ∆GA 2 are for the other layer colored in red (FE2), for which the critical fields are ET 2 and EA 2 , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' In the case that ∆GT 1 < ∆GA 1 < ∆GT 2 < ∆GA 2 , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=', ET 1 < EA 1 < ET 2 < EA 2 , a layer-by-layer flipping mech- anism for the FE contacts can be achieved by properly controlling the electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' As a result, one can have quadruple magnetic states based on the polarization- dependent IMCs in MAT heterostructures (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' 3d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Our calculations find that the barrier heights of In2Se3 and In2SSe2 monolayers fit the above requirement for the quadruple magnetic states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Specifically, we obtain 245 meV (∆GT 2 ) and 308 meV (∆GA 2 ) for In2SSe2 with po- larizations pointing toward and away from the MnBi2Te4 layer (see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' S9 and S10), respectively, which are larger than those of In2Se3 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' 3a, 152 meV for ∆GT 1 and 220 meV for ∆GA 1 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' 3e, we show the kinetic pathway of transforming the polarization states, which suggests that the quadruple states are ferroelectri- cally switchable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' The ferroelectrically tunable magnetic multistates give rise to a variety of distinct topological properties the MAT thin films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' We perform calculations of the Chern number (C) for the MnBi2Te4 multilayers with the mag- netic states show in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' For the bilayer sys- tems, the topological properties of MnBi2Te4 remain un- changed upon interfacing, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=', C = 1 for the FM state and C = 0 for the AFM state, which is also supported by the results of edge states (Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' S11 and S12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' For MnBi2Te4-3L, we have C = 0, -1, and 0 for T1, T2, and T3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Whereas for MnBi2Te4-4L, there are three Chern numbers for the quadruple states, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=', 1, 0, and -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Table II summarizes the Chern numbers for the studied MnBi2Te4 thin films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' We expect that such a TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Chern number (C) for MnBi2Te4 multilayers with different magnetic states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Arrows denote the magnetizations on Mn ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Systems IMCs C MnBi2Te4-2L M1 (↑↓) 0 M2 (↓↓) 1 MnBi2Te4-3L T1 (↓↓↑) 0 T2 (↓↓↓) 1 T3 (↑↓↓) 0 MnBi2Te4-4L Q1 (↑↑↓↑) 1 Q2 (↑↑↓↓) 0 Q3 (↓↑↓↓) 1 Q4 (↓↑↓↑) 0 ferroelectrically tunable multiplet for the Chern number may be seen in other MAT multilayers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' In conclusion, we have proposed to ferroelectrically tune the magnetism of MAT thin films using model and first-principles calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' The scheme is based on the fact that the IMCs are strongly dependent on the oc- cupation of d-orbitals of the Mn2+ ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' The variation in the occupation can be controlled by interfacing the films with a FE layer with appropriate band alignments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' We have demonstrated the concept in MAT/In2Se3 het- erostructures by performing first-principles calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' We find that the interfacing effect mainly has an im- pact on the interfacial MAT layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Specifically, there is spin-flipping in the interfacial layer when polarizations of the In2Se3 are reversed, which results in ferroelectrically switchable IMCs and an AFM-to-FM transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' On the other hand, the interfacing effect leads to asymmetric en- ergy barrier heights, which means that different electric fields are needed to switch the polarizations for the two states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' We further show that this physics can be used to build magnetic multistates in their sandwich structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Our calculations suggest that triple and quadruple mag- netic states with tunable Chern number can be obtained for MnBi2Te4 thin films by sandwiching them in between appropriate FE layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Our results will not only attract experimental interest in FE control of the magnetism and topological properties of MAT thin films, but also inspire designing novel magnetism in other 2D materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Acknowledgments We thank Haijun Zhang and Zhixin Guo for useful dis- cussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' This work was supported by the National Nat- ural Science Foundation of China (Grants No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' 12174098, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' 11774084, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' U19A2090 and No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' 91833302) and project supported by State Key Laboratory of Powder Metallurgy, Central South 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J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Park, Nature 563, 47 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' 5 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Gong and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Zhang, Science 363, eaav4450 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' 6 L.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Gao, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Chen, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Ren, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Cheng, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Li, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='- Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Chen, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' 12, 2361 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' 7 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Ni, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Li, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Amoroso, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' He, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Feng, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Kan, S.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Huang, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Clark, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Klein, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' MacNeill, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Navarro- Moratalla, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Xu and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Zou, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' 11, 3152 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' 10 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Jiang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Li, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Wang, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Mak, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Shan, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Nanotechnol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' 13, 549 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' 11 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Soriano and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Katsnelson, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' 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+page_content=' Yang, ACS Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Interfaces 12, 6243 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' 13 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Cheng, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Zhou, C.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Yang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Shao, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Li, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Yam, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Zhang, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Huang, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' B 103, L201405 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' 15 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Shen, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Wu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Dai, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} 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Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Wang, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' 19, 522 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' 27 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Deng, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Yu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Z.' 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367, 895 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' 28 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content='-Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Yan, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Zhang, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Heitmann, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Huang, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Liu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Brawer, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Ramirez, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Ding, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Cao, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Liu, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Dessau, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Ni, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' 11, 97 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' 31 Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Li, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Li, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' He, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Wan, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Duan, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Xu, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' B 102, 081107 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Liu, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Zhao, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' X 10, 031013 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' 34 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Lian, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Figgemeier, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Peixoto, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Vasili, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Valvidares, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Jung, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Cacho, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Alfonsov, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Mehlawat, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Kataev, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Hess, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} 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Gu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Peng, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Yang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Xu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Li, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Han, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Liu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Wang, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Gao, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Yang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Lei, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Huang, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Ye, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf'} +page_content=' 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0000000000000000000000000000000000000000..511fb6072156d486381f6c1e22f2767e592131e9 --- /dev/null +++ b/3NFQT4oBgHgl3EQf3DZF/content/tmp_files/2301.13426v1.pdf.txt @@ -0,0 +1,611 @@ +Discrete Search in Heterogeneous Integer Spaces +for Automated Choice of Parameters using +Correct-by-Construction Methods +Omar Radwan +oradwan@usc.edu +oradwan@alumni.usc.edu +Viterbi School of Engineering +University of Southern California +Yilin Zhang +yilinz80@usc.edu +Viterbi School of Engineering +University of Southern California +Luca Geretti +geretti@usc.edu +Viterbi School of Engineering +University of Southern California +Abstract—Discrete Search of integer spaces for tool parame- +ter values provides a powerful methodology for modeling and +finding a heuristically optimal parameter list for a given system. +Current tools and implementations that exist focus primarily +on homogeneous tool parameters, and the implementations for +heterogeneous tool parameters is lacking. In this paper we +introduce a correct-by-construction method of heterogeneous +parameter reachability and validity search, and further outline +the implementation as well as a demonstration using examples +of heterogeneous systems that this tool can be used for. +I. INTRODUCTION +A. Premise +Discrete Search of integer spaces provides a powerful mech- +anism through which to explore the reachable set of a given +system. Current design cools work primarily for homogeneous +parameter spaces, and mapping a heterogeneous parameter +space into the integer domain would provide a strong backbone +for both performance and allow for a wide range of uses in +many hybrid systems as well as hybrid parameters that are +contained within a single system. There are precautions that +would need to be taken for hybrid systems, which primarily +consist of having unsafe states, that even though they are +reachable, they would be considered to be unsafe in a real- +world implementation, as well dependencies between vari- +ables, that could transcend the homogeneous dependencies that +are trivial (i.e. comparing two integers together as compared +to a comparison between floating point and Boolean). There +also would exist optimal state locations of the parameter set, +and those would be modeled using an arbitrary cost function. +B. Related Work +Related work consists primarily of homogeneous tool pa- +rameter exploration implementations, and those concern them- +selves primarily with arriving at the reachable set primarily +for homogeneous parameter sets. This would include the tool +Ariadne [1], which has features built-in that allow it to find +an approximation of the given reachable set by giving by +controlling the growth of the approximation error. +One other concern that arises when attempting to model +heterogeneous parameters in integer spaces is the problem +of solvability within bounded time with close approximation, +and as outlined in [2], there does exist a finite bound for +finite discovery. There was a foray into unbounded analysis, +but that is infeasible given the constraints and would be too +computationally exhaustive. Another issue that comes up is +discrete versus non-discrete evolution in terms of time, and +this was a problem resolved by setting as a condition that +there can only exist discrete time steps and discrete evolution. +Fig. 1. From Citation [3] +C. Our Approach +For the implementation demonstrated in this paper, we focus +on a number of contributions that create a fast and efficient +method of finding the optimal set given existing constraints +and cost. We also define a semantic format that supports the +representation of heterogeneous parameters, which better suits +it for discrete search along hybrid domains. +For exploring the adjacent set space from our beginning +iteration point(initial state), there are a number of possible +implementation decisions that would need to be made on how +best to explore the reachable set given the constraints. The +path that we decided on was to create a correct by construction +approach, that would allow the exploration tool to only explore +the reachable set that is also valid given the constraints and +dependencies that are supplied. Our flow is as follows: given +a parameter list which can consist of integer, Boolean, and +composite parameters, as well as a list of constraints and +dependencies between variables, and a cost function, we aim +to find a valid parameter state that satisfies all of our given +requirements. +For our implementation, we split our computational engine +into two general algorithms. Our first algorithm involves com- +arXiv:2301.13426v1 [eess.SY] 31 Jan 2023 + +Fig. 7. Evader keeps pursuer from entering reachable set, and hence avoids collision (animation +at [44]puting a correct-by-construction interval for a given parameter +given our requirements, and our current state when it comes +to other parameters that exist within our set space. The second +algorithm is our step-by-step evolution iteration across the set +space of the parameter list based on the computation of local +optimal cost. +Compared to existing and related works, our approach has +the following contributions: +• Developed a representation for heterogeneous parameter +sets that allows for the discretization of all parameters +and results in the ability for integer space exploration for +all relevant types +• Created a correct-by-construction approach to not only +finding the reachable set of a given parameter set, but also +allowing the inclusion of heterogeneous inter-parameter +dependencies and assertions. +• Designed a method of evolution that allows for quick +computation of adjacent states for a given set of already +locally-optimal parameter instances with a method of +back-tracing and reset if arriving at an invalid location +• Demonstrate the applicability and the versatility of our +implementation on two examples that involve computing +minimum cost for a computer architecture design and a +re-programmable logic circuit with a demonstration of the +implementation of pseudo-Boolean constraints +II. IMPLEMENTATION +A. Environment and Language Considerations +We decided on implementing our design in Python [4], the +reason for that being that Python allows a host of libraries +and type-interfacing that would allow us to quickly prototype, +verify, and extend during testing. We also chose Python for +the reason of being able to interface easily with JSON [5], +which is our input-format of choice. JSON was chosen due +to its status as being very well-adopted and would provide an +easy interface for other CAD tools to create tool-parameter +sets for analysis using our program. +We also use a number of Python libraries to do the necessary +computations that are required for our implementation. A spe- +cial recognition is deserved of Numpy [6], which is a library +that allows for very quick computation of intervals, arrays, +and sets. Since we are operating in the integer domain, integer +arrays using Numpy libraries make the cost of computation a +significantly smaller area of concern during implementation. +B. Motivation for Design +To better improve the performance of discrete search in +heterogeneous space, there do exist a number of limitations. +Firstly, a slight weakness exists in parsing string type asser- +tions and evaluating them in a computationally static format +as opposed to extensive abstract syntax trees and symbolic +interval computation. Secondly, considering various typed +parameters and assertion relations, it is necessary to have a +uniform interface design such that algorithm implementation +is isolated with complicated typed transformation, which is +why JSON was selected, which could become unwieldy if +enumerated or vector parameters which to be considered. In +this case, a tool that would generate a statically-enumerated +JSON format that is acceptable to our program would be +required. +C. Evolution Algorithm +In this section, we introduce how the program will explore +feasible set constrained by assertions. The JSON format input +will be interpreted and loaded into our program. For the sake +of generality, we assume that there are n parameters denoted +as x1, · · · xn. First of all, for each parameter xi, we randomly +generate N − 1 valid neighboring points. For the random +sampling of these points, we experimented with a couple +methods. One was uniform sampling from the valid interval, +the other two where linear and square weighted sampling with +respect of distance from the interval. After these were tested, +we found that square weighting was the most effective, and +we will demonstrate these findings during our examples. With +xi itself, these N points form a list {xj +i}N +j=1. In total there are +n lists. +During the evolution process, each point will randomly +generate a neighboring point from its valid set. Therefore, all +n·N points will generate another n new lists. Without loss of +generality, we denote these n new lists as {xj +i}2N +j=N+1. Next +we the original list and new list with the same footnote i to get +n new list {xj +i}2N +j=1. From these n list, we evaluate 2N cost +function values as {cj = F(xj +1, xj +2, · · · xj +n) | j = 1, · · · , 2N}. +For these 2N cost values, we keep the smaller half and +corresponding parameter values to form n new lists. Repeat +the above steps until the ending requirements are satisfied. A +pseudo-code for this algorithm can be found at Algorithm 1 +D. Approach for feasibility checking between heterogeneous +parameters +For defining the the set of parameters that would exist for a +given system, we supply two atomic types and one composite +type: +1) Integer type +2) Boolean type +3) Composite type +Integers exist in the Integer domain, and Boolean’s likewise +in the Boolean domain. Composites are different in that they +are modeled like an array, given a composite parameter C, +C can contain any number of composites, Boolean’s, and +integers. This allows the modeling of parameters that cannot +be modeled as strictly scalar integer or Boolean values. Floats, +complex numbers, and vectors are all examples of what can +be modeled as a composite set. Furthermore, to maintain the +desired behavior of these parameters, the constraint paradigm +that we introduce allows us to describe the behavior of how +these composite parameters undergo evolution. +As an example, take Cube, which of type composite, and it +is defined by 3-equal length sides x, y, z, such that Cube(t) = +{x, y, z ∈ Z, x == y == z}∀t where t is time-step during +evolution. For the case of this parameter, the instantiation of +the of the domain of each sub-parameter would go with the +2 + +parameter declarations, while the instantiation of the constraint +that is intrinsic to cubes would be added to the constraints field +that is given. +This paradigm of allowing composite parameters to have +unique behaviors could lead to invalid states during evolution, +if one sub-parameter undergoes evolution independently and +is now not equal to the other two, that would lead to an unde- +sirable state. For this reason correct-by-construction interval +generation for each of the sub-parameters is done with all +assertions and constraints in mind. +One note on using composite parameters to model floating +point numbers. Initially during development we had planned +to incorporate a floating point type, however the tedious- +ness of setting properties for floating point as an atomic +type is redundant as all the properties of a floating point +value(mantissa, exponent, significant figures) can be modeled +as sub-parameters of a composite value, and the user can +specify the desired constraints and behaviors for comparison +and incorporation between the composite-ized floating point +value and other parameters. +E. Feasibility Checking given Constraints +In this section, a detailed explanation about how to construct +valid neighboring set is given. Suppose that there are m +assertions {Ai}m +i=1 on n parameters. For each parameter xi, +assertions containing xi are selected out of m, which is +{Ak | xi ∈ Ak}. Next, iterate through other parameters and +apply their values into these assertions. Finally, Intersect all +the intervals after evaluating the assertions to get the final +interval. A new value for xi is sampled randomly from the +final interval based on the square of their distance to xi. By +default, values closer to xi have higher probabilities to be +selected. More details can be found in Algorithm 2 +F. Desired Implementation Aspects that Proved Infeasible +One initial idea that was considered well thought out and +feasible was the incorporation of symbolic computation for +our constraint and dependency valid interval generation. The +Sympy [7] library in Python was going to be utilized for this +purpose. Though the algorithm was functional, the symbolic +computation cost was extremely prohibitive, and was not +feasible for a general-use case. After doing much research +to attempt to make it feasible, we discovered that even Sympy +as an organization recognizes that the substitution and eval- +uation is cost-prohibitive, and recommends other avenues for +repetitive computation. For this reason we had to re-calibrate +and find another solution. This solution was to do string +replacement of our given parameters with their values into +the string representation of our constraints, dependencies, and +costs. Then these string representations would be converted +into lambda functions that would be operated on by the +Numpy array operations. Since Numpy on the back-end uses C +libraries to do computation, this lessened our computation time +by an order of magnitude, for mostly the same functionality. +The functionality that is missing is due to the inherent +behavioral properties of lambda functions. Symbolic compu- +Algorithm 1: Evolution of Adjacent Optimal Cost +Input: List of variables Lv, Iterating parameter T, List +of assertions La, Cost function F +Output: Optimal value of variables L∗ +v +1 //This is for initial value selection, since we need to +enter the set space is what we presume to be a valid +point foreach v in Lv do +2 +v := Sample Uniform Distribution(Lower Bound of +v, Upper Bound of v) +3 +Construct Vi as the set of n sample of vi +4 end +5 while T <= K do +6 +foreach variable vi in Lv do +7 +Vi is the set of n values of vi +Svi =get intersect of all valid intervals(La, Lv, vi). +8 +Svisorted = Arrange by incrementing closeness +to value of ak +9 +WeightsSvisorted = array from 0 to length of +Svisorted +10 +foreach w in WeightsSvisorted do +11 +w = (length of Svisorted - index of w)2 +12 +end +13 +Use weighted sampling of WeightsSvisorted to +randomly sample n new values of vi from +Svisorted. +14 +Append these n values into Vi +15 +Construct n new list of variables {Lj +v}n +j=1, +Lj +v[i] = Lv[i]. +16 +Pick Lk +v with minimum F(Lk +v) in {Lj +v}n +i=j. +17 +Update Lv[i] = Lk +v[i]. +18 +Delete vi in Vi with n highest cost values. +19 +Update T. +20 +end +21 end +22 return Lv +tation was desired as it allowed the incorporation of very +rigorous Boolean SAT exploration, but this is not a feature +that is possible with the lambda paradigm. Therefore, to allow +the extend-ability of Boolean values, fuzzy pseudo-Boolean +logic [8] is implemented, which does allow for an adequate +semantic representation of Boolean logic. +III. EXAMPLES OF APPLICATION +For an example foray to explore what our program would +be able to handle, we decided on two different, yet related, +domains. +A. FPGA Synthesis +For our first example(outlined in 2), we decided on model- +ing our problem as an FPGA cost problem. Given a number of +constraints on an FPGA, i.e. memory size, available memory +ports, available input and output ports we have Routine1,2,3, +and only two of the previously mentioned three can be +3 + +Algorithm 2: Get Intersect of All Valid Points in +Bounds and Assertions +Input: List of assertions La, List of variables Lv, +Target variable vi +Output: All valid set Svi of vi +1 Initialize list of intervals Li = [] +2 foreach ak in La do +3 +if vi appears in ak then +4 +foreach vj in Lv do +5 +if vj ̸= vi then +6 +Plug in value of vj in ak. +7 +else +8 +continue +9 +end +10 +end +11 +Append ak into Li +12 +else +13 +continue +14 +end +15 end +16 Transform the intersection of Li into valid set Svi +17 return Svi +installed on the FPGA fabric, and depending on which two +are loaded onto the fabric, we then must enable a minimum +number of memory, I/O, and interconnection ports, as well +as have different memory properties. We then created a +polynomial cost function of these constraints, in an aim of +it becoming nonlinear and make the algorithm demonstrate its +effectiveness in traversing the set space while attempting to +find the given most optimal cost. +One highlight of this example is the inclusion of pseudo- +Boolean constraints, which manifest in the requirement that +only two of the three routines can function at any time, which +in terms of cost, creates a piece-wise function. The parameter +variation that is generated during random sampling is able +to traverse this piece wise function, because even though we +generate points using a correct-by-construction approach, in +some cases there is no valid interval, and in that case we reset +for that specific parameter back to the largest valid interval, +and randomly sample that. This allows the program to exit +any possible rut that it enters while making an early decision +on which Routine set to choose, and so it can backtrack +as necessary and choose another Routine set if the specific +parameter space undergoing evolution is no longer valid. The +results for these are demonstrated in Figure 3 with different +weights for random sampling methods from the valid intervals +generated. +B. Computer Architecture Design +Another example that we used is the creation of of a +computer architecture system. During the creation of a new +computer architecture, or the generation of a new implementa- +tion of an architecture, multiple design decisions must be made +Fig. 2. Illustration of FPGA Paradigm for Testing Our Implementation +with respect to area, inter-connectivity, interface requirements, +and transistor count. In this example, we model a simple +multi-fetch, multi-execution, processor design. We drafted the +requirements in terms of dependencies and constraints, and +given the constraints and requirements for the interfaces and +inter-connectivity between components, we aim to find the +minimal transistor count. This was a more rudimentary design, +and it aimed to find the computation limit of our implemen- +tation. One thing that we attempted to model was having +very large integer sets, and exploring those. Emulating design +space exploration for computer architectures with such large +intervals was the reason we had to refactor our computation +engine from purely symbolic to the lambda paradigm, as +the symbolic computation was not able to run search space +exploration and computation in a reasonable amount of time +with this example. The results for those example are posted in +Figure 4, along with the variation between random sampling +methods from the valid intervals generated. +C. Performance and Efficacy +As aforementioned during the discussion on the implemen- +tation, performance was a major bottleneck in our implemen- +tation, and there were a number of features that needed to be +added to be able to guarantee reasonable performance. The +first was the use of lambdas to calculate the valid interval set. +The second, which is outlined in the algorithm, is keeping a +short list of the least-cost neighbors that exist, and generating +new random neighbors from that list. This allows us to have +multiple different forays into the search space, and we could +possibly arrive to many local minima’s, but we only choose +the most optimal local minima. Computation time is static +across iterations, and there are parameter options to increase +or decrease the exhaustiveness of the search depending on the +intended use cases. +4 + +L2Cache +L1Cache +品品 +00001 +0000 +Memory Ports +Process 1 +Process 2 +indino +Input +Ports +Ports +Process 3We also wanted to verify the efficacy of our design and do +the best possible effort into generating the most optimal point. +To verify that our results where sane, we ran multiple differ- +ent instances of both the FPGA and Computer Architecture +description JSON files, and averaged those results out, and did +this for three different weights for random sampling(uniform, +linear weighted, square weighted), and what we found that in +all cases, our results for all runs where fairly similar, but there +are some noticeable differences worth discussion. +Firstly, the uniform random search has better performance +for lower iterations, and this is because during early stages +of evolution, a majority portion of the set space has yet +to be explored, and uniform sampling allows us to traverse +the majority of the set space early. However after a lot of +iterations, the square weighted random sampling from the +interval eventually makes us arrive to a more optimal cost, and +this is because as more and more of the set space is invalidated, +the parameters that are undergoing evolution get much closer +to the local optima, and square weighting allows us to more +likely sample these local optima and arrive at them at a quicker +rate than both uniform and linear random sampling. +Fig. 3. Table of the Impact of Different Weights and Effect on Set Exploration +for FPGA Example +IV. SUMMARY +To reiterate the major points that have been mentioned +throughout this paper, we have created a tool that performs +discrete search of integer spaces of mapped heterogeneous +parameters to the integer domain, and we utilized correct- +by-construction methods to ensure that given constraints and +dependencies are met, while attempting to find the most opti- +mal cost. This differs from the previous literature in that it is +able to accommodate for heterogeneous data structures and is +able to model hybrid systems, while comparatively the existing +literature exists primarily for reachability and homogeneous +parameter exploration. The main takeaways from this endeavor +include that there is a significant divide between the tools that +are used in industry, and the potential for tools that could be +used to better-optimize processes and methods that are used. +Fig. 4. Table of the Impact of Different Weights and Effect on Set Exploration +for Architecture Example +The main hurdle for widespread adoption of these methods +includes a difficulty of understanding and use, as well as +a computational cost-barrier that is evident in very complex +systems. +A. Wish-list of additional features +One feature that would have been useful to incorporate +would have been incorporating a Boolean SAT or SMT solver +[9], which would have allowed us to bypass pseudo-Boolean +constraints entirely, which are generated heuristically, and +instead rigorously solve Boolean equations for all possible +solutions. Incorporation a Boolean SAT solver such as Z3 +would’ve been time-prohibitive, but would’ve allowed for a +greater range of expressively for constraints. +B. Application Files +Due to space reasons, we do not go into detail on the +specifics of the Computer Architecture Example and the FPGA +Example. Please contact the authors for more information. +V. SOME THOUGHTS ON OPTIMIZATION AND USE CASES +Optimization aims at searching for values of x which +minimizes the objective function f bounded by constraints. +A general formula of optimization problem is in equation (1). +arg +x min f(x) +s.t. Constraints on x +(1) +In addition to existing gradient based methods which re- +quires the objective function to be differentiable or even +more smooth, discrete search algorithm proposed in this paper +achieves a high degree of performance on all kinds of objective +functions. +One of the most important features of cyber-physical sys- +tems is that they contains both continuous system components +and discrete system components. In this case, the constraints +may include discrete forms like SATs, and continuous forms +like inequalities. Our discrete search algorithm can be used to +choose optimal parameters for a cyber-physical system. +5 + +Different WeightTypesandCostper IterationForFPGA +Example +UniformWeightFPGA +Linear Weight FPGA Square Weight FPGA +40000000 +20000000 +10000000 +8000000 +6000000 +4000000 +1 +5 +10 +50 +100 +IterationDifferent Weight Types and Costper Iteration For Architecture +Example +Uniform Weight Arch +Linear Weight Arch +Square Weight Arch +5000000000000000 +1000000000000000 +500000000000000 +100000000000000 +50000000000000 +10000000000000 +5 +10 +50 +100VI. FURTHER POSSIBLE WORK +We would like to explore more about the background of +reachability analysis. Where does this problem rise from. +Moreover, as for existing optimization algorithms like heuristic +algorithms, gradient based methods and interior point methods, +what are the bottlenecks on applying these algorithms on +hybrid system reachability analysis. +Another topic is the connection between reachability anal- +ysis and optimization algorithm. If the reachability problem +can be formulated into an optimization problem, then it will +be easier to understand the problem from the mathematical +properties of objective function. +REFERENCES +[1] Luca Geretti, Pieter Collins, Davide Bresolin, and Tiziano Villa. Automat- +ing numerical parameters along the evolution of a nonlinear system. In +Runtime Verification: 22nd International Conference, RV 2022, Tbilisi, +Georgia, September 28–30, 2022, Proceedings, page 336–345, Berlin, +Heidelberg, 2022. Springer-Verlag. +[2] Michele Conforti, Gerard Cornuejols, and Giacomo Zambelli. +Integer +Programming / Michele Conforti, G´erard Cornu´ejols, Giacomo Zambelli. +Springer, Cham, 2014. +[3] Ian M. Mitchell and Claire J. Tomlin. +Overapproximating reachable +sets by hamilton-jacobi projections. +Journal of Scientific Computing, +19(1):323–346, 2003. +[4] Guido Van Rossum and Fred L Drake Jr. +Python reference manual. +Centrum voor Wiskunde en Informatica Amsterdam, 1995. +[5] Felipe Pezoa, Juan L Reutter, Fernando Suarez, Mart´ın Ugarte, and +Domagoj Vrgoˇc. Foundations of json schema. In Proceedings of the +25th International Conference on World Wide Web, pages 263–273. +International World Wide Web Conferences Steering Committee, 2016. +[6] Charles R. Harris, K. Jarrod Millman, St´efan J. van der Walt, Ralf +Gommers, Pauli Virtanen, David Cournapeau, Eric Wieser, Julian Tay- +lor, Sebastian Berg, Nathaniel J. Smith, Robert Kern, Matti Picus, +Stephan Hoyer, Marten H. van Kerkwijk, Matthew Brett, Allan Haldane, +Jaime Fern´andez del R´ıo, Mark Wiebe, Pearu Peterson, Pierre G´erard- +Marchant, Kevin Sheppard, Tyler Reddy, Warren Weckesser, Hameer +Abbasi, Christoph Gohlke, and Travis E. Oliphant. Array programming +with NumPy. Nature, 585(7825):357–362, September 2020. +[7] Sympy Foundation. +[8] Y. Dote. Introduction to fuzzy logic. In Proceedings of IECON ’95 - +21st Annual Conference on IEEE Industrial Electronics, volume 1, pages +50–56 vol.1, 1995. +[9] Leonardo De Moura and Nikolaj Bjørner. Satisfiability modulo theories: +Introduction and applications. Commun. ACM, 54(9):69–77, sep 2011. +6 + diff --git a/3NFQT4oBgHgl3EQf3DZF/content/tmp_files/load_file.txt b/3NFQT4oBgHgl3EQf3DZF/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c43ac2e7b9948c7d26db0de52dbbb33ebc649f7f --- /dev/null +++ b/3NFQT4oBgHgl3EQf3DZF/content/tmp_files/load_file.txt @@ -0,0 +1,228 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf,len=227 +page_content='Discrete Search in Heterogeneous Integer Spaces for Automated Choice of Parameters using Correct-by-Construction Methods Omar Radwan oradwan@usc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content='edu oradwan@alumni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content='usc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content='edu Viterbi School of Engineering University of Southern California Yilin Zhang yilinz80@usc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content='edu Viterbi School of Engineering University of Southern California Luca Geretti geretti@usc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content='edu Viterbi School of Engineering University of Southern California Abstract—Discrete Search of integer spaces for tool parame- ter values provides a powerful methodology for modeling and finding a heuristically optimal parameter list for a given system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' Current tools and implementations that exist focus primarily on homogeneous tool parameters, and the implementations for heterogeneous tool parameters is lacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' In this paper we introduce a correct-by-construction method of heterogeneous parameter reachability and validity search, and further outline the implementation as well as a demonstration using examples of heterogeneous systems that this tool can be used for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' INTRODUCTION A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' Premise Discrete Search of integer spaces provides a powerful mech- anism through which to explore the reachable set of a given system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' Current design cools work primarily for homogeneous parameter spaces, and mapping a heterogeneous parameter space into the integer domain would provide a strong backbone for both performance and allow for a wide range of uses in many hybrid systems as well as hybrid parameters that are contained within a single system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' There are precautions that would need to be taken for hybrid systems, which primarily consist of having unsafe states, that even though they are reachable, they would be considered to be unsafe in a real- world implementation, as well dependencies between vari- ables, that could transcend the homogeneous dependencies that are trivial (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' comparing two integers together as compared to a comparison between floating point and Boolean).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' There also would exist optimal state locations of the parameter set, and those would be modeled using an arbitrary cost function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' Related Work Related work consists primarily of homogeneous tool pa- rameter exploration implementations, and those concern them- selves primarily with arriving at the reachable set primarily for homogeneous parameter sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' This would include the tool Ariadne [1], which has features built-in that allow it to find an approximation of the given reachable set by giving by controlling the growth of the approximation error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' One other concern that arises when attempting to model heterogeneous parameters in integer spaces is the problem of solvability within bounded time with close approximation, and as outlined in [2], there does exist a finite bound for finite discovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' There was a foray into unbounded analysis, but that is infeasible given the constraints and would be too computationally exhaustive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' Another issue that comes up is discrete versus non-discrete evolution in terms of time, and this was a problem resolved by setting as a condition that there can only exist discrete time steps and discrete evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' From Citation [3] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' Our Approach For the implementation demonstrated in this paper, we focus on a number of contributions that create a fast and efficient method of finding the optimal set given existing constraints and cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' We also define a semantic format that supports the representation of heterogeneous parameters, which better suits it for discrete search along hybrid domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' For exploring the adjacent set space from our beginning iteration point(initial state), there are a number of possible implementation decisions that would need to be made on how best to explore the reachable set given the constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' The path that we decided on was to create a correct by construction approach, that would allow the exploration tool to only explore the reachable set that is also valid given the constraints and dependencies that are supplied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' Our flow is as follows: given a parameter list which can consist of integer, Boolean, and composite parameters, as well as a list of constraints and dependencies between variables, and a cost function, we aim to find a valid parameter state that satisfies all of our given requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' For our implementation, we split our computational engine into two general algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' Our first algorithm involves com- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content='13426v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content='SY] 31 Jan 2023 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' Evader keeps pursuer from entering reachable set, and hence avoids collision (animation at [44]puting a correct-by-construction interval for a given parameter given our requirements, and our current state when it comes to other parameters that exist within our set space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' The second algorithm is our step-by-step evolution iteration across the set space of the parameter list based on the computation of local optimal cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' Compared to existing and related works, our approach has the following contributions: Developed a representation for heterogeneous parameter sets that allows for the discretization of all parameters and results in the ability for integer space exploration for all relevant types Created a correct-by-construction approach to not only finding the reachable set of a given parameter set, but also allowing the inclusion of heterogeneous inter-parameter dependencies and assertions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content='Designed a method of evolution that allows for quick ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content='computation of adjacent states for a given set of already ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content='locally-optimal parameter instances with a method of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content='back-tracing and reset if arriving at an invalid location ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content='Demonstrate the applicability and the versatility of our ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content='implementation on two examples that involve computing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content='minimum cost for a computer architecture design and a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content='re-programmable logic circuit with a demonstration of the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content='implementation of pseudo-Boolean constraints ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content='II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' IMPLEMENTATION A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' Environment and Language Considerations We decided on implementing our design in Python [4], the reason for that being that Python allows a host of libraries and type-interfacing that would allow us to quickly prototype, verify, and extend during testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' We also chose Python for the reason of being able to interface easily with JSON [5], which is our input-format of choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' JSON was chosen due to its status as being very well-adopted and would provide an easy interface for other CAD tools to create tool-parameter sets for analysis using our program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' We also use a number of Python libraries to do the necessary computations that are required for our implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' A spe- cial recognition is deserved of Numpy [6], which is a library that allows for very quick computation of intervals, arrays, and sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' Since we are operating in the integer domain, integer arrays using Numpy libraries make the cost of computation a significantly smaller area of concern during implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' Motivation for Design To better improve the performance of discrete search in heterogeneous space, there do exist a number of limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' Firstly, a slight weakness exists in parsing string type asser- tions and evaluating them in a computationally static format as opposed to extensive abstract syntax trees and symbolic interval computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' Secondly, considering various typed parameters and assertion relations, it is necessary to have a uniform interface design such that algorithm implementation is isolated with complicated typed transformation, which is why JSON was selected, which could become unwieldy if enumerated or vector parameters which to be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' In this case, a tool that would generate a statically-enumerated JSON format that is acceptable to our program would be required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' Evolution Algorithm In this section, we introduce how the program will explore feasible set constrained by assertions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' The JSON format input will be interpreted and loaded into our program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' For the sake of generality, we assume that there are n parameters denoted as x1, · · · xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' First of all, for each parameter xi, we randomly generate N − 1 valid neighboring points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' For the random sampling of these points, we experimented with a couple methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' One was uniform sampling from the valid interval, the other two where linear and square weighted sampling with respect of distance from the interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' After these were tested, we found that square weighting was the most effective, and we will demonstrate these findings during our examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' With xi itself, these N points form a list {xj i}N j=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' In total there are n lists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' During the evolution process, each point will randomly generate a neighboring point from its valid set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' Therefore, all n·N points will generate another n new lists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' Without loss of generality, we denote these n new lists as {xj i}2N j=N+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' Next we the original list and new list with the same footnote i to get n new list {xj i}2N j=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' From these n list, we evaluate 2N cost function values as {cj = F(xj 1, xj 2, · · · xj n) | j = 1, · · · , 2N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' For these 2N cost values, we keep the smaller half and corresponding parameter values to form n new lists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' Repeat the above steps until the ending requirements are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' A pseudo-code for this algorithm can be found at Algorithm 1 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' Approach for feasibility checking between heterogeneous parameters For defining the the set of parameters that would exist for a given system, we supply two atomic types and one composite type: 1) Integer type 2) Boolean type 3) Composite type Integers exist in the Integer domain, and Boolean’s likewise in the Boolean domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' Composites are different in that they are modeled like an array, given a composite parameter C, C can contain any number of composites, Boolean’s, and integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' This allows the modeling of parameters that cannot be modeled as strictly scalar integer or Boolean values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' Floats, complex numbers, and vectors are all examples of what can be modeled as a composite set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' Furthermore, to maintain the desired behavior of these parameters, the constraint paradigm that we introduce allows us to describe the behavior of how these composite parameters undergo evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' As an example, take Cube, which of type composite, and it is defined by 3-equal length sides x, y, z, such that Cube(t) = {x, y, z ∈ Z, x == y == z}∀t where t is time-step during evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' For the case of this parameter, the instantiation of the of the domain of each sub-parameter would go with the 2 parameter declarations, while the instantiation of the constraint that is intrinsic to cubes would be added to the constraints field that is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' This paradigm of allowing composite parameters to have unique behaviors could lead to invalid states during evolution, if one sub-parameter undergoes evolution independently and is now not equal to the other two, that would lead to an unde- sirable state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' For this reason correct-by-construction interval generation for each of the sub-parameters is done with all assertions and constraints in mind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' One note on using composite parameters to model floating point numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' Initially during development we had planned to incorporate a floating point type, however the tedious- ness of setting properties for floating point as an atomic type is redundant as all the properties of a floating point value(mantissa, exponent, significant figures) can be modeled as sub-parameters of a composite value, and the user can specify the desired constraints and behaviors for comparison and incorporation between the composite-ized floating point value and other parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' Feasibility Checking given Constraints In this section, a detailed explanation about how to construct valid neighboring set is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' Suppose that there are m assertions {Ai}m i=1 on n parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' For each parameter xi, assertions containing xi are selected out of m, which is {Ak | xi ∈ Ak}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' Next, iterate through other parameters and apply their values into these assertions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' Finally, Intersect all the intervals after evaluating the assertions to get the final interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' A new value for xi is sampled randomly from the final interval based on the square of their distance to xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' By default, values closer to xi have higher probabilities to be selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' More details can be found in Algorithm 2 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' Desired Implementation Aspects that Proved Infeasible One initial idea that was considered well thought out and feasible was the incorporation of symbolic computation for our constraint and dependency valid interval generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' The Sympy [7] library in Python was going to be utilized for this purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' Though the algorithm was functional, the symbolic computation cost was extremely prohibitive, and was not feasible for a general-use case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' After doing much research to attempt to make it feasible, we discovered that even Sympy as an organization recognizes that the substitution and eval- uation is cost-prohibitive, and recommends other avenues for repetitive computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' For this reason we had to re-calibrate and find another solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' This solution was to do string replacement of our given parameters with their values into the string representation of our constraints, dependencies, and costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' Then these string representations would be converted into lambda functions that would be operated on by the Numpy array operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' Since Numpy on the back-end uses C libraries to do computation, this lessened our computation time by an order of magnitude, for mostly the same functionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' The functionality that is missing is due to the inherent behavioral properties of lambda functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' Symbolic compu- Algorithm 1: Evolution of Adjacent Optimal Cost Input: List of variables Lv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' Iterating parameter T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' List of assertions La,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' Cost function F Output: Optimal value of variables L∗ v 1 //This is for initial value selection,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' since we need to enter the set space is what we presume to be a valid point foreach v in Lv do 2 v := Sample Uniform Distribution(Lower Bound of v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' Upper Bound of v) 3 Construct Vi as the set of n sample of vi 4 end 5 while T <= K do 6 foreach variable vi in Lv do 7 Vi is the set of n values of vi Svi =get intersect of all valid intervals(La,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' Lv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' vi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' 8 Svisorted = Arrange by incrementing closeness to value of ak 9 WeightsSvisorted = array from 0 to length of Svisorted 10 foreach w in WeightsSvisorted do 11 w = (length of Svisorted - index of w)2 12 end 13 Use weighted sampling of WeightsSvisorted to randomly sample n new values of vi from Svisorted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' 14 Append these n values into Vi 15 Construct n new list of variables {Lj v}n j=1, Lj v[i] = Lv[i].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' 16 Pick Lk v with minimum F(Lk v) in {Lj v}n i=j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' 17 Update Lv[i] = Lk v[i].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' 18 Delete vi in Vi with n highest cost values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' 19 Update T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' 20 end 21 end 22 return Lv tation was desired as it allowed the incorporation of very rigorous Boolean SAT exploration, but this is not a feature that is possible with the lambda paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' Therefore, to allow the extend-ability of Boolean values, fuzzy pseudo-Boolean logic [8] is implemented, which does allow for an adequate semantic representation of Boolean logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' EXAMPLES OF APPLICATION For an example foray to explore what our program would be able to handle, we decided on two different, yet related, domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' FPGA Synthesis For our first example(outlined in 2), we decided on model- ing our problem as an FPGA cost problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' Given a number of constraints on an FPGA, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' memory size, available memory ports, available input and output ports we have Routine1,2,3, and only two of the previously mentioned three can be 3 Algorithm 2: Get Intersect of All Valid Points in Bounds and Assertions Input: List of assertions La, List of variables Lv, Target variable vi Output: All valid set Svi of vi 1 Initialize list of intervals Li = [] 2 foreach ak in La do 3 if vi appears in ak then 4 foreach vj in Lv do 5 if vj ̸= vi then 6 Plug in value of vj in ak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' 7 else 8 continue 9 end 10 end 11 Append ak into Li 12 else 13 continue 14 end 15 end 16 Transform the intersection of Li into valid set Svi 17 return Svi installed on the FPGA fabric, and depending on which two are loaded onto the fabric, we then must enable a minimum number of memory, I/O, and interconnection ports, as well as have different memory properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' We then created a polynomial cost function of these constraints, in an aim of it becoming nonlinear and make the algorithm demonstrate its effectiveness in traversing the set space while attempting to find the given most optimal cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' One highlight of this example is the inclusion of pseudo- Boolean constraints, which manifest in the requirement that only two of the three routines can function at any time, which in terms of cost, creates a piece-wise function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' The parameter variation that is generated during random sampling is able to traverse this piece wise function, because even though we generate points using a correct-by-construction approach, in some cases there is no valid interval, and in that case we reset for that specific parameter back to the largest valid interval, and randomly sample that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' This allows the program to exit any possible rut that it enters while making an early decision on which Routine set to choose, and so it can backtrack as necessary and choose another Routine set if the specific parameter space undergoing evolution is no longer valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' The results for these are demonstrated in Figure 3 with different weights for random sampling methods from the valid intervals generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' Computer Architecture Design Another example that we used is the creation of of a computer architecture system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' During the creation of a new computer architecture, or the generation of a new implementa- tion of an architecture, multiple design decisions must be made Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' Illustration of FPGA Paradigm for Testing Our Implementation with respect to area, inter-connectivity, interface requirements, and transistor count.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' In this example, we model a simple multi-fetch, multi-execution, processor design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' We drafted the requirements in terms of dependencies and constraints, and given the constraints and requirements for the interfaces and inter-connectivity between components, we aim to find the minimal transistor count.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' This was a more rudimentary design, and it aimed to find the computation limit of our implemen- tation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' One thing that we attempted to model was having very large integer sets, and exploring those.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' Emulating design space exploration for computer architectures with such large intervals was the reason we had to refactor our computation engine from purely symbolic to the lambda paradigm, as the symbolic computation was not able to run search space exploration and computation in a reasonable amount of time with this example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' The results for those example are posted in Figure 4, along with the variation between random sampling methods from the valid intervals generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' Performance and Efficacy As aforementioned during the discussion on the implemen- tation, performance was a major bottleneck in our implemen- tation, and there were a number of features that needed to be added to be able to guarantee reasonable performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' The first was the use of lambdas to calculate the valid interval set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' The second, which is outlined in the algorithm, is keeping a short list of the least-cost neighbors that exist, and generating new random neighbors from that list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' This allows us to have multiple different forays into the search space, and we could possibly arrive to many local minima’s, but we only choose the most optimal local minima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' Computation time is static across iterations, and there are parameter options to increase or decrease the exhaustiveness of the search depending on the intended use cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' 4 L2Cache L1Cache 品品 00001 0000 Memory Ports Process 1 Process 2 indino Input Ports Ports Process 3We also wanted to verify the efficacy of our design and do the best possible effort into generating the most optimal point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' To verify that our results where sane, we ran multiple differ- ent instances of both the FPGA and Computer Architecture description JSON files, and averaged those results out, and did this for three different weights for random sampling(uniform, linear weighted, square weighted), and what we found that in all cases, our results for all runs where fairly similar, but there are some noticeable differences worth discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' Firstly, the uniform random search has better performance for lower iterations, and this is because during early stages of evolution, a majority portion of the set space has yet to be explored, and uniform sampling allows us to traverse the majority of the set space early.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' However after a lot of iterations, the square weighted random sampling from the interval eventually makes us arrive to a more optimal cost, and this is because as more and more of the set space is invalidated, the parameters that are undergoing evolution get much closer to the local optima, and square weighting allows us to more likely sample these local optima and arrive at them at a quicker rate than both uniform and linear random sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' Table of the Impact of Different Weights and Effect on Set Exploration for FPGA Example IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' SUMMARY To reiterate the major points that have been mentioned throughout this paper, we have created a tool that performs discrete search of integer spaces of mapped heterogeneous parameters to the integer domain, and we utilized correct- by-construction methods to ensure that given constraints and dependencies are met, while attempting to find the most opti- mal cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' This differs from the previous literature in that it is able to accommodate for heterogeneous data structures and is able to model hybrid systems, while comparatively the existing literature exists primarily for reachability and homogeneous parameter exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' The main takeaways from this endeavor include that there is a significant divide between the tools that are used in industry, and the potential for tools that could be used to better-optimize processes and methods that are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' Table of the Impact of Different Weights and Effect on Set Exploration for Architecture Example The main hurdle for widespread adoption of these methods includes a difficulty of understanding and use, as well as a computational cost-barrier that is evident in very complex systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' Wish-list of additional features One feature that would have been useful to incorporate would have been incorporating a Boolean SAT or SMT solver [9], which would have allowed us to bypass pseudo-Boolean constraints entirely, which are generated heuristically, and instead rigorously solve Boolean equations for all possible solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' Incorporation a Boolean SAT solver such as Z3 would’ve been time-prohibitive, but would’ve allowed for a greater range of expressively for constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' Application Files Due to space reasons, we do not go into detail on the specifics of the Computer Architecture Example and the FPGA Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' Please contact the authors for more information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' SOME THOUGHTS ON OPTIMIZATION AND USE CASES Optimization aims at searching for values of x which minimizes the objective function f bounded by constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' A general formula of optimization problem is in equation (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' arg x min f(x) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' Constraints on x (1) In addition to existing gradient based methods which re- quires the objective function to be differentiable or even more smooth, discrete search algorithm proposed in this paper achieves a high degree of performance on all kinds of objective functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' One of the most important features of cyber-physical sys- tems is that they contains both continuous system components and discrete system components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' In this case, the constraints may include discrete forms like SATs, and continuous forms like inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' Our discrete search algorithm can be used to choose optimal parameters for a cyber-physical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' 5 Different WeightTypesandCostper IterationForFPGA Example UniformWeightFPGA Linear Weight FPGA Square Weight FPGA 40000000 20000000 10000000 8000000 6000000 4000000 1 5 10 50 100 IterationDifferent Weight Types and Costper Iteration For Architecture Example Uniform Weight Arch Linear Weight Arch Square Weight Arch 5000000000000000 1000000000000000 500000000000000 100000000000000 50000000000000 10000000000000 5 10 50 100VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' FURTHER POSSIBLE WORK We would like to explore more about the background of reachability analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' Where does this problem rise from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' Moreover, as for existing optimization algorithms like heuristic algorithms, gradient based methods and interior point methods, what are the bottlenecks on applying these algorithms on hybrid system reachability analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' Another topic is the connection between reachability anal- ysis and optimization algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' If the reachability problem can be formulated into an optimization problem, then it will be easier to 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' ACM, 54(9):69–77, sep 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} +page_content=' 6' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFQT4oBgHgl3EQf3DZF/content/2301.13426v1.pdf'} diff --git a/3NFST4oBgHgl3EQfYjhQ/content/tmp_files/2301.13788v1.pdf.txt b/3NFST4oBgHgl3EQfYjhQ/content/tmp_files/2301.13788v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..3d295145cbd21f48c4d9c8da21843fb6faf89575 --- /dev/null +++ b/3NFST4oBgHgl3EQfYjhQ/content/tmp_files/2301.13788v1.pdf.txt @@ -0,0 +1,828 @@ +1 + +Synthesis and characterization of PEG-coated Zn0.3MnxFe2.7-xO4 nanoparticles as +the dual T1/T2-weighted MRI contrast agent +Bahareh Rezaei, Ahmad Kermanpur*, Sheyda Labbaf +Department of Materials Engineering, Isfahan University of Technology, Isfahan 84156-83111, Iran + + + + + +Abstract +Super-paramagnetic nanoparticles (NPs) have been widely explored as magnetic resonance imaging +(MRI) contrast agents because of a combination of favorable magnetic properties, biocompability and +ease of fabrication. MRI using traditional T1- or T2-weighted single mode contrast-enhanced +techniques may yield inaccurate imaging results. In the present work, a T1/T2 dual mode contrast agent +based on the super-paramagnetic zinc-manganese ferrite (Zn0.3MnxFe2.7-xO4, x= 0, 0.25, 0.75 and 1) +NPs with small core size and a hydrophilic PEG surface coating is reported. The TEM, TGA and FTIR +results confirmed the formation of a uniform coating on the NPs surface. The MRI analysis revealed +that the Zn0.3Mn0.5Fe2.2O4 NPs had the maximum image contrast compared to other zinc-manganese +ferrite samples. Cell viability evaluations revealed that the coated and uncoated particles did not +inhibit cell growth pattern. The present PEG-coated Zn0.3Mn0.5Fe2.2O4 NPs can be utilized as a suitable +T1/T2-weighted MRI contrast agent for better diagnostic of abnormalities in the organs or tissues. + +Keywords +Magnetic Resonance Imaging (MRI); Super-paramagnetic nanoparticles; Zn0.3MnxFe2.7-xO4 +nanoparticles; Polyethylene Glycol (PEG) coating + +1. Introduction + +The most potent and painless test that gives extremely clear images of the internal organs in +the body is the magnetic resonance imaging (MRI) scan [1, 2]. Based on the magnetic relaxation +processes of water protons on soft tissue of nearly every internal structure in the human body [1, 3-5], +this method is a sort of diagnostic test that generates detailed images and functional information in a +non-invasive and real-time monitoring manner [6, 7]. It is a distinguished device since there is no +ionizing radiation during the imaging process and obviously reduces harmful side effects [2, 4, 8, 9]. +However, this test typically provides poor anatomical details, and clinicians have some difficulties to +distinguish between normal and abnormal tissues due to its low sensitivity [9, 10]. Hence, the clinical + +* Corresponding author; Tel. (+98)3133915738; Fax (+98)3133912752; Email: ahmad_k@iut.ac.ir + +2 + +domains urgently require more reliable MR images. There is a potential to create more accurate and +crisper images by adding contrast agents, which enables physicians to detect organs or in-vivo systems +more clear. This opens up a wide range of MRI applications for therapeutic medicine in addition to +diagnostic radiology. Despite the fact of shorter circulation time of Gd3+ ions as a T1-weighted MRI +contrast agent, which renders them useless for high-resolution and/or targeted MRI [9, 11] and many +concerns about potential trace deposition of Gd ions in the body, known as Nephrogenic Systemic +Fibrosis (NSF) [12-14], which is a rare disease that frequently develops in patients with severe renal +failure or after liver transplantation [15], Gd-based contrast agents can shorten the T1 relaxation time +effectively and provide brighter images in the regions of interest [16]. Following the increased +awareness of this side effect, researchers have much more emphasis on alternative methods based on +Mn-based complexes [15]. Although no scientific relationship has been proved between the NSF side +effect and Mn so far, the metal is still known to pose some toxicity when inhaled. However, small +amounts are essential to human health, but overexposure to free Mn ions may result in the +neurodegenerative disorder known as ‘Manganism’ with symptoms similar Parkinson’s disease [11]. +Unlike Gd3+ and Mn2+ chelates, iron oxide nanoparticles (NPs) have achieved great attention +due to the outstanding properties they exhibit at the nano-metric scale. A large number of benefits +including biocompatibility, superparamagnetic behavior at room temperature, high saturation +magnetization that can be tailored by size, shape, composition and assembly, tunable cellular uptake, +biodispersibility, and large surface areas that make them a good candidate for polymer coating, +conjugation with targeting molecules and other probes for achieving targeting and multimodal agents +[17, 18] is reported for the iron oxide NPs. Super-paramagnetic NPs can be employed as T2-weighted +MRI contrast agents since they are more sensitive in the micro- or nano-molar range than Gd +complexes [17]. Clinical MR imaging applications often use iron oxide-based NPs with strong +magnetic moments as T2-weighted MRI contrast agents. The limited usage of iron oxide NPs as T1 +contrast agents is due to their high transverse to longitudinal relaxivity ratio [19]. However, the use of +superparamagnetic NPs in MRI is constrained by a negative contrast effect and magnetic susceptibility +artifacts. Because the signal is frequently confused with signals from bleeding, calcification, or metal +deposits and the susceptibility artifacts alter the background image, the resulting dark signal in T2- +weighted MRI may be exploited to mislead clinical diagnosis [18]. The T1-weighted MRI contrast +agents, however, have advantages over T2-weighted MRI contrast agents. These advantages include +better imaging quality, brighter images that can more effectively distinguish between normal and +lesion tissues, and also the ability to provide better resolution for blood imaging. Nonetheless, in T1- + +3 + +weighted MR imaging, some normal tissues (such as fatty tissue) may be mistaken for bright lesions +that have been increased by T1 contrast agents [20]. Therefore, efforts to integrate T1 and T2 imaging +to prevent probable MRI artifacts and produce superior clinical images have been made as a result of +the rising demand in the clinical diagnosis for both T1- and T2-weighted MR images. [18, 21]. +Additionally, when several organ scans are required, injecting one dosage offers unmatched benefits +to patients and doctors [16]. Super-paramagnetic NPs have the potential to exhibit significant dual +T1/T2 relaxation performances when their sizes are decreased to less than 10 nm, according to some +theoretical investigations [21-24]. Recently, super-paramagnetic iron oxide-gold composite NPs is +synthesized by a green method [25]. It is shown that the NPs exhibited a high relaxivities ratio (r2/r1) +of 13.20, indicating the potential as a T2 contrast agent. +Surface modification is often practical to provide better stability under physiological +conditions and prolong bloodstream circulation time, thereby increasing MR imaging quality [26]. +This surface modification is known to restrict the uptake of plasma proteins (i.e., corona proteins), +which lowers the likelihood that macrophages will recognize and remove them [27]. In order to +overcome the aforementioned difficulties, polymeric coatings on the surface of magnetic NPs are +recommended [28]. In a recent work [29], iron oxide ferrofluid is synthesized by thermal +decomposition using poly (maleic anhydride-alt-1-octadecene, noted as PMAO) as a phase +transferring ligand. The results have demonstrated that the magnetic particles were fully covered at +high coverage by long non-magnetic polymeric chains. It is shown that this ligand could improve the +ferrofluid stability up to as long as 6 months. The MR images in solution and in rabbit using the +prepared PMAO-coated magnetic NPs had the best contrast effect on T2 weighted maps. +Polyethylene glycol (PEG) is a highly water soluble, hydrophilic, biocompatible, non- +antigenic, and protein-resistant polymer that is easily eliminated through the kidneys and is not +absorbed by humans' immune systems among all forms of polymeric coatings. PEG has also been +frequently employed for linking anticancer medications to proteins to prolong their half-life, as well +as for organ preservation [30. It also functions as an antibacterial, non-toxic lubricant and binder that +is frequently used in a variety of medicinal applications [31, 32]. Additionally, PEG-capped magnetic +NPs have demonstrated promise as effective and efficient magnetic hyperthermia candidates as well +as multifunctional nano-carriers for the encapsulation of hydrophobic medicines [28]. In our previous +work, we successfully synthesized Zn0.3MnxFe2.7-xO4 (x=0, 0.25, 0.5, 0.75 and 1) NPs by a one-step +citric acid-assistant hydrothermal method and reported the effect of citric acid concentration, pH of +the medium and the amount of Mn addition on the structure, purity, and magnetic properties of the + +4 + +synthesized NPs [33]. According to the author’s knowledge, citric acid-assistant hydrothermal +synthesis of PEG-6000 coated Zn0.3Mn0.5Fe2.2O4 NPs as a dual mode T1/T2 imaging contrast agent +have not been previously reported. In the present study, PEG surface coating is applied on the surface +of the zinc-manganese ferrite NPs and then physiochemical properties of the optimized sample is +thoroughly investigated. The mono-dispersed magnetic PEG-coated and uncoated Zn-Mn ferrite NPs +containing different levels of Mn content is synthesized and the MR imaging of the NPs in the presence +of external magnetic field is investigated. +2. Materials and Experimental Techniques +2.1. Materials +All raw materials, including Fe (NO3)3.9 H2O, NH4OH 25%, Zn (NO3)2.4H2O, Mn (NO3)2.4H2O and +C6H8O7.H2O (citric acid), CH3OH, and PEG (MW=6000 g/mol) were purchased from Merck Co. with +minimum purity of 99%. +2.2. Synthesis of Mn-Zn NPs +In order to synthesize Zn0.3MnxF2.7-xO4 NPs, where x is the molar fraction of manganese ions (Mn2+) +from 0 to 1, various amounts of manganese iron nitrate, zinc nitrate and manganese nitrate were +dissolved in 25 ml of distilled water. A reddish brown slurry was formed after adding a solution of +25% NH4OH which was added for the purpose of adjusting the pH of the media to 10. The resulting +slurry was then washed with the deionized distilled water three times. Following the addition of the +citric acid (CA), the mixture was rapidly stirred for 30 minutes before being placed to a 350 ml Teflon- +lined autoclave with a 65% fill level. The autoclave was kept at 185 °C for 15 h and then cooled to +room temperature [33]. Table 1 shows the experimental conditions of the synthesized samples. The +uncoated samples were coded as NCZMX in which X is the molar fraction of Mn2+ ions. +Table 1: The hydrothermal process parameters and the corresponding sample codes in the present work +Sample code +Temperature (℃) +Time (h) +Citric acid (mmol) +pH +Molar fraction of Mn2+(x) +NCZM +185 +15 +3.5 +10.5 +0 +NCZM25 +185 +15 +3.5 +10 +0.25 +NCZM50 +185 +15 +3.5 +10 +0.5 +NCZM75 +185 +15 +3.5 +10 +0.75 +NCZM100 +185 +15 +3.5 +10 +1 + +5 + +2.3. Coating of Mn-Zn NPs +15 mg of NCZM50 and NCZM25 NPs were added to 1 ml deionized distilled water and then placed +in an ultrasonic bath for 30 min. A polymeric solution containing 3 wt% PEG was dissolved in 1.5 ml +of deionized distilled water and stirred for 30 min. The prepared magnetic ferro-fluid placed on a +magnetic stirrer and then, the PEG solution were slowly added. This mixture was stirred at room +temperature for another 1 h at ambient temperature (25 °C). Finally, the coated NPs were magnetically +collected, washed with distilled water and dried in a vacuum oven at 40 °C for 24 h. The synthesized +coated NPs are named as CZM25 and CZM50. +2.4. Cell viability +The MCF-7 cells were cultured in Dulbecco’s modified Eagle’s medium DMEM (Gibco 12800, UK) +supplemented with 10% fetal bovine serum, 100 U/ml penicillin, 100 μg/ml streptomycin and 2 mM +L-glutamine at 37 °C in a humidified atmosphere of 5% CO2. The MG-63 osteoblast-like-cells were +seeded at a density of 10,000 cells/well in a 96 well plate and cultured with complete medium +containing NPs at concentrations of 50, 100 and 250 g/ml. MCF-7 cells were exposed to particles +for 24 h, after which Alamar Blue cytotoxicity assay was conducted and absorbance was measured at +450 nm using a micro-plate reader. The results represent the mean values ± SD of two individual +experiments each performed in quadruplicate. Differences between groups were determined by +student’s t test with values of p<0.05 considered significant [34, 35]. +2.5. Characterizations +Philips diffractometer, MPD-XPERT model, using CuKα radiation (λ = 1.5406 Å), was used for phase +identification. Estimation of the average crystallite size (L) of the samples, using the full width at half +maximum value (β) obtained from the spinel peaks located at every 2θ in the pattern, was carried out +by the modified Scherer’s formula. According to Scherer's modified formula, Lnβ (β in radians) is +plotted against Ln(1/cosθ). A linear plot is obtained using the linear regression which is defined as Eq. +(1). The intercept of the line would be Ln(kλ/L) (k=0.9); the value of L (mean crystallite size) can be +obtained using all the peaks: [33, 36]. +𝐋𝐧𝛃 = 𝐋𝐧 ((𝟎. 𝟗𝟒𝛌 +𝐋 +) + 𝐋𝐧 ( 𝟏 +𝐜𝐨𝐬𝛉)) +(1) + +The miller indices of the planes were extracted from the cards in the X’Pert software. Then, the mean +lattice parameter was calculated based on Eq. (2) [37]: + +6 + + + (2) +The shape, size, and size distribution of NPs were investigated using transmission electron microscopy +(TEM) with energy of 200 kV at Arya Rastak company in Tehran. A droplet of diluted magnetic flux +was placed on a carbon coated copper mesh and placed at room temperature to allow water to +evaporate. The average particle size of the produced zinc-manganese ferrite NPs from the TEM and +SEM data was calculated by measuring the diameter of at least 100 NPs with ImageJ software. The +data were fitted by a log-normal distribution curve and then the mean size was obtained. +Fourier transform infrared spectra (FTIR) were recorded in the range of 4000-400 cm-1 to detect +functional groups. +Saturation magnetization (Ms) values were conducted from the high field part of the measured +magnetization curves, where the magnetization curve becomes linear and line’s slope reaches to zero. +Colloidal properties of the aqueous magnetic ferro-fluids were investigated using a Zeta Potential +Estimator to measure the surface charge of NPs, hydrodynamic size, zeta potential and poly-dispersity +index of NPs (in pH=7) under different conditions. +Thermo-gravimetric analysis (TGA) was used to investigate the presence of polymer coating on the +surface of NPs. +MRI tests were performed with a 1.5 T clinical MRI instrument with a head coil working at 37 ℃. For +T1 and T2-weighted MRI of in-vitro cells at 1.5 T, the following parameters were adopted: [Mat +(320*192), FoV (184*230), and TR (407)], [Mat (256*192), FoV (260*260), and TR (7)], [Mat +(320*192), FoV (184*230), TR (2570)]. In order to simulate the physiological state of the body, PBS +solution and water was used to create a positive and negative contrast in the images. + +aj = d; × Jh,? +k;? + ?7 + + +Fig. 1. Image of the prepared instrument for MRI imaging. +3. Results and Discussion +3.1. Structural properties +Fig. 2. shows XRD pattern of the NCZM50 NPs in which the diffraction peaks are in good agreement +with planes (220), (311), (222), (400), (422), (511), (440), (620), (533) and (444) representing +synthesis of pure spinel phase without the need for any calcination step. The crystallite size of the +sample was estimated as 22 nm. + +Fig. 2. The XRD pattern of the NCZM50 sample. +Surface coating is important in preventing NPs from agglomeration in physiological environment +which also act as a barrier, effectively shielding the magnetic core against the attack of chemical + +140- +S +S +NC7.M50 +120- +Spincl:01-086-510 +100 - +80 +S +ntensi +F09三 +S +S +S +40- +S +S +S +20 - +0: +- +20 +40 +60 +80 +208 + +species in the aqueous solution. Here, PEG was utilized to coat the optimized NPs. The FT-IR spectra +of the pure NCZM50, the PEG-coated CZM50 NPs and the PEG are shown in Fig. 3. For the pure +NPs, at around 3300 cm-1, a strong wide band exists which is attributed to the O-H stretching vibrations +of water molecules which are assigned to –OH group of CA absorbed by NCZM50 NPs (a structural +bond). The stretching vibration of C-H corresponds to the peak at ~2925 cm- 1 [38, 39]. The absorption +band at 1690-1760 cm-1 is due to the vibration of asymmetric carboxyl group (-COOH) [28, 40]. +Hence, it is suggested that CA binds to the NPs surface through carboxylate groups of citrate ions +[28]. Furthermore, Fe-O stretching band as the characteristic peak of magnetite NPs was located at +around 520 cm−1 which is attributed to the Fe-O stretching vibration bond in tetrahedral sites and the +absorption band in the 437 cm-1 corresponds to a Fe-O vibrating bond in octahedral sites of ferrite +phase [41]. Hydroxyl groups (-OH) of PEG are linked to the carboxyl group (-COOH) of citric acid +(CA) for coating of Zn0.3Mn0.5Fe2.2O4 NPs. As it can be seen in Fig. 3, the highest peak for PEG curve +showed a very small shift in PEG-coated sample. The peak at 1105 cm-1 for pure PEG were shifted to +lower frequencies which is a proof of C-O-C and C-O-H groups bonding with Zn0.3Mn0.5Fe2.2O4 NPs. +The absorption band at 2884 cm-1 can also be due to the H-C bonds stretching vibrations of the +polymeric chain. The peaks corresponding to the bonds, C-H and C-O-C are the strong evidence to +show that the synthesized magnetite NPs surface has been coated with PEG [38, 40]. + +Fig. 3. The FT-IR spectra of the pure NCZM50 and PEG-coated CZM50 NPs along with the PEG coating and +citric acid. + +citricacid +2.4 +Zn0.3Mn0.5Fe2.204 +PEG-Zn0.3Mn0.5Fe2.204 +PEG +2.2 +2.0- +1.8 +-COOH +.6 +1.4 +C-H +HO +1.2 +1.0 +0.8 +0.6- +C-O-C groups +C-H groups +0.4 +500 +1000 +1500 +2000 +2500 +3000 +3500 +4000 +Wave number (cm-')9 + + +The presence of PEG layer on the NPs surface was also characterized by TGA which is presented in +Fig. 4. The first stage of weight loss at a temperature about 32-35 °C can be related to the removal of +water molecules (hydroxyl ions) that are physically absorbed to the surface of the NPs. This weight +loss in the uncoated sample is 2.45% and in the coated sample is equal to 2.15%. The comparison of +the first weight loss in the two samples shows that the total water loss of the NPs is more than coated +NPs which is due to the total absence of water from the magnetic material structure [42]. The second +step, starting at about 50-300 °C, results from the loss of organic groups that were conjugated to the +surface of the particles. PEG desorption and subsequent evaporation were the causes of this weight +loss. When 7.5 mg of PEG 6000 were used, the weight loss for particles was almost 24%, indicating +76% iron oxide in the polymer-coated NPs. Weight losses less than 15–20% can imply that the +coverage of particle surface by the polymer is not complete [40]. + +Fig. 4. The TGA result of the NCZM50 and CZM50 samples. +3.2. Microstructural analysis +Fig. 5 shows TEM micrograph and particle size distribution curve of the coated and uncoated samples. +By using ImageJ software to measure the diameter of at least 100 NPs, the average particle size and +the standard deviation was determined. As it can be seen, the synthesized NPs exhibit a rather uniform +size distribution, shape, and morphology. The mean particle size of the coated NPs is a bit greater than +that of the uncoated ones. It can be seen that NPs have become more dispersed after applying the + +100 +- +95 +Weight Percent (%) +90 +NCZM50 +85 +CZM50 + 08 +75 +- +- +1 +50 +100 +150 +200 +250 +300 +0 +Temperature (°C)10 + +coating in an aqueous medium. The average size of NPs obtained from the results of the TEM images +before and after coating was 6.9±1.54 nm and 9.25±1.6 nm, respectively, which indicates that the +polymer coating is applied on the surface of NPs at a low thickness [39]. + +Fig. 5. (a, c) TEM images and (b, d) particle size distribution histogram of the (a, b) uncoated NCZM50 and +(c, d) coated CZM50 NPs. +3.3. Stability and colloidal properties +The colloidal stability of magnetic fluids of NCZM50 sample was investigated using a zeta potential +measurement at pH=7 and various time points. The result indicated that the NCZM50 NPs sample had +a mean zeta potential of -48.86± 0.70 mV and a mean hydrodynamic size of 104 nm. According to +the results, the strong negative charge of the NPs (caused by the presence of citrate ions on their +surface) and the steric and electrostatic forces ensure their long-term stability in aqueous media [43]. +c +d + +(a) +70 +60 +50 +Frequency +40 +30 +20 +10 +0 +5 +6 +7 +8 +9 +10 +11 +12 +13 +14 +100nm +15 +16 +size (nm) +(d) +100 +80- +60 +40- +20- +5 +6 +7 +8 +9 +10 +11 +12 +13 +14 +15 +16 +size (nm) +75nm11 + +Poly-dispersity index (PDI) of NCZM50 sample was found to be 0.306. PDI is a parameter for +determining the particle size distribution of different NPs, which is obtained from photon correlation +spectroscopic analysis. It is a dimensionless number calculated from the autocorrelation function and +ranges from a value of 0.01 up to 0.7 for mono-dispersed and greater than 0.7 for poly-dispersed +particles [44]. In general, the particle size between 10 and 100 nm have the longest circulation time; +by contrast, it has been reported that particles of more than 200 nm tend to be immediately destroyed +by one of the MPS organs [43] and tend to be eliminated by the RES [9, 45], those with diameters <10 +nm are removed mainly by renal filtration, and particles larger than 400 nm (minimum diameter of +capillaries) will be filtered by the lung [46]. + +3.4. Magnetic properties +The particle size and magnetization saturation values of different NPs are presented in Table 2. The +room temperature M–H curve for NCZM50 and CZM50 samples is shown in Fig. 6. No hysteresis +loop can be seen and the value of magnetization sharply increases with the external magnetic field +strength. The M–H curve has an S-shape at the low field region, and the high field side of the curve is +almost linear with the external field [47]. Saturation magnetization for the NCZM50 and CZM50 NPs +is 55 emu/g and 38 emu/g respectively. The difference in particle size and the softening of the +magnetization caused by the presence of PEG can both be used to explain this mismatch [38]. The +magnetization curve of the CZM50 sample also revealed a negligible remnant magnetization at zero +field, reflecting the super-paramagnetic behavior of the ferro-fluid. Since magnetic powder has a +diameter much below the 20 nm cut-off expected for magnetite to show super-paramagnetic behavior, +the lack of hysteresis at ambient temperature is consistent with this theory [48]. + +Table 2. The size and Ms values of different NPs +Code +Chemistry +Size (nm) +Ms (emu/gr) +NCZM0 +Zn0.3Fe2.7O4 +14.5±2.7 +47 +NCZM25 +Zn0.3Mn0.25Fe2.45O4 +23.6±2.3 +47 +NCZM50 +Zn0.3Mn0. 5Fe2.2O4 +6.9±1.5 +55 +NCZM75 +Zn0.3Mn0. 75Fe1.95O4 +11.3 ±2.3 +41 +NCZM100 +Zn0.3Mn1Fe1.7O4 +6.7±2.4 +37 +CZM50 +PEG coated- Zn0.3Mn0.5Fe2.2O4 +9.3±1.6 +38 + + +12 + + +Fig. 6. The M vs H curves of the synthesized NCZM50 and CZM50 NPs. + +3.5. MRI analysis +MRI examination of the body can be performed with several coil types, depending on the design of +the MRI unit and the information required. Figs. 7(a-b) show T1– and T2-weighted MR images of +Fe3O4 and Zn-Mn ferrite solutions recorded on a 1.5-T MRI scanner at room temperature at different +concentrations (0.1, 0.15 and 0.2 mg/ml). As it can be seen, both T1 and T2-weighted MR images show +a strong dependence of signal intensity on manganese concentrations and among the Fe3O4 control +sample, Zn-based and Mn-Zn-based super-paramagnetic NPs, Mn-Zn ferrites represent a better MRI +contrast [49]. This is due to the fact that Mn2+ with five unpaired electrons, after Gd3+, is the most +powerful cation used as a MRI contrast agent [50]. Due to their greater paramagnetism and five +unpaired electrons, divalent manganese ions (Mn2+) have been shown to be a successful method of +increasing the r1 of ultra-small iron oxide NPs. A peculiar mixed spinel structure, a greater saturation +magnetization (Ms), and a high r2 of manganese doped iron oxide NPs result from the doped Mn2+ +with a higher magnetic moment (B=5.92) being able to fill both the tetrahedral (Td) and octahedral +(Oh) sites in the crystal lattice. The doped Mn2+ and ultra-small iron oxide NPs also exhibit synergetic + +60 - +CZM50 +NCZM50 +40- +Magnetization (emu/g) +20 +0 +40 +Magnetization (emu/g) +20 +20 +0 +-40 - +20 +-40 +09- +-400 +-200 +0 +200 +400 +Applied field (Oe) +-15000 +-10000 +-5000 +0 +5000 +10000 +15000 +Applied field (Oe)13 + +enhancement, which will further enhance both r1 and r2 of Mn-iron oxide NPs, according to the +embedding logic. The Mn-iron oxide NPs may therefore make superior candidates for dual-contrast +CA [20]. Indeed, it has recently been discovered that decreasing iron oxide NPs below 10 nm improves +their effectiveness as T1 contrast agents, suggesting that this approach could be employed to create +dual contrast agents. The utility of these NPs as T1 contrast agents is unfortunately limited by the low +r2/r1 values caused by the large decrease in r2 that occurred along with the increase in r1. To get over +this restriction, alloy-based NPs which has a high Ms are a suitable candidate to achieve NPs with +high MRI sensitivity [51]. The addition of Mn2+ and Zn2+ divalent cation ions to the spinel ferrite +structure causes the mass magnetization of the material to rise, which enhances the magnetic +characteristics. Therefore, the higher contrast in Zn0.3Mn0.5Fe2.2O4 NPs with higher saturation +magnetization can be justified [52]. As it is presented in Fig. 7, NCZM50 sample with core diameters +about 6.7±1.54 nm and saturation magnetization about 55 emu/g is capable of producing dual positive +and negative contrast in images [26, 53]. However, the length of the polymer chain, which relates to +coating thickness, has a substantial impact on relaxivity as well. According to computer simulations, +the physical exclusion of protons from the super-paramagnetic iron oxide magnetic field and the +protons' residence period within the coating zone compete to decide the influence of coating thickness +on relaxivity. +As it can be seen in the Fig. 8, the surface coatings also affect the relaxivity of NPs. Laconte et al. +reported that the increased coating thickness would dramatically decrease the r2 and r1 relaxivity of +mono-crystalline magnetic NPs. Therefore it is important to note that both the chemistry of coating +and its thickness affect the value of r2 and r1 in which as the coating thickness increases, the ratio r2/r1 +decreases. This is due to the inner hydrophobic layer excluding water, while the outer hydrophilic +PEG layer allows water to diffuse within the coating zone [53] . + + + + +14 + + +Fig. 7. (a) T1-weighted and (b) T2-weighted MR images of the uncoated Fe3O4 and Mn-Zn ferrite NPs at +different concentrations indicated by different numbers: (1) Fe3O4 control sample, (2) NCZM0, (3) NCZM25, +(4) NCZM50, (5) NCZM75, and (6) NCZM100. + + + + + +Fig. 8. (a) T1-weighted and (b) T2-weighted MR images of un-coated and uncoated samples indicated by +different numbers: (1) Fe3O4 control sample, (2) NCZM50, and (3) CZM50. + +3.6. Cell viability +Cytotoxicity evaluations of the uncoated and coated NPs were investigated by evaluating their +cytotoxicity using MCF-7 cell line. The results of Alamar blue cytotoxicity assay are presented in Fig. +9. According to the results, a similar trend is observed in the activity of cells affected by different +concentrations of NPs after 24 h compared with the control group. In general, coated and uncoated +particles did not negatively change the cell growth process, and did not result significant reduction in +cell viability. In fact, a better growth was observed in the presence of coated NPs. + + +(a) +(b)(a) +(b) +215 + + +Fig. 9. The cytotoxicity assays performed on MCF-7 cells in the presence of coated and uncoated NPs after +24 h. +4. Conclusions +Mono-dispersed Zn0.3Mn0.5Fe2.2O4 NPs with an average size of about 6.9±1.5 nm were successfully +synthesized by a facile, one step citric acid-assisted hydrothermal method. The NPs were stabilized +with a layer of hydrophilic PEG and exhibited long-term colloidal stability in aqueous media at pH=7. +The magnetic properties of the uncoated and coated Zn-Mn ferrite NPs were measured as 55 and 38 +emu/g, respectively, showing super-paramagnetic behavior at room temperature. More significantly, +the synthesized NPs displayed unexpectedly high T1 and T2 imaging contrast due to Zn2+ and Mn2+ +doping and PEG-6000 coating. The present zinc manganese iron oxide NPs coated by PEG +(ZnMnIONPs@PEG) are supposed to be a suitable candidate for application as T1/T2 dual contrast +agent, as shown by in-vitro MR imaging. Interestingly, applying low thickness of PEG layer on the +surface of the Zn0.3Mn0.5Fe2.2O4 NPs had no significant effect on the MR imaging. + +References +[1] +S. Mornet, S. Vasseur, F. Grasset, and E. Duguet, Magnetic nanoparticle design for medical diagnosis +and therapy, J. Mater. Chem., 14 (2004) 2161-2175. +[2] +Y.-M. 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Zhang, Magnetite nanoparticles for medical MR imaging, Mater. +Tod., 14 (2011) 330-338. + + + + + diff --git a/3NFST4oBgHgl3EQfYjhQ/content/tmp_files/load_file.txt b/3NFST4oBgHgl3EQfYjhQ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..680a6686c6d47de711d3256bcc544c7341c4e5cf --- /dev/null +++ b/3NFST4oBgHgl3EQfYjhQ/content/tmp_files/load_file.txt @@ -0,0 +1,851 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf,len=850 +page_content='1 Synthesis and characterization of PEG-coated Zn0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='3MnxFe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='7-xO4 nanoparticles as the dual T1/T2-weighted MRI contrast agent Bahareh Rezaei, Ahmad Kermanpur*, Sheyda Labbaf Department of Materials Engineering, Isfahan University of Technology, Isfahan 84156-83111, Iran Abstract Super-paramagnetic nanoparticles (NPs) have been widely explored as magnetic resonance imaging (MRI) contrast agents because of a combination of favorable magnetic properties, biocompability and ease of fabrication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' MRI using traditional T1- or T2-weighted single mode contrast-enhanced techniques may yield inaccurate imaging results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' In the present work, a T1/T2 dual mode contrast agent based on the super-paramagnetic zinc-manganese ferrite (Zn0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='3MnxFe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='7-xO4, x= 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='75 and 1) NPs with small core size and a hydrophilic PEG surface coating is reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' The TEM, TGA and FTIR results confirmed the formation of a uniform coating on the NPs surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' The MRI analysis revealed that the Zn0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='3Mn0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='5Fe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='2O4 NPs had the maximum image contrast compared to other zinc-manganese ferrite samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' Cell viability evaluations revealed that the coated and uncoated particles did not inhibit cell growth pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' The present PEG-coated Zn0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='3Mn0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='5Fe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='2O4 NPs can be utilized as a suitable T1/T2-weighted MRI contrast agent for better diagnostic of abnormalities in the organs or tissues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' Keywords Magnetic Resonance Imaging (MRI);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' Super-paramagnetic nanoparticles;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' Zn0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='3MnxFe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='7-xO4 nanoparticles;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' Polyethylene Glycol (PEG) coating 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' Introduction The most potent and painless test that gives extremely clear images of the internal organs in the body is the magnetic resonance imaging (MRI) scan [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' Based on the magnetic relaxation processes of water protons on soft tissue of nearly every internal structure in the human body [1, 3-5], this method is a sort of diagnostic test that generates detailed images and functional information in a non-invasive and real-time monitoring manner [6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' It is a distinguished device since there is no ionizing radiation during the imaging process and obviously reduces harmful side effects [2, 4, 8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' However, this test typically provides poor anatomical details, and clinicians have some difficulties to distinguish between normal and abnormal tissues due to its low sensitivity [9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' Hence, the clinical Corresponding author;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' Tel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' (+98)3133915738;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' Fax (+98)3133912752;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' Email: ahmad_k@iut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='ir 2 domains urgently require more reliable MR images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' There is a potential to create more accurate and crisper images by adding contrast agents, which enables physicians to detect organs or in-vivo systems more clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' This opens up a wide range of MRI applications for therapeutic medicine in addition to diagnostic radiology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' Despite the fact of shorter circulation time of Gd3+ ions as a T1-weighted MRI contrast agent,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' which renders them useless for high-resolution and/or targeted MRI [9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' 11] and many concerns about potential trace deposition of Gd ions in the body,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' known as Nephrogenic Systemic Fibrosis (NSF) [12-14],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' which is a rare disease that frequently develops in patients with severe renal failure or after liver transplantation [15],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' Gd-based contrast agents can shorten the T1 relaxation time effectively and provide brighter images in the regions of interest [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' Following the increased awareness of this side effect, researchers have much more emphasis on alternative methods based on Mn-based complexes [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' Although no scientific relationship has been proved between the NSF side effect and Mn so far, the metal is still known to pose some toxicity when inhaled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' However, small amounts are essential to human health, but overexposure to free Mn ions may result in the neurodegenerative disorder known as ‘Manganism’ with symptoms similar Parkinson’s disease [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' Unlike Gd3+ and Mn2+ chelates, iron oxide nanoparticles (NPs) have achieved great attention due to the outstanding properties they exhibit at the nano-metric scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' A large number of benefits including biocompatibility, superparamagnetic behavior at room temperature, high saturation magnetization that can be tailored by size, shape, composition and assembly, tunable cellular uptake, biodispersibility, and large surface areas that make them a good candidate for polymer coating, conjugation with targeting molecules and other probes for achieving targeting and multimodal agents [17, 18] is reported for the iron oxide NPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' Super-paramagnetic NPs can be employed as T2-weighted MRI contrast agents since they are more sensitive in the micro- or nano-molar range than Gd complexes [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' Clinical MR imaging applications often use iron oxide-based NPs with strong magnetic moments as T2-weighted MRI contrast agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' The limited usage of iron oxide NPs as T1 contrast agents is due to their high transverse to longitudinal relaxivity ratio [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' However, the use of superparamagnetic NPs in MRI is constrained by a negative contrast effect and magnetic susceptibility artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' Because the signal is frequently confused with signals from bleeding, calcification, or metal deposits and the susceptibility artifacts alter the background image, the resulting dark signal in T2- weighted MRI may be exploited to mislead clinical diagnosis [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' The T1-weighted MRI contrast agents, however, have advantages over T2-weighted MRI contrast agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' These advantages include better imaging quality, brighter images that can more effectively distinguish between normal and lesion tissues, and also the ability to provide better resolution for blood imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' Nonetheless, in T1- 3 weighted MR imaging, some normal tissues (such as fatty tissue) may be mistaken for bright lesions that have been increased by T1 contrast agents [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' Therefore, efforts to integrate T1 and T2 imaging to prevent probable MRI artifacts and produce superior clinical images have been made as a result of the rising demand in the clinical diagnosis for both T1- and T2-weighted MR images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' [18, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' Additionally, when several organ scans are required, injecting one dosage offers unmatched benefits to patients and doctors [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' Super-paramagnetic NPs have the potential to exhibit significant dual T1/T2 relaxation performances when their sizes are decreased to less than 10 nm, according to some theoretical investigations [21-24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' Recently, super-paramagnetic iron oxide-gold composite NPs is synthesized by a green method [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' It is shown that the NPs exhibited a high relaxivities ratio (r2/r1) of 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='20, indicating the potential as a T2 contrast agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' Surface modification is often practical to provide better stability under physiological conditions and prolong bloodstream circulation time, thereby increasing MR imaging quality [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' This surface modification is known to restrict the uptake of plasma proteins (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=', corona proteins), which lowers the likelihood that macrophages will recognize and remove them [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' In order to overcome the aforementioned difficulties, polymeric coatings on the surface of magnetic NPs are recommended [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' In a recent work [29], iron oxide ferrofluid is synthesized by thermal decomposition using poly (maleic anhydride-alt-1-octadecene, noted as PMAO) as a phase transferring ligand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' The results have demonstrated that the magnetic particles were fully covered at high coverage by long non-magnetic polymeric chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' It is shown that this ligand could improve the ferrofluid stability up to as long as 6 months.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' The MR images in solution and in rabbit using the prepared PMAO-coated magnetic NPs had the best contrast effect on T2 weighted maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=" Polyethylene glycol (PEG) is a highly water soluble, hydrophilic, biocompatible, non- antigenic, and protein-resistant polymer that is easily eliminated through the kidneys and is not absorbed by humans' immune systems among all forms of polymeric coatings." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' PEG has also been frequently employed for linking anticancer medications to proteins to prolong their half-life, as well as for organ preservation [30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' It also functions as an antibacterial, non-toxic lubricant and binder that is frequently used in a variety of medicinal applications [31, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' Additionally, PEG-capped magnetic NPs have demonstrated promise as effective and efficient magnetic hyperthermia candidates as well as multifunctional nano-carriers for the encapsulation of hydrophobic medicines [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' In our previous work, we successfully synthesized Zn0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='3MnxFe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='7-xO4 (x=0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='75 and 1) NPs by a one-step citric acid-assistant hydrothermal method and reported the effect of citric acid concentration, pH of the medium and the amount of Mn addition on the structure, purity, and magnetic properties of the 4 synthesized NPs [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' According to the author’s knowledge, citric acid-assistant hydrothermal synthesis of PEG-6000 coated Zn0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='3Mn0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='5Fe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='2O4 NPs as a dual mode T1/T2 imaging contrast agent have not been previously reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' In the present study, PEG surface coating is applied on the surface of the zinc-manganese ferrite NPs and then physiochemical properties of the optimized sample is thoroughly investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' The mono-dispersed magnetic PEG-coated and uncoated Zn-Mn ferrite NPs containing different levels of Mn content is synthesized and the MR imaging of the NPs in the presence of external magnetic field is investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' Materials and Experimental Techniques 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' Materials All raw materials, including Fe (NO3)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='9 H2O, NH4OH 25%, Zn (NO3)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='4H2O, Mn (NO3)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='4H2O and C6H8O7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='H2O (citric acid), CH3OH, and PEG (MW=6000 g/mol) were purchased from Merck Co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' with minimum purity of 99%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' Synthesis of Mn-Zn NPs In order to synthesize Zn0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='3MnxF2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='7-xO4 NPs, where x is the molar fraction of manganese ions (Mn2+) from 0 to 1, various amounts of manganese iron nitrate, zinc nitrate and manganese nitrate were dissolved in 25 ml of distilled water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' A reddish brown slurry was formed after adding a solution of 25% NH4OH which was added for the purpose of adjusting the pH of the media to 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' The resulting slurry was then washed with the deionized distilled water three times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' Following the addition of the citric acid (CA), the mixture was rapidly stirred for 30 minutes before being placed to a 350 ml Teflon- lined autoclave with a 65% fill level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' The autoclave was kept at 185 °C for 15 h and then cooled to room temperature [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' Table 1 shows the experimental conditions of the synthesized samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' The uncoated samples were coded as NCZMX in which X is the molar fraction of Mn2+ ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' Table 1: The hydrothermal process parameters and the corresponding sample codes in the present work Sample code Temperature (℃) Time (h) Citric acid (mmol) pH Molar fraction of Mn2+(x) NCZM 185 15 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='5 0 NCZM25 185 15 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='5 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='25 NCZM50 185 15 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='5 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='5 NCZM75 185 15 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='5 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='75 NCZM100 185 15 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='5 10 1 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' Coating of Mn-Zn NPs 15 mg of NCZM50 and NCZM25 NPs were added to 1 ml deionized distilled water and then placed in an ultrasonic bath for 30 min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' A polymeric solution containing 3 wt% PEG was dissolved in 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='5 ml of deionized distilled water and stirred for 30 min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' The prepared magnetic ferro-fluid placed on a magnetic stirrer and then, the PEG solution were slowly added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' This mixture was stirred at room temperature for another 1 h at ambient temperature (25 °C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' Finally, the coated NPs were magnetically collected, washed with distilled water and dried in a vacuum oven at 40 °C for 24 h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' The synthesized coated NPs are named as CZM25 and CZM50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' Cell viability The MCF-7 cells were cultured in Dulbecco’s modified Eagle’s medium DMEM (Gibco 12800, UK) supplemented with 10% fetal bovine serum, 100 U/ml penicillin, 100 μg/ml streptomycin and 2 mM L-glutamine at 37 °C in a humidified atmosphere of 5% CO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' The MG-63 osteoblast-like-cells were seeded at a density of 10,000 cells/well in a 96 well plate and cultured with complete medium containing NPs at concentrations of 50, 100 and 250 \uf06dg/ml.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' MCF-7 cells were exposed to particles for 24 h, after which Alamar Blue cytotoxicity assay was conducted and absorbance was measured at 450 nm using a micro-plate reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' The results represent the mean values ± SD of two individual experiments each performed in quadruplicate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' Differences between groups were determined by student’s t test with values of p<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='05 considered significant [34, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' Characterizations Philips diffractometer, MPD-XPERT model, using CuKα radiation (λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='5406 Å), was used for phase identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' Estimation of the average crystallite size (L) of the samples, using the full width at half maximum value (β) obtained from the spinel peaks located at every 2θ in the pattern, was carried out by the modified Scherer’s formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=" According to Scherer's modified formula, Lnβ (β in radians) is plotted against Ln(1/cosθ)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' A linear plot is obtained using the linear regression which is defined as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' The intercept of the line would be Ln(kλ/L) (k=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='9);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' the value of L (mean crystallite size) can be obtained using all the peaks: [33, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' 𝐋𝐧𝛃 = 𝐋𝐧 ((𝟎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' 𝟗𝟒𝛌 𝐋 ) + 𝐋𝐧 ( 𝟏 𝐜𝐨𝐬𝛉)) (1) The miller indices of the planes were extracted from the cards in the X’Pert software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' Then, the mean lattice parameter was calculated based on Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' (2) [37]: 6 (2) The shape, size, and size distribution of NPs were investigated using transmission electron microscopy (TEM) with energy of 200 kV at Arya Rastak company in Tehran.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' A droplet of diluted magnetic flux was placed on a carbon coated copper mesh and placed at room temperature to allow water to evaporate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' The average particle size of the produced zinc-manganese ferrite NPs from the TEM and SEM data was calculated by measuring the diameter of at least 100 NPs with ImageJ software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' The data were fitted by a log-normal distribution curve and then the mean size was obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' Fourier transform infrared spectra (FTIR) were recorded in the range of 4000-400 cm-1 to detect functional groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' Saturation magnetization (Ms) values were conducted from the high field part of the measured magnetization curves, where the magnetization curve becomes linear and line’s slope reaches to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' Colloidal properties of the aqueous magnetic ferro-fluids were investigated using a Zeta Potential Estimator to measure the surface charge of NPs, hydrodynamic size, zeta potential and poly-dispersity index of NPs (in pH=7) under different conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' Thermo-gravimetric analysis (TGA) was used to investigate the presence of polymer coating on the surface of NPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' MRI tests were performed with a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='5 T clinical MRI instrument with a head coil working at 37 ℃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' For T1 and T2-weighted MRI of in-vitro cells at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='5 T, the following parameters were adopted: [Mat (320*192), FoV (184*230), and TR (407)], [Mat (256*192), FoV (260*260), and TR (7)], [Mat (320*192), FoV (184*230), TR (2570)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' In order to simulate the physiological state of the body, PBS solution and water was used to create a positive and negative contrast in the images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' aj = d;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' × Jh,?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' +k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' + ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='7 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' Image of the prepared instrument for MRI imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' Results and Discussion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' Structural properties Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' shows XRD pattern of the NCZM50 NPs in which the diffraction peaks are in good agreement with planes (220), (311), (222), (400), (422), (511), (440), (620), (533) and (444) representing synthesis of pure spinel phase without the need for any calcination step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' The crystallite size of the sample was estimated as 22 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' The XRD pattern of the NCZM50 sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' Surface coating is important in preventing NPs from agglomeration in physiological environment which also act as a barrier, effectively shielding the magnetic core against the attack of chemical 140- S S NC7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='M50 120- Spincl:01-086-510 100 - 80 S ntensi F09三 S S S 40- S S S 20 - 0: 20 40 60 80 208 species in the aqueous solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' Here, PEG was utilized to coat the optimized NPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' The FT-IR spectra of the pure NCZM50, the PEG-coated CZM50 NPs and the PEG are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' For the pure NPs, at around 3300 cm-1, a strong wide band exists which is attributed to the O-H stretching vibrations of water molecules which are assigned to –OH group of CA absorbed by NCZM50 NPs (a structural bond).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' The stretching vibration of C-H corresponds to the peak at ~2925 cm- 1 [38, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' The absorption band at 1690-1760 cm-1 is due to the vibration of asymmetric carboxyl group (-COOH) [28, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' Hence, it is suggested that CA binds to the NPs surface through carboxylate groups of citrate ions [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' Furthermore, Fe-O stretching band as the characteristic peak of magnetite NPs was located at around 520 cm−1 which is attributed to the Fe-O stretching vibration bond in tetrahedral sites and the absorption band in the 437 cm-1 corresponds to a Fe-O vibrating bond in octahedral sites of ferrite phase [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' Hydroxyl groups (-OH) of PEG are linked to the carboxyl group (-COOH) of citric acid (CA) for coating of Zn0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='3Mn0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='5Fe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='2O4 NPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' As it can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' 3, the highest peak for PEG curve showed a very small shift in PEG-coated sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' The peak at 1105 cm-1 for pure PEG were shifted to lower frequencies which is a proof of C-O-C and C-O-H groups bonding with Zn0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='3Mn0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='5Fe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='2O4 NPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' The absorption band at 2884 cm-1 can also be due to the H-C bonds stretching vibrations of the polymeric chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' The peaks corresponding to the bonds, C-H and C-O-C are the strong evidence to show that the synthesized magnetite NPs surface has been coated with PEG [38, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' The FT-IR spectra of the pure NCZM50 and PEG-coated CZM50 NPs along with the PEG coating and citric acid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' citricacid 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='4 Zn0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='3Mn0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='5Fe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='204 PEG-Zn0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='3Mn0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='5Fe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='204 PEG 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='0- 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='8 COOH .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='4 C-H HO 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='6- C-O-C groups C-H groups 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content="4 500 1000 1500 2000 2500 3000 3500 4000 Wave number (cm-')9 The presence of PEG layer on the NPs surface was also characterized by TGA which is presented in Fig." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' The first stage of weight loss at a temperature about 32-35 °C can be related to the removal of water molecules (hydroxyl ions) that are physically absorbed to the surface of the NPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' This weight loss in the uncoated sample is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='45% and in the coated sample is equal to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='15%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' The comparison of the first weight loss in the two samples shows that the total water loss of the NPs is more than coated NPs which is due to the total absence of water from the magnetic material structure [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' The second step, starting at about 50-300 °C, results from the loss of organic groups that were conjugated to the surface of the particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' PEG desorption and subsequent evaporation were the causes of this weight loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' When 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='5 mg of PEG 6000 were used, the weight loss for particles was almost 24%, indicating 76% iron oxide in the polymer-coated NPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' Weight losses less than 15–20% can imply that the coverage of particle surface by the polymer is not complete [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' The TGA result of the NCZM50 and CZM50 samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' Microstructural analysis Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' 5 shows TEM micrograph and particle size distribution curve of the coated and uncoated samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' By using ImageJ software to measure the diameter of at least 100 NPs, the average particle size and the standard deviation was determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' As it can be seen, the synthesized NPs exhibit a rather uniform size distribution, shape, and morphology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' The mean particle size of the coated NPs is a bit greater than that of the uncoated ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' It can be seen that NPs have become more dispersed after applying the 100 95 Weight Percent (%) 90 NCZM50 85 CZM50 08 75 1 50 100 150 200 250 300 0 Temperature (°C)10 coating in an aqueous medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' The average size of NPs obtained from the results of the TEM images before and after coating was 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='9±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='54 nm and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='25±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='6 nm, respectively, which indicates that the polymer coating is applied on the surface of NPs at a low thickness [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' (a, c) TEM images and (b, d) particle size distribution histogram of the (a, b) uncoated NCZM50 and (c, d) coated CZM50 NPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' Stability and colloidal properties The colloidal stability of magnetic fluids of NCZM50 sample was investigated using a zeta potential measurement at pH=7 and various time points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' The result indicated that the NCZM50 NPs sample had a mean zeta potential of -48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='86± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='70 mV and a mean hydrodynamic size of 104 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' According to the results, the strong negative charge of the NPs (caused by the presence of citrate ions on their surface) and the steric and electrostatic forces ensure their long-term stability in aqueous media [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' c d (a) 70 60 50 Frequency 40 30 20 10 0 5 6 7 8 9 10 11 12 13 14 100nm 15 16 size (nm) (d) 100 80- 60 40- 20- 5 6 7 8 9 10 11 12 13 14 15 16 size (nm) 75nm11 Poly-dispersity index (PDI) of NCZM50 sample was found to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='306.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' PDI is a parameter for determining the particle size distribution of different NPs, which is obtained from photon correlation spectroscopic analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' It is a dimensionless number calculated from the autocorrelation function and ranges from a value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='01 up to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='7 for mono-dispersed and greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='7 for poly-dispersed particles [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' In general, the particle size between 10 and 100 nm have the longest circulation time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' by contrast, it has been reported that particles of more than 200 nm tend to be immediately destroyed by one of the MPS organs [43] and tend to be eliminated by the RES [9, 45], those with diameters <10 nm are removed mainly by renal filtration, and particles larger than 400 nm (minimum diameter of capillaries) will be filtered by the lung [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' Magnetic properties The particle size and magnetization saturation values of different NPs are presented in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' The room temperature M–H curve for NCZM50 and CZM50 samples is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' No hysteresis loop can be seen and the value of magnetization sharply increases with the external magnetic field strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' The M–H curve has an S-shape at the low field region, and the high field side of the curve is almost linear with the external field [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' Saturation magnetization for the NCZM50 and CZM50 NPs is 55 emu/g and 38 emu/g respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' The difference in particle size and the softening of the magnetization caused by the presence of PEG can both be used to explain this mismatch [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' The magnetization curve of the CZM50 sample also revealed a negligible remnant magnetization at zero field, reflecting the super-paramagnetic behavior of the ferro-fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' Since magnetic powder has a diameter much below the 20 nm cut-off expected for magnetite to show super-paramagnetic behavior, the lack of hysteresis at ambient temperature is consistent with this theory [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' The size and Ms values of different NPs Code Chemistry Size (nm) Ms (emu/gr) NCZM0 Zn0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='3Fe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='7O4 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='5±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='7 47 NCZM25 Zn0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='3Mn0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='25Fe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='45O4 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='6±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='3 47 NCZM50 Zn0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='3Mn0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' 5Fe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='2O4 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='9±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='5 55 NCZM75 Zn0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='3Mn0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' 75Fe1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='95O4 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='3 ±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='3 41 NCZM100 Zn0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='3Mn1Fe1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='7O4 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='7±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='4 37 CZM50 PEG coated- Zn0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='3Mn0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='5Fe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='2O4 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='3±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='6 38 12 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' The M vs H curves of the synthesized NCZM50 and CZM50 NPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' MRI analysis MRI examination of the body can be performed with several coil types, depending on the design of the MRI unit and the information required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' 7(a-b) show T1– and T2-weighted MR images of Fe3O4 and Zn-Mn ferrite solutions recorded on a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='5-T MRI scanner at room temperature at different concentrations (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='15 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='2 mg/ml).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' As it can be seen, both T1 and T2-weighted MR images show a strong dependence of signal intensity on manganese concentrations and among the Fe3O4 control sample, Zn-based and Mn-Zn-based super-paramagnetic NPs, Mn-Zn ferrites represent a better MRI contrast [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' This is due to the fact that Mn2+ with five unpaired electrons, after Gd3+, is the most powerful cation used as a MRI contrast agent [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' Due to their greater paramagnetism and five unpaired electrons, divalent manganese ions (Mn2+) have been shown to be a successful method of increasing the r1 of ultra-small iron oxide NPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' A peculiar mixed spinel structure, a greater saturation magnetization (Ms), and a high r2 of manganese doped iron oxide NPs result from the doped Mn2+ with a higher magnetic moment (B=5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='92) being able to fill both the tetrahedral (Td) and octahedral (Oh) sites in the crystal lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' The doped Mn2+ and ultra-small iron oxide NPs also exhibit synergetic 60 - CZM50 NCZM50 40- Magnetization (emu/g) 20 0 40 Magnetization (emu/g) 20 20 0 40 - 20 40 09- 400 200 0 200 400 Applied field (Oe) 15000 10000 5000 0 5000 10000 15000 Applied field (Oe)13 enhancement, which will further enhance both r1 and r2 of Mn-iron oxide NPs, according to the embedding logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' The Mn-iron oxide NPs may therefore make superior candidates for dual-contrast CA [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' Indeed, it has recently been discovered that decreasing iron oxide NPs below 10 nm improves their effectiveness as T1 contrast agents, suggesting that this approach could be employed to create dual contrast agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' The utility of these NPs as T1 contrast agents is unfortunately limited by the low r2/r1 values caused by the large decrease in r2 that occurred along with the increase in r1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' To get over this restriction, alloy-based NPs which has a high Ms are a suitable candidate to achieve NPs with high MRI sensitivity [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' The addition of Mn2+ and Zn2+ divalent cation ions to the spinel ferrite structure causes the mass magnetization of the material to rise, which enhances the magnetic characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' Therefore, the higher contrast in Zn0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='3Mn0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='5Fe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='2O4 NPs with higher saturation magnetization can be justified [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' As it is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' 7, NCZM50 sample with core diameters about 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='7±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='54 nm and saturation magnetization about 55 emu/g is capable of producing dual positive and negative contrast in images [26, 53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' However, the length of the polymer chain, which relates to coating thickness, has a substantial impact on relaxivity as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=" According to computer simulations, the physical exclusion of protons from the super-paramagnetic iron oxide magnetic field and the protons' residence period within the coating zone compete to decide the influence of coating thickness on relaxivity." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' As it can be seen in the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' 8, the surface coatings also affect the relaxivity of NPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' Laconte et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' reported that the increased coating thickness would dramatically decrease the r2 and r1 relaxivity of mono-crystalline magnetic NPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' Therefore it is important to note that both the chemistry of coating and its thickness affect the value of r2 and r1 in which as the coating thickness increases, the ratio r2/r1 decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' This is due to the inner hydrophobic layer excluding water, while the outer hydrophilic PEG layer allows water to diffuse within the coating zone [53] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' 14 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' (a) T1-weighted and (b) T2-weighted MR images of the uncoated Fe3O4 and Mn-Zn ferrite NPs at different concentrations indicated by different numbers: (1) Fe3O4 control sample, (2) NCZM0, (3) NCZM25, (4) NCZM50, (5) NCZM75, and (6) NCZM100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' (a) T1-weighted and (b) T2-weighted MR images of un-coated and uncoated samples indicated by different numbers: (1) Fe3O4 control sample, (2) NCZM50, and (3) CZM50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' Cell viability Cytotoxicity evaluations of the uncoated and coated NPs were investigated by evaluating their cytotoxicity using MCF-7 cell line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' The results of Alamar blue cytotoxicity assay are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' According to the results, a similar trend is observed in the activity of cells affected by different concentrations of NPs after 24 h compared with the control group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' In general, coated and uncoated particles did not negatively change the cell growth process, and did not result significant reduction in cell viability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' In fact, a better growth was observed in the presence of coated NPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' (a) (b)(a) (b) 215 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' The cytotoxicity assays performed on MCF-7 cells in the presence of coated and uncoated NPs after 24 h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' Conclusions Mono-dispersed Zn0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='3Mn0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='5Fe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='2O4 NPs with an average size of about 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='9±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='5 nm were successfully synthesized by a facile, one step citric acid-assisted hydrothermal method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' The NPs were stabilized with a layer of hydrophilic PEG and exhibited long-term colloidal stability in aqueous media at pH=7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' The magnetic properties of the uncoated and coated Zn-Mn ferrite NPs were measured as 55 and 38 emu/g, respectively, showing super-paramagnetic behavior at room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' More significantly, the synthesized NPs displayed unexpectedly high T1 and T2 imaging contrast due to Zn2+ and Mn2+ doping and PEG-6000 coating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' The present zinc manganese iron oxide NPs coated by PEG (ZnMnIONPs@PEG) are supposed to be a suitable candidate for application as T1/T2 dual contrast agent, as shown by in-vitro MR imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' Interestingly, applying low thickness of PEG layer on the surface of the Zn0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='3Mn0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='5Fe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content='2O4 NPs had no significant effect on the MR imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NFST4oBgHgl3EQfYjhQ/content/2301.13788v1.pdf'} +page_content=' References [1] S.' 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a/79E1T4oBgHgl3EQfnQTu/content/tmp_files/2301.03308v1.pdf.txt b/79E1T4oBgHgl3EQfnQTu/content/tmp_files/2301.03308v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..389bf6ead2538fea285ad7fd302cc370b4a8a19f --- /dev/null +++ b/79E1T4oBgHgl3EQfnQTu/content/tmp_files/2301.03308v1.pdf.txt @@ -0,0 +1,1377 @@ +Efficient Design of Helical Higher-Order Topological Insulators +in 3D Elastic Medium +Jiachen Luo1, Zongliang Du1,2*, Hui Chen3, Xianggui Ding1, Chang Liu1,2, +Weisheng Zhang1,2, Xu Guo1,2* +1State Key Laboratory of Structural Analysis for Industrial Equipment, Department of Engineering +Mechanics, Dalian University of Technology, Dalian, 116023, China +2Ningbo Institute of Dalian University of Technology, Ningbo, 315016, China +3Piezoelectric Device Laboratory, School of Mechanical Engineering and Mechanics, +Ningbo University, Ningbo 315211, China +E-mail: zldu@dlut.edu.cn (ZD); guoxu@dlut.edu.cn (XG) +Abstract +Topological materials (TMs) are well-known for their topological protected properties. +Phononic system has the advantage of direct observation and engineering of topological +phenomena on the macroscopic scale. For the inverse design of 3D TMs in continuum +medium, however, it would be extremely difficult to classify the topological properties, +tackle the computational complexity, and search solutions in an infinite parameter space. +This work proposed a systematic design framework for the 3D mechanical higher-order +topological insulators (HOTIs) by combining the symmetry indicators (SI) method and the +moving morphable components (MMC) method. The 3D unit cells are described by the +MMC method with only tens of design variables. By evaluating the inherent singularity +properties in the 3D mechanical system, the classic formulas of topological invariants are +modified accordingly for elastic waves. Then a mathematical formulation is proposed for +designing the helical multipole topological insulators (MTIs) featured corner states and +helical energy fluxes, by constraining the corresponding topological invariants and +maximizing the width of band gap. Mechanical helical HOTIs with different symmetries +are obtained by this method and verified by full wave simulations. This design paradigm +can be further extended to design 3D TMs among different symmetry classes and space +groups, and different physical systems. +Keywords: Topological materials, Mechanical higher-order topological insulators, +Topology optimization, Symmetry indicators + + +1. Introduction +Metamaterials are well-known for its novel modulation of photons, phonons, and matter +waves in various applications. Enriching with the topological characteristics, it gives out a +new innovative material—topological materials (TMs), which are robust to various +defects1–3. Recently, the photonic and phononic TMs have attracted a great research interest +in engineering topological phenomena on the macroscopic scale2–12. Related topologically +protected states revealed some prospective applications2,3,12–23. +For example, the quantum spin/valley Hall topological insulators guide the energy flux in +a spin-locked transmission, which alternatively switches the one-way tunnel for the +propagated waves with immunity to defects6,8,9,24–28. That is an ideal way to improve the +effectivity of applications in opto-mechanics, current semiconductor and integrated circuit +industry13–22. For the higher-order topological materials (HOTIs), it is characterized by an +intensively localized topological phase within the lower dimensional domain such as the +edges and corners3,7,29–31. As a pioneering example of the HOTIs, the multipole topological +insulators (MTIs) can also provide a multipole moment enhanced topological phases, +where the bulk dipole is vanished3,30,31. Together with the pseudo-spin phenomenon, a +helical multipole-induced topological phase is inherited in the helical MTI32,33. Those +topological phases in the HOTIs are robust to various defects in manufacture, and show a +promising prospective in optical/acoustic subwavelength imaging, microelectronics, laser +aspects3,19,23,31,34–36. +Although the theoretical tight binding models have been developed, how to efficiently +design 3D unit cells with the demanded topological behaviors is still a crucial challenge. +Some typical options include tracing the featured degenerated states near the Dirac points, +restricting a special band structure from the band folded mechanism, keeping an obvious +Berry curvature (a quantity to topology), or realizing the maximal pseudo-spin energy +fluxes in the crossing waveguide37–44. To calculate the topological invariants, however, it +is very expensive to integrate the Berry curvature or its related terms in the whole Brillouin +zone. This issue would be more pronounced for the 3D continuous TMs, which generally +cover an infinite design parameter space and are more computationally expensive for +analysis and design optimization. +Luckily, the theoretical breakthrough in topological quantum chemistry gives new insight +into this bottleneck, from the fruitful meeting between chemistry and physics (in the real +and momentum space)45–48. The fundamental tool is calculating the real space orbits for +every band, with the aids of the elementary band representations (EBRs) or the symmetry +indicators (SIs). It gives out the topology in a simple linear function of the symmetry + +characters at some listed point29,30,46,48. Successful applications of the SI method include +the classification of TMs among the whole 230 space and 1651 magnetic groups, and the +discovery of thousands of TMs with many uncovered for the first time47,49,50. Most recently, +the catalogue of topological phononic materials becomes an attractive focus4,5,11. This +inspires us to efficiently identify the topological properties of the 3D mechanical unit cells +using the SI method. It is worth to note that, in the mechanical system, rigid body motions +corresponding to zero energy states yield the singularity at some high symmetry point. As +a result, the classic formulas derived in the quantum mechanics system need to be modified +at first. +Furthermore, how to describe the 3D unit cells is an essential factor for choosing design +optimization method39,41,43,44,51. This is because the topological invariants are discrete +variables, which cannot in general effectively handled by the gradient-based algorithms. +The Moving Morphable Component (MMC) method could describe 3D unit cells using +only a few explicit geometry parameters, and this makes it suitable to guarantee a +computationally tractable solution process of the inverse design formulation52,53. In general, +we summarize three characteristics of a desired optimization framework of the TMs as: (i) +effective to identify the topological characters for arbitrary unit cells; (ii) suitable to +topological materials in different classes and different physical systems; (iii) efficient to +execute the solution procedure. +In this study, we proposed a unified optimization framework for the 3D continuum TMs +by combining the SI method and the MMC method. In this design framework, the helical +MTIs with the helical edge and helical corner states can be effectively obtained by +simultaneously constraining the modified fractional corner charge and pseudo-spin +invariants. The proposed method thoughtfully modified the topological invariants for 3D +elastic HOTIs according to the singularity points with zero energy, and successfully +obtained optimized 3D helical MTIs in different symmetries, where the in-gap corner states +are derived from the quadrupole moment. The numerical simulations of the transmission +spectra and crossing waveguide applications validated the intensive corner energy and the +spin-locked energy flux. +The rest of the paper is organized as follows: in Section 2, the governing equations for +elastic waves are introduced. And then based on the description method of 3D elastic unit +cells using the MMC method and the specialized formulas of topological invariants in +Sections 3 and 4, an efficient deign paradigm is proposed for the mechanical helical MTIs. +Optimized designs with different symmetries are presented in Section 6 together with the +applications in a novel crossing elastic waveguide. Finally, some concluding remarks are +discussed in Section 7. + +2. The governing equations for elastic waves +In 3D elastic mechanics, the harmonic wave formulation is expressed as54 +(������������ + ������������)∇(∇ ⋅ ������������) + ������������∇2������������ = −������������2������������������������ +(1) +where ������������ is the angular frequency, ������������, ������������, and ������������ are the Lame’s parameters and mass density, +and ������������ = (������������, ������������, ������������)⊤ denotes the displacement field. +Since the considered phononic crystal is periodic, the displacement field satisfies the +translation condition as ������������(������������ + ������������) = ������������(������������) with ������������ denoting the primitive lattice vector. +According to the Bloch theorem, the harmonic elastic wave propagation can be determined +by the following discretized equations +������������������������ = −������������2������������������������ +������������(������������0 + ������������)|BC = ������������(������������0)|BC ������������i������������⋅������������ +(2) +Here, the matrices ������������ and ������������ refer to the stiffness and mass matrixes, and ������������ is the +eigenvector. +In order to handle the periodic constraints in the above eigenvalue problem, the standard +Lagrange multiplier method is adopted55. By defining a Lagrange multiplier ������������, Eq. (2) can +be reformulated as +������������� + ������������2������������ +������������f +������������ +������������ � ������������� +������������� = ������������ +(3) +in which the constraint matrices ������������ and ������������f are used to homogenize the eigenvalue problem. +Now, let us decompose the eigenvector with the solution ������������c as ������������ = ������������null������������c + ������������0, where +the matrix ������������null and vector ������������0 belong to the null space of ������������. An allowable value is ������������0 = +������������. After left multiplying Eq. (3) by ������������nullf +⊤ + (������������nullf is the null space of ������������f +⊤), we have the final +governing equation of the 3D elastic wave as +������������c������������c = −������������2������������c������������c +(4) +where the eliminated stiffness matrix is ������������c = ������������nullf +⊤ +������������������������null, and the eliminated mass matrix +is ������������c = ������������nullf +⊤ +������������������������null. Because Eq. (3) requires ������������f ������������ = ������������, it is needless to solve for it since +������������ is useless, and a practical choice is setting ������������f = ������������⊤, then ������������nullf = ������������null. + +3. Description of the 3D unit cells via the MMC method +Structural topology optimization has been successfully applied to inverse design various +topological metamaterials11,38–44,56. For designing the 3D elastic topological insulators, we +adopt the Moving Morphable Component (MMC) method40,42,52,53,56, which has the +advantages of the explicit geometry description and improved computational efficiency. +The building block in the MMC method is a set of morphable components, described by +some geometry parameters, such as the center coordinate, length, width, and thickness. As +a result, through updating those geometry parameters, every component can move, morph, +merge or disappear to form the optimized structure, as shown in Fig. 1. In this way, the +optimal parameter space will be deeply shrunk, and the solution efficiency will be +significantly improved. +In our work, each 3D component (the inclusion phase) is explicitly characterized by the +ellipsoid with a design variable vector ������������������������ = (������������0������������ +⊤ , ������������������������ +⊤, ������������������������ +⊤)⊤, i.e., the center coordinate +������������0 = (������������0, ������������0, ������������0), the length vector of semi-axes ������������ = (������������1, ������������2, ������������3), and the Euler rotation +angles ������������ = (������������, ������������, ������������), as shown in Fig. 1(a). In this manner, each MMC can be explicitly +determined by only 9 design variables. Furthermore, in a unit cell, the inclusion phase is +identified by the topology description function ������������������������(������������, ������������������������) for each component expressed +with its covered region Ω������������ as +������������������������(������������, ������������������������) = ‖������������′‖2 +2 − 1 = � +> 0 +if ������������ ∈ Ω������������ += 0 +if ������������ ∈ ������������Ω������������ +< 0 +else +(5) +In Eq. (5), the local coordinates are determined by the global coordinates ������������ and the rotation +matrix ������������(������������) as +������������������������ +′ = 1 +������������������������ +R������������������������(������������)������������������������� − ������������0������������� +(6) +According to the symmetry requirement of the unit cells, only the MMCs in a reduced +design domain need to be optimized and they can be transformed to the rest part. +Furthermore, all the inclusions represented by MMCs in the design domain can be +smoothed by the K-S aggregation technique 57 or the Boolean operation (adopted by this +work). + + + +(a) +(b) +Fig. 1. An illustration of a 3D unit cell described by the MMC method. (a) The geometric +description of the 3D components and (b) some representative configurations in the +optimization. +4. The SI induced topological invariants of the helical MTIs +The helical multipole topological insulators (MTIs)32,33, as a compound topological +material, should simultaneously hold the characters (or the topological invariants) from the +multipole moment and the pseudo-spin. In general, the calculation of topological invariants +is computationally expensive for the continuum unit cells. Nevertheless, recent work in +topological quantum chemistry reveals a rapid approach to identify topological invariants +through its symmetry indicators (SIs)4,5,46,49,50,58. Next, we will introduce the SI method +into the calculation of topological invariants, and then give out the method to design helical +MTIs. +Based on the SI method, for a spinless ������������������������=3,6-symmetric mechanic system with the time +reversal symmetry (TRS), we identify the eigenvalue Π������������ +(������������) of the ������������̂������������ rotational operator as +Π������������ +(������������) = ������������i2������������(������������−1)/������������ = �������������(Π)�������������̂�������������������������(Π)�, ������������ ∈ [1, ������������] +(7) +where ������������(Π) denotes the ������������-component of the displacement at the high-symmetry point Π. +The symbol #Π������������ +(������������) counts the number of Π������������ +(������������) below the target band gap. Compared to the +reference point Γ = (0,0), we define the SI at Π as �Π������������ +(������������)� = #Π������������ +(������������) − #Γ������������ +(������������). At the high- +symmetry points, it satisfies ������������̂������������������������ = ������������ + ������������ with ������������ denoting the reciprocal lattice vector. +For the ������������3-symmetric hexagonal unit cells, the high-symmetry points include Γ and K in +the ������������3 symmetry, while for the ������������6-symmetric hexagonal unit cells, they include Γ in the ������������6 +symmetry, K in the ������������3 symmetry, and M in the ������������2 symmetry, respectively. In this manner, + +AZ +Z'RY +X +L3 +L2Original +Morphing +Moving +Mergingthe topological classification is determined completely by the corresponding SIs, such as +the fractional corner charge and the pseudo-spin invariants in the following contents. +4.1 The fractional corner charge invariants +For the MTIs, the fractional corner charge ������������(������������) is an effective topological invariant to +determine the topological corner states29–31. For the 3D mechanical topological system with +the TRS and ������������3 symmetry, we propose the following formulas +������������(3) = �1 +3 �#K������������≠1 +(3) − 1 +2 #Γ(3)� mod 1� × ��#Γ(3) + 1�mod 2� +������������(6) = �1 +4 �#M1 +(2)� + 1 +6 �#K1 +(3)�� mod 2 +(8) +Here, the red terms in the function of ������������(3) are introduced to avoid the confused distinction +of the unpaired degenerate states6,9,10,25. As an alternative strategy, #Γ(3) counts the two- +order degeneracy at the Γ point, and #Γ(3)/2 identically equals the #Γ2 +(3) or #Γ3 +(3). The red +modulo term in ������������(3) guarantees the degenerate states to be in pairs (i.e., #Γ(3) is an even +number). For a visualization, the Fig. 2 shows that our modification successful avoids the +confused distinction of degenerate states when ������������ = 4. For more details, refer to Appendix +A. + + + +(a) +(b) +(c) +Fig. 2. The script of the modification procedure. (a) The singularity in mechanics and +degenerate states in a hexagonal unit cell. (b) and (c) The vector transformation of +displacement component (������������, ������������) and ������������. Here, only the degenerate states are calculated +under the ������������̂3 operator, and denoted as 1/2 = (������������ + ������������∗)/2 with its eigenvalues ������������ and its +conjugation ������������∗; the other states are calculated under the ������������̂2 operator. +4.2 The pseudo-spin invariants +For the photonic/phononic quantum spin/valley Hall effects, the protected chiral energy +flux can be well identified from the pseudo-spin vortex phenomenon and can be well + +Degenerate +-m=5 +1/2 +1/2 +States +m=4 +-1 +m=3 +-1 ++1 +个个 +00 +Singularity +ky个 +(u, v) +(-xo,yo) +(xo, yo) x +(-u, v)w +(xo, Yo. +(-xo,-yo) +x +wquantified by the Chen-spin or ������������2 invariants6,9,24,25. In our spinless 3D mechanical system +without spatial inversion symmetry, an alternative approach is adopted through tracing the +(broken) Dirac cone and band inversion9,10,25,38,40,42–44. +Practically, for the 3D ������������3 and ������������6 symmetric unit cells, the pseudo-spin invariants are +modified as10,29 +������������(3) = sgn�#K2 +(3) − #K3 +(3)� +������������(6) = sgn�#Γp +(6) − #Γd +(6) − 2� +(9) +Here, terms #Γp +(6) and #Γd +(6) count the orbits p and d under the ������������̂6 operator for the Γ point, +respectively. Notably, the subtracted red term is introduced to correct the #Γp +(6) due to the +singularity in the 3D elastic wave or the transverse electromagnetic wave (the singularity +has the same eigenvalue as the orbit p)5,59. For the 3D elastic wave, the singularity relates +to three translational motions, their displacement and vector transformation are shown in +Fig. 2. This modification is based on counting all occupied bands below the target band +gap, including the first three bands crossed through the singularity. Furthermore, the +counting idea keeps target bands isolated from other bands, as the SI method requires. For +more details, refer to Appendix B. +5. An efficient design paradigm of 3D mechanical helical MTIs +With the above topological invariants presented in Eqs. (8) and (9), we can now design the +helical MTIs using explicit topology optimization method. The corresponding optimization +formulation and solution process are introduced as follows. +5.1 Mathematical formulation +Combining the MMC-based description method and the modified formulas of topological +invariants in elastic medium, optimized 3D helical MTIs can be obtained by solving the +following mathematical formulation: +find +������������ = (������������1 +⊤, … , ������������������������ +⊤, ������������)⊤ +max +min(������������ref − max������������������������������������ +������������ , min������������������������������������ +������������+1 − ������������ref) +s. t. +������������c������������c = −������������2������������c������������c +�������������(������������), ������������(������������)� = �������������ref +(������������), ������������ref +(������������)� +������������min ≤ ������������ ≤ ������������max +(10) + +In the design variable vector, ������������������������ describes the ������������th component in the slab with a thickness +������������ (in the ������������-axis), as illustrated in Fig. 1. By denoting the eigenfrequency of the ������������th band +as ������������������������, the gap width between the ������������th and (������������ + 1)th bands is maximized with a target +mid-frequency ������������ref. The third equation in Eq. (10) is the governing equation for the 3D +elastic waves. Since the fractional corner charge and the pseudo-spin invariants +simultaneously contribute to the existence of helical corner states, the target topology +invariants �������������ref +(������������), ������������ref +(������������)� is introduced as a constraint. The last inequality persists the lower +and upper bounds of the design variable vector. +In principle, by updating the governing equation and target topological invariants, the +mathematical formulation in Eq. (10) can be applied for designing TMs among different +symmetry classes, and different physical systems. In this work, we focused on the inverse +design of 3D helical MTIs in elastic medium with ������������3 and ������������6 symmetries. + +Fig. 3. The scheme of optimization for the helical MTIs. +5.2 Solution process +Since the topological invariants are quantized, gradient-based optimization algorithms +would be ineffective for solving Eq. (10). Thanks to the advantage of a fewer number of +design variables in the MMC method, the genetic algorithm (GA) is adopted here and the +settings are presented in Appendix C. To be specific, the flowchart for the rational design +of helical MTIs is shown in Fig. 3, and its solution process is summarized as follows: +• +STEP 1: Initialization of the MMC method and the GA solver. +The gap label ������������, the mid-frequency ������������ref, and the nonzero topological invariants +�������������ref +(������������), ������������ref +(������������)� are initialized first through a trial process, starting from ������������ = 3; +• +STEP 2: Optimal design of the first MTI. + +Init. +STEP1 +Band Order +200 Random +FEA +Count Cases +If No Case +(Q(n), z(n)) +Q(n) ± 0&z(n) ±0? +m=3 +Unit Cells +(COMSOL) +N +Y +m=m+1 + STEP2 +1st MTI +GA +Generate +Set Valid Para. +'ref +FEA +Calculate +Unit Cell +Conv.? +fref, m +(COMSOL) +Q(n), z(n) +MTI Partner +Y +MMC +N +STEP3 +0, +ref +EndWith the parameters determined in STEP 1, solve the mathematical programming +Eq. (10) to obtain the first optimized MTI with the predefined invariant +�������������ref +(������������), ������������ref +(������������)� and mid-frequency ������������ref; +• +STEP 3: Optimal design of the MTI partner (if necessary). +With the desired topological invariants setting as �0, −������������ref +(������������)� and the other +parameters the same as STEP 2, solve Eq. (10) to obtain the optimized MTI partner +with the inverted pseudo-spin effect. + + +(a) +(b) +Fig. 4. The statistical charts (b) of different states at the Γ point (the partitions of p and d +would be decomposed into the boxed partitions without the modification in Eq. (8)) and +(c) of different TMs. Hint: s.p. —singularity point. +To illustrate the effectiveness of the proposed design framework, the statistical charts of +the states at the Γ point (6000 samples) and of different TMs (8000 samples) are illustrated +in Fig. 4(a) and 4(b), respectively. It can be found that, using Eq. (8), the states p and d are +successfully identified, and they take about 21.6% and 26.2% of the whole set as shown in +Fig. 4(a). Without the modification in Eq. (8), however, such states would be decomposed +to Γ2 +(3) state (22.1%), Γ3 +(3) (22.1%), and an unpaired set of state (3.5%). This unpaired set +would further make troubles for the calculation of the fractional corner charge invariant. In +Fig. 4(b), 6.6% of 8000 samples are four typical TMs (quantum valley/spin Hall +topological insulators (QVTIs/QSTIs), MTIs and helical MTIs), while the desired helical +MTIs only account for 4.0%. This validates the necessity of developing inverse design +paradigm for the helical MTIs. + +s.p. +31.1% +d +11.1% +21.6% +9.5% +0.6% +26.2% +3 +Other +22.1% +Unpaired +3.5% +22.1% +p +** +（3） +S +pOther +93.4% +0.7% +QVTI +2.4% +4.0% +Helical MTI +QSTI +0.2% +MTI6. Applications of the MMC-based design framework for 3D helical MTIs +in elastic medium +In the present work, the helical MTIs are periodic in the in-plane direction and made of the +basic medium EP and scattering medium Fe (materials parameters and more setup details +are referred to Appendix C). +6.1 Optimal design of ������������3-symmetric mechanical helical MTIs +Under the optimization framework, the optimized ������������3-symmetric helical MTIs are obtained +in Fig. 5(a). And there is a normalized bulk band gap at 0.741-1.069 between the 6th and +7th bands (colored in grey in Fig. 5(a)). The symmetry behaviors of the high-symmetry +points are shown in the Fig. 5(b). There are three broken degenerate states (from the Dirac +cone) at the K point below the target bandgap, while only the third one is unpaired, and +implies the possibility of a pseudo-spin vortex. The phase field of this unpaired state is also +inserted in Fig. 5(a). The corner charge and the pseudo-spin invariants are (2/3,1). +In order to realize the band inversion, the corresponding MTI partner can be easily +constructed by applying the spatial reversal operation, or in other words, its invariants are +set as (0, −1). An opposite pair of ������������(3) invariants would produce a helical topological state +from the bulk-boundary correspondence. Moreover, a pair of zero and nonzero fractional +corner charges reveal the appearance of corner states18, as shown in Fig. 6(a) around the +normalized frequencies of 0.894 and 0.966. The latter localized corner mode is displayed +in the inserted diagram. + + + +(a) + +(b) +Fig. 5. The optimized ������������3-symmetric mechanical helical MTIs. (a) The band structure +inserted with the unit cell and the phase field of the unpaired state. (b) The symmetry- + +1.2 +(2A/2πC) +1 +0.8 +0.6 +Freq +0.4 +0.2 +0 +T +M +K +Singularity(2) +Band +-(3) +b +b +1 +-1 +3 +wt +2 +-1 ++1 +3 ++1 ++1 +m +4 ++1 ++1 +5 ++1 ++1 +6 ++1 ++1 +mbehavior-table, in which the degenerate states are tagged as ������������ = ������������i2������������/3 for the K point, +while for the Γ point they are tagged as ������������. + + +(a) +(b) + + +(c) +(d) +Fig. 6. Simulation results of the optimized ������������3-symmetric helical MTIs. (a) The eigenvalue +spectrum (points are colored according to the corner energy intensity) and the energy field +of a corner state. (b) The transmission spectra from the probes in bulk, edge, and corner +area (colored in legend). (c) Energy fields tagged in (b) at the normalized frequencies of +0.793, 0.879, 0.966, and 1.121. (d) The energy flux and their zoom-in views of helical edge +states at the normalized frequency of 0.862. +Besides, the full-wave transmission is presented in Fig. 6(b), where energy is captured from +different regions around the outer bulk, the interface edges, and the interface corners. A +spin-down (clockwise) helical source is excited near the supercell’s center, shown as the +star in Fig. 6(c). The transmission reveals some edge energy peaks around the normalized +frequencies of 0.793 and 1.001, and some intensively localized corner states around the +normalized frequencies of 0.879 and 0.966. For a clear visualization, the corresponding + +1.02 +0.98 +(S2A/2 TC) +0.94 +0.9 +Freq ( +0.86 +0.82 +Index(dB) +3 +Transmission ( +2 +-60 +Corner +-120 +Edge +Bulk +0.7 +0.8 +0.9 +1 +1.1 +Freg (2 A/2πc)1 +2 +3Spin-Down +Spin-Upbulk, edge, and corner energy fields are displayed in Fig. 6(c). In contrast to the edge gap +around the normalized frequency range of 0.872-1.001 (colored in light-green), those in- +gap corner states are derived from the quadrupole moment. +For the verification of the helical behavior, a biased helical source off the center is excited +additionally, as shown in Fig. 6(d). The inserted arrow diagrams displayed the energy flux +near their corners and edges. We found that these two supercells had significant opposite +responses under different exciting helical sources (spin-up or spin-down). All their corners +held a clear energy vortex (clockwise or anticlockwise). Their edge energy fluxes are +locked by their exciting sources and only could flow forward or backward. +6.2 Optimal design of ������������6-symmetric mechanical helical MTIs +For the optimized ������������6-symmetric MTI pairs, as illustrated by the band structures shown in +Fig. 7(a), band gaps are observed in the normalized frequency ranges of 1.344-1.489 (up) +and 1.332-1.450 (below), respectively. Below the gap, there are four degenerate states +found at the Γ points for both cases, but only the last two states formed an unpaired double +Dirac cone, which features the pseudo-spin vortex. The phase fields of these unpaired states +are inserted in Fig. 7(a), from which the band inversion is clearly displayed. The symmetry +behaviors in Fig. 7(a) show that the corner charges and the modified pseudo-spin invariants +are (1/2,1)and (0, −1), respectively. Specifically, the pair of opposite ������������(6) invariants lock +the energy flux by the pseudo-spin phenomenon. In contrast, the pair of zero and nonzero +corner charges predict the topological corner states (according to the vanished bulk +polarization in ������������6-symmetric unit cells, these nonzero corner charges are only derived from +the quadrupole moment30). By combining these two topological characters, the topological +corner state will also have pseudo-spin behaviors and present as a helical corner state. For +a verification of this helical corner state, the eigenvalue spectrum of the supercell’s +simulation is shown in Fig. 7(b), and its energy density distribution, at the normalized +frequency of 1.426, is highly localized at corners. +6.3 Applications of the optimized helical MTIs in a crossing waveguide +As an application of the helical MTIs, a crossing waveguide (a single layer) composed of +the two optimized ������������3-symmetric helical TMIs in Subsection 6.1 (colored blue/yellow for +the original/inversed TMIs mentioned above) is developed in Fig. 8(a). Since the additional +pseudo-spin freedom locks the energy flux in the waveguide, two opposite transmissions +would be discovered when we sequentially excited the Port 1 and Port 2. By gradually +modulating the exciting frequency, the energy will spread through the center wall and +induce the output corner states. + + + +(a) +(b) +Fig. 7. Simulation results of the optimized ������������6-symmetric helical MTIs. (a) The band +structures and the symmetry-behavior-tables of the optimized MTI pairs. The inserted +diagrams include the optimized unit cells and the ������������-directional displacement fields of the +unpaired states. In those tables, the degenerate states for the K point are tagged as ������������ = +������������i2������������/3, while for the Γ point they are tagged as ������������. (b) The eigenvalue spectrum and the +inserted energy field of the corner state (points are colored according to the corner energy +intensity). +The simulations in the normalized frequency range of 0.7-1.1 are processed to test the +performance of the waveguide, as shown in Fig. 8(b). It is clear that when Port 1 is excited +at the normalized frequency of 0.776, the energy only transmits to Port 2 and Port 3, yet it +only transmits to Port 1 and Port 4 from Port 2. This phenomenon reveals the locked helical +energy flux as expected. At the normalized frequency of 0.897, the corner states in the +lower half of the waveguide are excited in both cases. Here, these states stay in the band +gap of the edge states (i.e., 0.872-1.001, refer to Appendix D for more details), and their +energy only localizes at corners, and no edge states exist. +To test the working range of the one-way transmission in this waveguide, we distinguished +the energy from the different ports (Port 3 or Port 4), as shown in Figs. 8(c) and 8(d). Here +the light-green area and yellow-solid points refer to the band gap of the edge states and the +states in Fig. 8(b). In this much wider frequency range of 0.749-0.861, the average +difference between both ports is higher than 10dB. When we reverse the exciting port, the +output port, which has a higher transmission, is also turned, as shown in Fig. 8(d). In this +frequency range, the first two edge bands, as illustrated in Appendix D, will be excited. + +1.6 +Band +a +(2A/2πc) +1 +-1 ++1 +m +1.2 +2 +-1 +-1 +wt +1 +3 ++1 ++1 ++1 ++1 +0.8 +Freq ( +4 ++1 ++1 ++1 +at +0.6 +5 ++1 +1 +m +0.4 +6 +-1 +-1 ++1 +0.2 +7 +0 +M +K +Singularity +1.6 +r(2) r(3) m(2) k(3) +Band +a +1.4 +b +9 +b +Freq (2A/2πc) +1 +w) +1 +2 +-1 ++1 +wt +1 +3 ++1 ++1 ++1 ++1 +0.8 +4 ++1 +wt +0.6 +5 ++1 ++1 ++1 +m +0.4 +6 ++1 +-1 ++1 +0.2 +7 ++1 ++1 +0 +M +K +Singularity1.47 +(S2A/2 TC) +1.44 +1.41 +Freq ( +1.38 +1.35 +IndexHence, these one-way transmission results from the helical edge states. Moreover, the +corner states tagged with the number 2 and 4 are in the gap of the edge state but display an +apparent energy concentration from the exciting source. + + +(a) +(b) + + +(c) +(d) +Fig. 8. The crossing waveguide made of the optimized ������������3-symmetric helical MTIs. (a) The +sketches of the waveguide and its energy fluxes in different exciting cases (the exciting +line sources are tagged as stars). (b) The energy fields at the normalized frequencies of +0.776 and 0.897. The measured transmission of Ports 3 and 4 (c) from the exciting Port 1 +or (d) from the exciting Port 2. Here the band gap of the edge states (light-green region) +and the typical states (yellow-solid points) are colored. +7. Concluding remarks +In this work, we proposed an optimization framework for the inverse design of multi- +functional topological materials in the 3D continuous medium. By carefully manifesting +the degenerate states and singularity points in the elastic waves, the 3D helical multipole + +Port1 +Port 2 +Port 4 +Port 3 +C +Q2 +32 +0 +(dB) +-50 +Transmission ( +-100 +-150 +-200 +Port 3 +-250 +Port 4 +0.7 +0.8 +0.9 +1 +1.1 +Freq (2 A/2πc)3 +Transmission (dB) +-50 +-100 +-150 +-200 +Port 3 +-250 +-Port 4 +0.7 +0.8 +0.9 +1 +1.1 +Freq (2 A/2πc)topological insulators are well-classified by the fractional corner charge and the pseudo- +spin invariants. With the explicit topology optimization and the symmetry indicator +methods, the proposed design paradigm has the advantages of (1) rapid classification of +the 3D topological materials and (2) efficient optimization of the 3D continuum unit cells +in a smaller explicit parameter space. This framework shows outstanding suitability to the +3D topological system and can also be generalized to other symmetry classes and space +groups. Besides, building up a topological materials library in continuous medium would +be an exciting topic for further research. +Methods +The solid mechanic simulation is performed in the commercial software COMSOL +MULTIPHYSICS. The default open surfaces are set as free boundaries. The Bloch theorem +is numerical expressed by the Floquet periodic boundaries. In common, the energy in solid +mechanics is consistent in distribution as the amplitude of total displacement ‖(������������, ������������, ������������)‖2 +2. +Acknowledgements +The financial supports from the National Natural Science Foundation (11821202, +11732004, 12002073, 12002077, 12272075, 11922204), the National Key Research and +Development Plan (2020YFB1709401), Dalian Talent Innovation Program (2020RQ099), +the Fundamental Research Funds for the Central Universities (DUT20RC(3)020, +DUT21RC(3)076), and 111 Project (B14013) are gratefully acknowledged. +Author contributions +X. G. and Z. D. conceived the idea and initiated the project. J. L. and Z. D. established the +theory. J. L and X. D. performed the numerical calculations and simulations. All the other +authors contributed to the discussions of the results and the manuscript preparation. +Declaration of competing interest +There are no conflicts to declare. +Data availability +Data will be made available on request. + + +Appendix +Appendix A: Modification of the fractional corner charge invariant +According to the results in literature30, the fractional corner charge invariant of the ������������3- +symmetric hexagonal unit cells is +������������������������ +′(3) = 1 +3 �K������������≠1 +(3) � mod 1 +(A. 1) +where subscript ������������ equals 2 or 3 depending on the symmetry of the constructed supercell. +Due to the TRS and ������������3 symmetry, some two-order degenerate states are protected at the Γ +point, such as states from the linear combination of the Γ2 +(3) and Γ3 +(3), and they are +computationally expensive to identify clearly, especially for the unpaired degenerate +states6,9,10,25,28. Instead, we termed the invariant with the number of the two-order +degenerate states #Γ(3). To be specific, the topological character of the ������������3-symmetric +hexagonal unit cell is given by +������������‾(3) = �#Γ(3), #K2 +(3), #K3 +(3)� +(A. 2) +Considering Eq. (A.2), the modified fractional corner charge invariants and the symmetry +behaviors are listed in Table A.1 for some possible cases. +Table A.1. The symmetry behaviors of the ������������3-symmetric unit cells with TRS +(for the fractional corner charge invariants) +������������������������=2 +(3) +������������������������=3 +(3) +#K2 +(3) +#K3 +(3) +#Γ(3) +#Γ2 +(3) +#Γ2 +(3) +1/3 +0 +1 +0 +0 +0 +0 +0 +1/3 +0 +1 +0 +0 +0 +0 +2/3 +1 +0 +2 +1 +1 +0 +0 +1 +1 +2 +1 +1 +0 +0 +1 +0 +1 +1 +0 +0 +0 +0 +1 +1 +0 +1 +0 +0 +1 +0 +1 +0 +1 +0 +0 +0 +1 +1 +1 +0 +In Table A.1, the red colored invariants ������������������������ +(3) are modified from Eq. (A.2). This +modification is derived from the fact that the degenerate states #Γ(3) always appear in a + +pair; or not, it is gapless 6,9,10,25,28. For the unpaired case, it is ambiguous to be tagged as +Γ2 +(3) or Γ3 +(3), hence the invariants ������������������������ +(3) in the last four cases cannot be solely identified by +the ������������‾(3) in Eq. (A.2). Therefore, when #Γ(3) is odd, the corresponding fractional corner +charge should be modified into zero with gapless band reality. When #Γ(3) is even, number +#Γ(3) can be equivalently divided as: #Γ2 +(3) = #Γ3 +(3) = #Γ(3)/2. +Thus, the modified fractional corner charge invariant gives +������������(3) = �1 +3 �#K������������≠1 +(3) − 1 +2 #Γ(3)� mod 1� × ��#Γ(3) + 1�mod 2� +(A. 3) +where the red module term aims to avoid the unpaired two-order degenerate states. +Appendix B: Modification of the pseudo-spin invariant +For the ������������3 or ������������6-symmetric unit cells with the TRS, an alternative approach to get the +pseudo-spin invariants is to trace the broken (double) Dirac cones at the K or Γ point +9,10,25,38,40,42–44. Thus, their topological characters are given by +������������‾(3) = �#K2 +(3), #K3 +(3)� +������������‾(6) = �#Γ1 +(2), #Γ2 +(2), #Γ(3)� +(B. 1) +Here, #Γ(3) counts the two-order degenerate states of the Γ point in the ������������3 operator. The +modified pseudo-spin invariants and their symmetry behaviors are listed in Table B.1 and +Table B.2 for some possible cases. +Different as scaling a function5,59, the operation of a symmetry operator ������������� on a vector +function ������������(������������) transforms as �������������������������(������������) = �������������������������(�������������−1������������), where ������������� is the rotational operator in �������������. +For the present symmetry groups (������������3 or ������������6) in our paper, all group elements behave as a +rotation around ������������ -axis, and the transformation can be simplified as �������������������������(������������) = +�������������������������T(�������������−1������������) + �������������������������L(�������������−1������������) , +where ������������ = ������������T + ������������L = (������������, ������������, 0)⊤ + (0,0, ������������)⊤ is +the +displacement vector in mechanics. This decomposed equation implies ������������T and ������������L hold the +same symmetry, except for the singular cases with displacement component ������������T = ������������ or +������������L = ������������. + + + + +Table B.1. The symmetry behaviors of the ������������3-symmetric unit cells with TRS +(for the pseudo-spin invariants) +������������(3) +#K2 +(3) +#K3 +(3) +1 +1 +0 +1 +2 +0 +-1 +0 +1 +0 +1 +1 +Table B.2. The symmetry behaviors of the ������������6-symmetric unit cells with TRS +(for the pseudo-spin invariants) +������������(6) +#Γ1 +(2) +#Γ2 +(2) +#Γ(3) +Orbits +-1 +2 +0 +2 +2d +1 +0 +2 +2 +2p +0 +2 +2 +4 +2p + 2d +0 +1 +0 +0 +1s +0 +0 +1 +0 +1f +0 +1 +2 +2 +2 s.p. +In the original SI theory29,30, the occupied bands counted in the SI method should be +isolated from others, and an alternative approach is to count all bands below the target band +gap. For the photonic and phononic systems, however, the first two or three bands always +converge to plane waves when |������������| → 0, where transverse modes produce two singularities +with ������������L = ������������5,59. In Table B.2, the red-colored data reveal the symmetry behaviors of the +first three bands that always cross through the singularities around the zero energy. Hence, +we defined the modified pseudo-spin invariants to overcount those singularities as +������������(3) = sgn�#K2 +(3) − #K3 +(3)� +������������(6) = sgn�#Γp +(6) − #Γd +(6) − 2� +(B. 2) +where the red term is the modification from the singularities. For the case of photonics, the +Eq. (B.2) should be further modified as the work5. And the terms #Γp +(6) and #Γd +(6) count +the p and d states at the Γ point. + +Table C.1. Some typical optimized unit cells for the ������������3 and the ������������6-symmetric TMs. +(Here, the red data refers to the examples presented in our paper) +������������������������ +������������{������������} +������������{������������} +������������{deg} +������������{������������} �������������(������������), ������������(������������)� ������������max +(������������) +∼ ������������min +(������������+1) +������������3 (0.7217,0.95,0.0275) +(0.3382,0.5073,0.1691) +(108,18,126) +0.55 +(2/3, 1) +0.741~1.069 + +(0.4041,1.00,0.0900) +(0.3082,0.3082,0.2568) +(144,72,90) +0.45 +(2/3, -1) +0.772~1.000 +������������6 (0.9238,0.90,0.3250) +(0.3682,0.2455,0.3068) +(90,108,0) +0.65 +(1/2, 1) +1.344~1.489 + +(0.8776,0.90,0.1750) +(0.2155,0.3232,0.2693) +(90, 144, 72) +0.50 +(0, -1) +1.332~1.450 + +(0.9584,0.80,0.1800) +(0.1766,0.4121,0.2355) +(126,126,72) +0.60 +(1/2, 1) +1.254~1.319 + +(0.8603,0.95,0.1575) +(0.2055,0.3596,0.3082) +(108,144,90) +0.45 +(0, -1) +1.249~1.344 + + +Fig. C.1. Some optimized unit cells with the ������������3 and ������������6 symmetries. The order refers to +the row number of Table C.1. +Appendix C: The setup of the optimization and the GA solver +For the parameters of material and optimization solver in our paper, the setup gives: the +basic medium is epoxy (EP) with the elastic modulus ������������0 = 4.35GPa, the Poisson’s ratio +������������0 = 0.37, and the mass density ������������0 = 1180kg ⋅ m−3. The scattering medium is steel (Fe) +with the elastic modulus ������������ = 200GPa, the Poisson’s ratio ������������ = 0.2, and the mass density +������������ = 7800kg ⋅ m−3. The genetic algorithm (GA) solver is set as: the population size of 100, +the crossover fraction of 0.9, the migration fraction of 0.3, the elite size of 5, the +objectivation tolerance of 1e-5, the stall generation limit of 15. The lattice constant is ������������ = +|������������| = 1m. + +3 +5 +2Table C.1 and Fig. C.1 list some typical optimized unit cells. The row number in Table +C.1 is consistent with the order of unit cells in Fig. C.1. For the examples in the main text, +we set their optimization procedure as +• +For the ������������3-symmetric helical MTIs, the broken Dirac cone appears at the K point. +The first unit cell is optimized with setting ������������ = 6, nonzero topological invariants +(2/3,1) and no specific ������������ref, which will auto-update as the mid-frequency of the +target gap. +• +For the ������������6-symmetric helical MTIs, the broken Dirac cone appears at the Γ point. +The first unit cell is optimized with setting ������������ = 7, �������������ref +(������������), ������������ref +(������������)� = (1/2,1), and +������������ref = 1.4. Then the MTI partner is optimized with setting ������������ = 7, �������������ref +(������������), ������������ref +(������������)� = +(0, −1), and ������������ref = 1.4. +Appendix D: The supercell’s setup for the edge and the corner states +The setups for two example supercells are detailed as +• +For the ������������3-symmetric unit cells in the main text, the script of the truncated supercell +is shown in Fig. D.1, where it provides an approach to adjust the frequency of the +edge states. This truncation does not break the crystalline symmetry, and the +topological edge states would not vanish. In Fig. D.1(a) and D.1(b), the truncation +is set as ������������ = 1/4 × 2������������/√3 , and the edge gap is between 0.872-1.001. In Fig. D.1(c) +and (d), the eigenvalue spectrum and the crossing waveguide are simulated with the +truncation ������������ = 1 × 2������������/√3. The light-blue areas in Fig. D.1 refer to the supercell’s +structures in the main text. +• +For the ������������6-symmetric unit cell, the inner interface between the unit cell pairs can +be alternatively truncated as Fig. D.2. In Fig. D.2 (a) and D.2(b), the gap of the +edge states is found between 1.398-1.425 with the truncation ������������ = 1/2 × ������������. The +supercell’s script for the eigenvalue spectrum is displayed in Fig. D.2(c) with the +truncation ������������ = 1 × 2������������/√3 . In Fig. D.2(c), except for the inner hexagonal interface, +six base medium cylinders with a diameter of 0.4������������ are added to adjust the +supercell’s corners. The light-blue areas in Fig. D.2 refer to the supercell’s +structures in the main text. + + + +(a) +(b) + + +(c) +(d) +Fig. D.1. The supercell’s scripts of the ������������3-symmetric unit cell. 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Phys Rev Lett 121, +263903 (2018). + + diff --git a/79E1T4oBgHgl3EQfnQTu/content/tmp_files/load_file.txt b/79E1T4oBgHgl3EQfnQTu/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..02468ad1e57be0858f6e6daf4c77004918834f22 --- /dev/null +++ b/79E1T4oBgHgl3EQfnQTu/content/tmp_files/load_file.txt @@ -0,0 +1,970 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf,len=969 +page_content='Efficient Design of Helical Higher-Order Topological Insulators in 3D Elastic Medium Jiachen Luo1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' Zongliang Du1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='2*,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' Hui Chen3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' Xianggui Ding1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' Chang Liu1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' Weisheng Zhang1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' Xu Guo1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='2* 1State Key Laboratory of Structural Analysis for Industrial Equipment,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' Department of Engineering Mechanics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' Dalian University of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' Dalian,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' 116023,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' China 2Ningbo Institute of Dalian University of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' Ningbo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' 315016,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' China 3Piezoelectric Device Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' School of Mechanical Engineering and Mechanics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' Ningbo University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' Ningbo 315211,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' China E-mail: zldu@dlut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='cn (ZD);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' guoxu@dlut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='cn (XG) Abstract Topological materials (TMs) are well-known for their topological protected properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' Phononic system has the advantage of direct observation and engineering of topological phenomena on the macroscopic scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' For the inverse design of 3D TMs in continuum medium, however, it would be extremely difficult to classify the topological properties, tackle the computational complexity, and search solutions in an infinite parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' This work proposed a systematic design framework for the 3D mechanical higher-order topological insulators (HOTIs) by combining the symmetry indicators (SI) method and the moving morphable components (MMC) method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' The 3D unit cells are described by the MMC method with only tens of design variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' By evaluating the inherent singularity properties in the 3D mechanical system, the classic formulas of topological invariants are modified accordingly for elastic waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' Then a mathematical formulation is proposed for designing the helical multipole topological insulators (MTIs) featured corner states and helical energy fluxes, by constraining the corresponding topological invariants and maximizing the width of band gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' Mechanical helical HOTIs with different symmetries are obtained by this method and verified by full wave simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' This design paradigm can be further extended to design 3D TMs among different symmetry classes and space groups, and different physical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' Keywords: Topological materials, Mechanical higher-order topological insulators, Topology optimization, Symmetry indicators 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' Introduction Metamaterials are well-known for its novel modulation of photons, phonons, and matter waves in various applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' Enriching with the topological characteristics, it gives out a new innovative material—topological materials (TMs), which are robust to various defects1–3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' Recently, the photonic and phononic TMs have attracted a great research interest in engineering topological phenomena on the macroscopic scale2–12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' Related topologically protected states revealed some prospective applications2,3,12–23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' For example, the quantum spin/valley Hall topological insulators guide the energy flux in a spin-locked transmission, which alternatively switches the one-way tunnel for the propagated waves with immunity to defects6,8,9,24–28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' That is an ideal way to improve the effectivity of applications in opto-mechanics, current semiconductor and integrated circuit industry13–22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' For the higher-order topological materials (HOTIs), it is characterized by an intensively localized topological phase within the lower dimensional domain such as the edges and corners3,7,29–31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' As a pioneering example of the HOTIs, the multipole topological insulators (MTIs) can also provide a multipole moment enhanced topological phases, where the bulk dipole is vanished3,30,31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' Together with the pseudo-spin phenomenon, a helical multipole-induced topological phase is inherited in the helical MTI32,33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' Those topological phases in the HOTIs are robust to various defects in manufacture, and show a promising prospective in optical/acoustic subwavelength imaging, microelectronics, laser aspects3,19,23,31,34–36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' Although the theoretical tight binding models have been developed, how to efficiently design 3D unit cells with the demanded topological behaviors is still a crucial challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' Some typical options include tracing the featured degenerated states near the Dirac points, restricting a special band structure from the band folded mechanism, keeping an obvious Berry curvature (a quantity to topology), or realizing the maximal pseudo-spin energy fluxes in the crossing waveguide37–44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' To calculate the topological invariants, however, it is very expensive to integrate the Berry curvature or its related terms in the whole Brillouin zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' This issue would be more pronounced for the 3D continuous TMs, which generally cover an infinite design parameter space and are more computationally expensive for analysis and design optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' Luckily, the theoretical breakthrough in topological quantum chemistry gives new insight into this bottleneck, from the fruitful meeting between chemistry and physics (in the real and momentum space)45–48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' The fundamental tool is calculating the real space orbits for every band, with the aids of the elementary band representations (EBRs) or the symmetry indicators (SIs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' It gives out the topology in a simple linear function of the symmetry characters at some listed point29,30,46,48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' Successful applications of the SI method include the classification of TMs among the whole 230 space and 1651 magnetic groups, and the discovery of thousands of TMs with many uncovered for the first time47,49,50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' Most recently, the catalogue of topological phononic materials becomes an attractive focus4,5,11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' This inspires us to efficiently identify the topological properties of the 3D mechanical unit cells using the SI method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' It is worth to note that, in the mechanical system, rigid body motions corresponding to zero energy states yield the singularity at some high symmetry point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' As a result, the classic formulas derived in the quantum mechanics system need to be modified at first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' Furthermore, how to describe the 3D unit cells is an essential factor for choosing design optimization method39,41,43,44,51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' This is because the topological invariants are discrete variables, which cannot in general effectively handled by the gradient-based algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' The Moving Morphable Component (MMC) method could describe 3D unit cells using only a few explicit geometry parameters, and this makes it suitable to guarantee a computationally tractable solution process of the inverse design formulation52,53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' In general, we summarize three characteristics of a desired optimization framework of the TMs as: (i) effective to identify the topological characters for arbitrary unit cells;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' (ii) suitable to topological materials in different classes and different physical systems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' (iii) efficient to execute the solution procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' In this study, we proposed a unified optimization framework for the 3D continuum TMs by combining the SI method and the MMC method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' In this design framework, the helical MTIs with the helical edge and helical corner states can be effectively obtained by simultaneously constraining the modified fractional corner charge and pseudo-spin invariants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' The proposed method thoughtfully modified the topological invariants for 3D elastic HOTIs according to the singularity points with zero energy, and successfully obtained optimized 3D helical MTIs in different symmetries, where the in-gap corner states are derived from the quadrupole moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' The numerical simulations of the transmission spectra and crossing waveguide applications validated the intensive corner energy and the spin-locked energy flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' The rest of the paper is organized as follows: in Section 2, the governing equations for elastic waves are introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' And then based on the description method of 3D elastic unit cells using the MMC method and the specialized formulas of topological invariants in Sections 3 and 4, an efficient deign paradigm is proposed for the mechanical helical MTIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' Optimized designs with different symmetries are presented in Section 6 together with the applications in a novel crossing elastic waveguide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' Finally, some concluding remarks are discussed in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' The governing equations for elastic waves In 3D elastic mechanics, the harmonic wave formulation is expressed as54 (������������ + ������������)∇(∇ ⋅ ������������) + ������������∇2������������ = −������������2������������������������ (1) where ������������ is the angular frequency, ������������, ������������, and ������������ are the Lame’s parameters and mass density, and ������������ = (������������, ������������, ������������)⊤ denotes the displacement field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' Since the considered phononic crystal is periodic, the displacement field satisfies the translation condition as ������������(������������ + ������������) = ������������(������������) with ������������ denoting the primitive lattice vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' According to the Bloch theorem, the harmonic elastic wave propagation can be determined by the following discretized equations ������������������������ = −������������2������������������������ ������������(������������0 + ������������)|BC = ������������(������������0)|BC ������������i������������⋅������������ (2) Here, the matrices ������������ and ������������ refer to the stiffness and mass matrixes, and ������������ is the eigenvector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' In order to handle the periodic constraints in the above eigenvalue problem, the standard Lagrange multiplier method is adopted55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' By defining a Lagrange multiplier ������������, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' (2) can be reformulated as ������������� + ������������2������������ ������������f ������������ ������������ � ������������� ������������� = ������������ (3) in which the constraint matrices ������������ and ������������f are used to homogenize the eigenvalue problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' Now, let us decompose the eigenvector with the solution ������������c as ������������ = ������������null������������c + ������������0, where the matrix ������������null and vector ������������0 belong to the null space of ������������.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' An allowable value is ������������0 = ������������.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' After left multiplying Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' (3) by ������������nullf ⊤ (������������nullf is the null space of ������������f ⊤), we have the final governing equation of the 3D elastic wave as ������������c������������c = −������������2������������c������������c (4) where the eliminated stiffness matrix is ������������c = ������������nullf ⊤ ������������������������null, and the eliminated mass matrix is ������������c = ������������nullf ⊤ ������������������������null.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' Because Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' (3) requires ������������f ������������ = ������������, it is needless to solve for it since ������������ is useless, and a practical choice is setting ������������f = ������������⊤, then ������������nullf = ������������null.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' Description of the 3D unit cells via the MMC method Structural topology optimization has been successfully applied to inverse design various topological metamaterials11,38–44,56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' For designing the 3D elastic topological insulators, we adopt the Moving Morphable Component (MMC) method40,42,52,53,56, which has the advantages of the explicit geometry description and improved computational efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' The building block in the MMC method is a set of morphable components, described by some geometry parameters, such as the center coordinate, length, width, and thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' As a result, through updating those geometry parameters, every component can move, morph, merge or disappear to form the optimized structure, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' In this way, the optimal parameter space will be deeply shrunk, and the solution efficiency will be significantly improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' In our work, each 3D component (the inclusion phase) is explicitly characterized by the ellipsoid with a design variable vector ������������������������ = (������������0������������ ⊤ , ������������������������ ⊤, ������������������������ ⊤)⊤, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=', the center coordinate ������������0 = (������������0, ������������0, ������������0), the length vector of semi-axes ������������ = (������������1, ������������2, ������������3), and the Euler rotation angles ������������ = (������������, ������������, ������������), as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' In this manner, each MMC can be explicitly determined by only 9 design variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' Furthermore, in a unit cell, the inclusion phase is identified by the topology description function ������������������������(������������, ������������������������) for each component expressed with its covered region Ω������������ as ������������������������(������������, ������������������������) = ‖������������′‖2 2 − 1 = � > 0 if ������������ ∈ Ω������������ = 0 if ������������ ∈ ������������Ω������������ < 0 else (5) In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' (5), the local coordinates are determined by the global coordinates ������������ and the rotation matrix ������������(������������) as ������������������������ ′ = 1 ������������������������ R������������������������(������������)������������������������� − ������������0������������� (6) According to the symmetry requirement of the unit cells, only the MMCs in a reduced design domain need to be optimized and they can be transformed to the rest part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' Furthermore, all the inclusions represented by MMCs in the design domain can be smoothed by the K-S aggregation technique 57 or the Boolean operation (adopted by this work).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' (a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' An illustration of a 3D unit cell described by the MMC method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' (a) The geometric description of the 3D components and (b) some representative configurations in the optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' The SI induced topological invariants of the helical MTIs The helical multipole topological insulators (MTIs)32,33, as a compound topological material, should simultaneously hold the characters (or the topological invariants) from the multipole moment and the pseudo-spin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' In general, the calculation of topological invariants is computationally expensive for the continuum unit cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' Nevertheless, recent work in topological quantum chemistry reveals a rapid approach to identify topological invariants through its symmetry indicators (SIs)4,5,46,49,50,58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' Next, we will introduce the SI method into the calculation of topological invariants, and then give out the method to design helical MTIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' Based on the SI method,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' for a spinless ������������������������=3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='6-symmetric mechanic system with the time reversal symmetry (TRS),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' we identify the eigenvalue Π������������ (������������) of the ������������̂������������ rotational operator as Π������������ (������������) = ������������i2������������(������������−1)/������������ = �������������(Π)�������������̂�������������������������(Π)�,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' ������������ ∈ [1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' ������������] (7) where ������������(Π) denotes the ������������-component of the displacement at the high-symmetry point Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' The symbol #Π������������ (������������) counts the number of Π������������ (������������) below the target band gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' Compared to the reference point Γ = (0,0), we define the SI at Π as �Π������������ (������������)� = #Π������������ (������������) − #Γ������������ (������������).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' At the high- symmetry points, it satisfies ������������̂������������������������ = ������������ + ������������ with ������������ denoting the reciprocal lattice vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' For the ������������3-symmetric hexagonal unit cells, the high-symmetry points include Γ and K in the ������������3 symmetry, while for the ������������6-symmetric hexagonal unit cells, they include Γ in the ������������6 symmetry, K in the ������������3 symmetry, and M in the ������������2 symmetry, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=" In this manner, AZ Z'RY X L3 L2Original Morphing Moving Mergingthe topological classification is determined completely by the corresponding SIs, such as the fractional corner charge and the pseudo-spin invariants in the following contents." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='1 The fractional corner charge invariants For the MTIs, the fractional corner charge ������������(������������) is an effective topological invariant to determine the topological corner states29–31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' For the 3D mechanical topological system with the TRS and ������������3 symmetry, we propose the following formulas ������������(3) = �1 3 �#K������������≠1 (3) − 1 2 #Γ(3)� mod 1� × ��#Γ(3) + 1�mod 2� ������������(6) = �1 4 �#M1 (2)� + 1 6 �#K1 (3)�� mod 2 (8) Here, the red terms in the function of ������������(3) are introduced to avoid the confused distinction of the unpaired degenerate states6,9,10,25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' As an alternative strategy, #Γ(3) counts the two- order degeneracy at the Γ point, and #Γ(3)/2 identically equals the #Γ2 (3) or #Γ3 (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' The red modulo term in ������������(3) guarantees the degenerate states to be in pairs (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=', #Γ(3) is an even number).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' For a visualization, the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' 2 shows that our modification successful avoids the confused distinction of degenerate states when ������������ = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' For more details, refer to Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' (a) (b) (c) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' The script of the modification procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' (a) The singularity in mechanics and degenerate states in a hexagonal unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' (b) and (c) The vector transformation of displacement component (������������, ������������) and ������������.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' Here, only the degenerate states are calculated under the ������������̂3 operator, and denoted as 1/2 = (������������ + ������������∗)/2 with its eigenvalues ������������ and its conjugation ������������∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' the other states are calculated under the ������������̂2 operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='2 The pseudo-spin invariants For the photonic/phononic quantum spin/valley Hall effects, the protected chiral energy flux can be well identified from the pseudo-spin vortex phenomenon and can be well Degenerate m=5 1/2 1/2 States m=4 1 m=3 1 +1 个个 00 Singularity ky个 (u, v) (-xo,yo) (xo, yo) x (-u, v)w (xo, Yo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' (-xo,-yo) x wquantified by the Chen-spin or ������������2 invariants6,9,24,25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' In our spinless 3D mechanical system without spatial inversion symmetry, an alternative approach is adopted through tracing the (broken) Dirac cone and band inversion9,10,25,38,40,42–44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' Practically, for the 3D ������������3 and ������������6 symmetric unit cells, the pseudo-spin invariants are modified as10,29 ������������(3) = sgn�#K2 (3) − #K3 (3)� ������������(6) = sgn�#Γp (6) − #Γd (6) − 2� (9) Here, terms #Γp (6) and #Γd (6) count the orbits p and d under the ������������̂6 operator for the Γ point, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' Notably, the subtracted red term is introduced to correct the #Γp (6) due to the singularity in the 3D elastic wave or the transverse electromagnetic wave (the singularity has the same eigenvalue as the orbit p)5,59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' For the 3D elastic wave, the singularity relates to three translational motions, their displacement and vector transformation are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' This modification is based on counting all occupied bands below the target band gap, including the first three bands crossed through the singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' Furthermore, the counting idea keeps target bands isolated from other bands, as the SI method requires.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' For more details, refer to Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' An efficient design paradigm of 3D mechanical helical MTIs With the above topological invariants presented in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' (8) and (9), we can now design the helical MTIs using explicit topology optimization method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' The corresponding optimization formulation and solution process are introduced as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='1 Mathematical formulation Combining the MMC-based description method and the modified formulas of topological invariants in elastic medium, optimized 3D helical MTIs can be obtained by solving the following mathematical formulation: find ������������ = (������������1 ⊤, … , ������������������������ ⊤, ������������)⊤ max min(������������ref − max������������������������������������ ������������ , min������������������������������������ ������������+1 − ������������ref) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' ������������c������������c = −������������2������������c������������c �������������(������������), ������������(������������)� = �������������ref (������������), ������������ref (������������)� ������������min ≤ ������������ ≤ ������������max (10) In the design variable vector, ������������������������ describes the ������������th component in the slab with a thickness ������������ (in the ������������-axis), as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' By denoting the eigenfrequency of the ������������th band as ������������������������, the gap width between the ������������th and (������������ + 1)th bands is maximized with a target mid-frequency ������������ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' The third equation in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' (10) is the governing equation for the 3D elastic waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' Since the fractional corner charge and the pseudo-spin invariants simultaneously contribute to the existence of helical corner states, the target topology invariants �������������ref (������������), ������������ref (������������)� is introduced as a constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' The last inequality persists the lower and upper bounds of the design variable vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' In principle, by updating the governing equation and target topological invariants, the mathematical formulation in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' (10) can be applied for designing TMs among different symmetry classes, and different physical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' In this work, we focused on the inverse design of 3D helical MTIs in elastic medium with ������������3 and ������������6 symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' The scheme of optimization for the helical MTIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='2 Solution process Since the topological invariants are quantized, gradient-based optimization algorithms would be ineffective for solving Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' Thanks to the advantage of a fewer number of design variables in the MMC method, the genetic algorithm (GA) is adopted here and the settings are presented in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' To be specific, the flowchart for the rational design of helical MTIs is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' 3, and its solution process is summarized as follows: STEP 1: Initialization of the MMC method and the GA solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' The gap label ������������, the mid-frequency ������������ref, and the nonzero topological invariants �������������ref (������������), ������������ref (������������)� are initialized first through a trial process, starting from ������������ = 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' STEP 2: Optimal design of the first MTI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' Init.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' STEP1 Band Order 200 Random FEA Count Cases If No Case (Q(n), z(n)) Q(n) ± 0&z(n) ±0?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' m=3 Unit Cells (COMSOL) N Y m=m+1 STEP2 1st MTI GA Generate Set Valid Para.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=" 'ref FEA Calculate Unit Cell Conv." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' fref, m (COMSOL) Q(n), z(n) MTI Partner Y MMC N STEP3 0, ref EndWith the parameters determined in STEP 1, solve the mathematical programming Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' (10) to obtain the first optimized MTI with the predefined invariant �������������ref (������������), ������������ref (������������)� and mid-frequency ������������ref;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' STEP 3: Optimal design of the MTI partner (if necessary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' With the desired topological invariants setting as �0, −������������ref (������������)� and the other parameters the same as STEP 2, solve Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' (10) to obtain the optimized MTI partner with the inverted pseudo-spin effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' (a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' The statistical charts (b) of different states at the Γ point (the partitions of p and d would be decomposed into the boxed partitions without the modification in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' (8)) and (c) of different TMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' Hint: s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' —singularity point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' To illustrate the effectiveness of the proposed design framework, the statistical charts of the states at the Γ point (6000 samples) and of different TMs (8000 samples) are illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' 4(a) and 4(b), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' It can be found that, using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' (8), the states p and d are successfully identified, and they take about 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='6% and 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='2% of the whole set as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' 4(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' Without the modification in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' (8), however, such states would be decomposed to Γ2 (3) state (22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='1%), Γ3 (3) (22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='1%), and an unpaired set of state (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='5%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' This unpaired set would further make troubles for the calculation of the fractional corner charge invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' 4(b), 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='6% of 8000 samples are four typical TMs (quantum valley/spin Hall topological insulators (QVTIs/QSTIs), MTIs and helical MTIs), while the desired helical MTIs only account for 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='0%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' This validates the necessity of developing inverse design paradigm for the helical MTIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='1% d 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='1% 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='6% 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='5% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='6% 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='2% 3 Other 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='1% Unpaired 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='5% 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='1% p ** （3） S pOther 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='4% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='7% QVTI 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='4% 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='0% Helical MTI QSTI 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='2% MTI6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' Applications of the MMC-based design framework for 3D helical MTIs in elastic medium In the present work, the helical MTIs are periodic in the in-plane direction and made of the basic medium EP and scattering medium Fe (materials parameters and more setup details are referred to Appendix C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='1 Optimal design of ������������3-symmetric mechanical helical MTIs Under the optimization framework, the optimized ������������3-symmetric helical MTIs are obtained in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' 5(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' And there is a normalized bulk band gap at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='741-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='069 between the 6th and 7th bands (colored in grey in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' 5(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' The symmetry behaviors of the high-symmetry points are shown in the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' 5(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' There are three broken degenerate states (from the Dirac cone) at the K point below the target bandgap, while only the third one is unpaired, and implies the possibility of a pseudo-spin vortex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' The phase field of this unpaired state is also inserted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' 5(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' The corner charge and the pseudo-spin invariants are (2/3,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' In order to realize the band inversion, the corresponding MTI partner can be easily constructed by applying the spatial reversal operation, or in other words, its invariants are set as (0, −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' An opposite pair of ������������(3) invariants would produce a helical topological state from the bulk-boundary correspondence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' Moreover, a pair of zero and nonzero fractional corner charges reveal the appearance of corner states18, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' 6(a) around the normalized frequencies of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='894 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='966.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' The latter localized corner mode is displayed in the inserted diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' (a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' The optimized ������������3-symmetric mechanical helical MTIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' (a) The band structure inserted with the unit cell and the phase field of the unpaired state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' (b) The symmetry- 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='2 (2A/2πC) 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='6 Freq 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='2 0 T M K Singularity(2) Band (3) b b 1 1 3 wt 2 1 +1 3 +1 +1 m 4 +1 +1 5 +1 +1 6 +1 +1 mbehavior-table, in which the degenerate states are tagged as ������������ = ������������i2������������/3 for the K point, while for the Γ point they are tagged as ������������.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' (a) (b) (c) (d) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' Simulation results of the optimized ������������3-symmetric helical MTIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' (a) The eigenvalue spectrum (points are colored according to the corner energy intensity) and the energy field of a corner state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' (b) The transmission spectra from the probes in bulk, edge, and corner area (colored in legend).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' (c) Energy fields tagged in (b) at the normalized frequencies of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='793, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='879, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='966, and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='121.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' (d) The energy flux and their zoom-in views of helical edge states at the normalized frequency of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='862.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' Besides, the full-wave transmission is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' 6(b), where energy is captured from different regions around the outer bulk, the interface edges, and the interface corners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' A spin-down (clockwise) helical source is excited near the supercell’s center, shown as the star in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' 6(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' The transmission reveals some edge energy peaks around the normalized frequencies of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='793 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='001, and some intensively localized corner states around the normalized frequencies of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='879 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='966.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' For a clear visualization, the corresponding 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='98 (S2A/2 TC) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='9 Freq ( 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='86 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='82 Index(dB) 3 Transmission ( 2 60 Corner 120 Edge Bulk 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='9 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='1 Freg (2 A/2πc)1 2 3Spin-Down Spin-Upbulk, edge, and corner energy fields are displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' 6(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' In contrast to the edge gap around the normalized frequency range of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='872-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='001 (colored in light-green), those in- gap corner states are derived from the quadrupole moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' For the verification of the helical behavior, a biased helical source off the center is excited additionally, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' 6(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' The inserted arrow diagrams displayed the energy flux near their corners and edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' We found that these two supercells had significant opposite responses under different exciting helical sources (spin-up or spin-down).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' All their corners held a clear energy vortex (clockwise or anticlockwise).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' Their edge energy fluxes are locked by their exciting sources and only could flow forward or backward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='2 Optimal design of ������������6-symmetric mechanical helical MTIs For the optimized ������������6-symmetric MTI pairs, as illustrated by the band structures shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' 7(a), band gaps are observed in the normalized frequency ranges of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='344-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='489 (up) and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='332-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='450 (below), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' Below the gap, there are four degenerate states found at the Γ points for both cases, but only the last two states formed an unpaired double Dirac cone, which features the pseudo-spin vortex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' The phase fields of these unpaired states are inserted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' 7(a), from which the band inversion is clearly displayed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' The symmetry behaviors in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' 7(a) show that the corner charges and the modified pseudo-spin invariants are (1/2,1)and (0, −1), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' Specifically, the pair of opposite ������������(6) invariants lock the energy flux by the pseudo-spin phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' In contrast, the pair of zero and nonzero corner charges predict the topological corner states (according to the vanished bulk polarization in ������������6-symmetric unit cells, these nonzero corner charges are only derived from the quadrupole moment30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' By combining these two topological characters, the topological corner state will also have pseudo-spin behaviors and present as a helical corner state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' For a verification of this helical corner state, the eigenvalue spectrum of the supercell’s simulation is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' 7(b), and its energy density distribution, at the normalized frequency of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='426, is highly localized at corners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='3 Applications of the optimized helical MTIs in a crossing waveguide As an application of the helical MTIs, a crossing waveguide (a single layer) composed of the two optimized ������������3-symmetric helical TMIs in Subsection 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='1 (colored blue/yellow for the original/inversed TMIs mentioned above) is developed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' 8(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' Since the additional pseudo-spin freedom locks the energy flux in the waveguide, two opposite transmissions would be discovered when we sequentially excited the Port 1 and Port 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' By gradually modulating the exciting frequency, the energy will spread through the center wall and induce the output corner states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' (a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' Simulation results of the optimized ������������6-symmetric helical MTIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' (a) The band structures and the symmetry-behavior-tables of the optimized MTI pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' The inserted diagrams include the optimized unit cells and the ������������-directional displacement fields of the unpaired states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' In those tables, the degenerate states for the K point are tagged as ������������ = ������������i2������������/3, while for the Γ point they are tagged as ������������.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' (b) The eigenvalue spectrum and the inserted energy field of the corner state (points are colored according to the corner energy intensity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' The simulations in the normalized frequency range of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='7-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='1 are processed to test the performance of the waveguide, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' 8(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' It is clear that when Port 1 is excited at the normalized frequency of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='776, the energy only transmits to Port 2 and Port 3, yet it only transmits to Port 1 and Port 4 from Port 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' This phenomenon reveals the locked helical energy flux as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' At the normalized frequency of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='897, the corner states in the lower half of the waveguide are excited in both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' Here, these states stay in the band gap of the edge states (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=', 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='872-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='001, refer to Appendix D for more details), and their energy only localizes at corners, and no edge states exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' To test the working range of the one-way transmission in this waveguide, we distinguished the energy from the different ports (Port 3 or Port 4), as shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' 8(c) and 8(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' Here the light-green area and yellow-solid points refer to the band gap of the edge states and the states in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' 8(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' In this much wider frequency range of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='749-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='861, the average difference between both ports is higher than 10dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' When we reverse the exciting port, the output port, which has a higher transmission, is also turned, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' 8(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' In this frequency range, the first two edge bands, as illustrated in Appendix D, will be excited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='6 Band a (2A/2πc) 1 1 +1 m 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='2 2 1 1 wt 1 3 +1 +1 +1 +1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='8 Freq ( 4 +1 +1 +1 at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='6 5 +1 1 m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='4 6 1 1 +1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='2 7 0 M K Singularity 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='6 r(2) r(3) m(2) k(3) Band a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='4 b 9 b Freq (2A/2πc) 1 w) 1 2 1 +1 wt 1 3 +1 +1 +1 +1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='8 4 +1 wt 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='6 5 +1 +1 +1 m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='4 6 +1 1 +1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='2 7 +1 +1 0 M K Singularity1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='47 (S2A/2 TC) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='44 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='41 Freq ( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='38 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='35 IndexHence, these one-way transmission results from the helical edge states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' Moreover, the corner states tagged with the number 2 and 4 are in the gap of the edge state but display an apparent energy concentration from the exciting source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' (a) (b) (c) (d) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' The crossing waveguide made of the optimized ������������3-symmetric helical MTIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' (a) The sketches of the waveguide and its energy fluxes in different exciting cases (the exciting line sources are tagged as stars).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' (b) The energy fields at the normalized frequencies of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='776 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='897.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' The measured transmission of Ports 3 and 4 (c) from the exciting Port 1 or (d) from the exciting Port 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' Here the band gap of the edge states (light-green region) and the typical states (yellow-solid points) are colored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' Concluding remarks In this work, we proposed an optimization framework for the inverse design of multi- functional topological materials in the 3D continuous medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' By carefully manifesting the degenerate states and singularity points in the elastic waves, the 3D helical multipole Port1 Port 2 Port 4 Port 3 C Q2 32 0 (dB) 50 Transmission ( 100 150 200 Port 3 250 Port 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='9 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='1 Freq (2 A/2πc)3 Transmission (dB) 50 100 150 200 Port 3 250 Port 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='9 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='1 Freq (2 A/2πc)topological insulators are well-classified by the fractional corner charge and the pseudo- spin invariants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' With the explicit topology optimization and the symmetry indicator methods, the proposed design paradigm has the advantages of (1) rapid classification of the 3D topological materials and (2) efficient optimization of the 3D continuum unit cells in a smaller explicit parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' This framework shows outstanding suitability to the 3D topological system and can also be generalized to other symmetry classes and space groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' Besides, building up a topological materials library in continuous medium would be an exciting topic for further research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' Methods The solid mechanic simulation is performed in the commercial software COMSOL MULTIPHYSICS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' The default open surfaces are set as free boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' The Bloch theorem is numerical expressed by the Floquet periodic boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' In common, the energy in solid mechanics is consistent in distribution as the amplitude of total displacement ‖(������������, ������������, ������������)‖2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' Acknowledgements The financial supports from the National Natural Science Foundation (11821202, 11732004, 12002073, 12002077, 12272075, 11922204), the National Key Research and Development Plan (2020YFB1709401), Dalian Talent Innovation Program (2020RQ099), the Fundamental Research Funds for the Central Universities (DUT20RC(3)020, DUT21RC(3)076), and 111 Project (B14013) are gratefully acknowledged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' Author contributions X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' conceived the idea and initiated the project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' established the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' L and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' performed the numerical calculations and simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' All the other authors contributed to the discussions of the results and the manuscript preparation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' Declaration of competing interest There are no conflicts to declare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' Data availability Data will be made available on request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' Appendix Appendix A: Modification of the fractional corner charge invariant According to the results in literature30, the fractional corner charge invariant of the ������������3- symmetric hexagonal unit cells is ������������������������ ′(3) = 1 3 �K������������≠1 (3) � mod 1 (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' 1) where subscript ������������ equals 2 or 3 depending on the symmetry of the constructed supercell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' Due to the TRS and ������������3 symmetry, some two-order degenerate states are protected at the Γ point, such as states from the linear combination of the Γ2 (3) and Γ3 (3), and they are computationally expensive to identify clearly, especially for the unpaired degenerate states6,9,10,25,28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' Instead, we termed the invariant with the number of the two-order degenerate states #Γ(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' To be specific, the topological character of the ������������3-symmetric hexagonal unit cell is given by ������������‾(3) = �#Γ(3), #K2 (3), #K3 (3)� (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' 2) Considering Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='2), the modified fractional corner charge invariants and the symmetry behaviors are listed in Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='1 for some possible cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' The symmetry behaviors of the ������������3-symmetric unit cells with TRS (for the fractional corner charge invariants) ������������������������=2 (3) ������������������������=3 (3) #K2 (3) #K3 (3) #Γ(3) #Γ2 (3) #Γ2 (3) 1/3 0 1 0 0 0 0 0 1/3 0 1 0 0 0 0 2/3 1 0 2 1 1 0 0 1 1 2 1 1 0 0 1 0 1 1 0 0 0 0 1 1 0 1 0 0 1 0 1 0 1 0 0 0 1 1 1 0 In Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='1, the red colored invariants ������������������������ (3) are modified from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' This modification is derived from the fact that the degenerate states #Γ(3) always appear in a pair;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' or not, it is gapless 6,9,10,25,28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' For the unpaired case, it is ambiguous to be tagged as Γ2 (3) or Γ3 (3), hence the invariants ������������������������ (3) in the last four cases cannot be solely identified by the ������������‾(3) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' Therefore, when #Γ(3) is odd, the corresponding fractional corner charge should be modified into zero with gapless band reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' When #Γ(3) is even, number #Γ(3) can be equivalently divided as: #Γ2 (3) = #Γ3 (3) = #Γ(3)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' Thus, the modified fractional corner charge invariant gives ������������(3) = �1 3 �#K������������≠1 (3) − 1 2 #Γ(3)� mod 1� × ��#Γ(3) + 1�mod 2� (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' 3) where the red module term aims to avoid the unpaired two-order degenerate states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' Appendix B: Modification of the pseudo-spin invariant For the ������������3 or ������������6-symmetric unit cells with the TRS, an alternative approach to get the pseudo-spin invariants is to trace the broken (double) Dirac cones at the K or Γ point 9,10,25,38,40,42–44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' Thus, their topological characters are given by ������������‾(3) = �#K2 (3), #K3 (3)� ������������‾(6) = �#Γ1 (2), #Γ2 (2), #Γ(3)� (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' 1) Here, #Γ(3) counts the two-order degenerate states of the Γ point in the ������������3 operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' The modified pseudo-spin invariants and their symmetry behaviors are listed in Table B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='1 and Table B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='2 for some possible cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' Different as scaling a function5,59, the operation of a symmetry operator ������������� on a vector function ������������(������������) transforms as �������������������������(������������) = �������������������������(�������������−1������������), where ������������� is the rotational operator in �������������.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' For the present symmetry groups (������������3 or ������������6) in our paper,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' all group elements behave as a rotation around ������������ -axis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' and the transformation can be simplified as �������������������������(������������) = �������������������������T(�������������−1������������) + �������������������������L(�������������−1������������) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' where ������������ = ������������T + ������������L = (������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' ������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' 0)⊤ + (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' ������������)⊤ is the displacement vector in mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' This decomposed equation implies ������������T and ������������L hold the same symmetry, except for the singular cases with displacement component ������������T = ������������ or ������������L = ������������.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' Table B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' The symmetry behaviors of the ������������3-symmetric unit cells with TRS (for the pseudo-spin invariants) ������������(3) #K2 (3) #K3 (3) 1 1 0 1 2 0 1 0 1 0 1 1 Table B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' The symmetry behaviors of the ������������6-symmetric unit cells with TRS (for the pseudo-spin invariants) ������������(6) #Γ1 (2) #Γ2 (2) #Γ(3) Orbits 1 2 0 2 2d 1 0 2 2 2p 0 2 2 4 2p + 2d 0 1 0 0 1s 0 0 1 0 1f 0 1 2 2 2 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' In the original SI theory29,30, the occupied bands counted in the SI method should be isolated from others, and an alternative approach is to count all bands below the target band gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' For the photonic and phononic systems, however, the first two or three bands always converge to plane waves when |������������| → 0, where transverse modes produce two singularities with ������������L = ������������5,59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' In Table B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='2, the red-colored data reveal the symmetry behaviors of the first three bands that always cross through the singularities around the zero energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' Hence, we defined the modified pseudo-spin invariants to overcount those singularities as ������������(3) = sgn�#K2 (3) − #K3 (3)� ������������(6) = sgn�#Γp (6) − #Γd (6) − 2� (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' 2) where the red term is the modification from the singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' For the case of photonics, the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='2) should be further modified as the work5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' And the terms #Γp (6) and #Γd (6) count the p and d states at the Γ point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' Table C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' Some typical optimized unit cells for the ������������3 and the ������������6-symmetric TMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' (Here, the red data refers to the examples presented in our paper) ������������������������ ������������{������������} ������������{������������} ������������{deg} ������������{������������} �������������(������������), ������������(������������)� ������������max (������������) ∼ ������������min (������������+1) ������������3 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='7217,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='95,0.' metadata={'source': 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1) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='254~1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='319 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='8603,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='95,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='1575) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='2055,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='3596,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='3082) (108,144,90) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='45 (0, -1) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='249~1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='344 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' Some optimized unit cells with the ������������3 and ������������6 symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' The order refers to the row number of Table C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' Appendix C: The setup of the optimization and the GA solver For the parameters of material and optimization solver in our paper, the setup gives: the basic medium is epoxy (EP) with the elastic modulus ������������0 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='35GPa, the Poisson’s ratio ������������0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='37, and the mass density ������������0 = 1180kg ⋅ m−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' The scattering medium is steel (Fe) with the elastic modulus ������������ = 200GPa, the Poisson’s ratio ������������ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='2, and the mass density ������������ = 7800kg ⋅ m−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' The genetic algorithm (GA) solver is set as: the population size of 100, the crossover fraction of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='9, the migration fraction of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='3, the elite size of 5, the objectivation tolerance of 1e-5, the stall generation limit of 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' The lattice constant is ������������ = |������������| = 1m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' 3 5 2Table C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='1 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='1 list some typical optimized unit cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' The row number in Table C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='1 is consistent with the order of unit cells in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' For the examples in the main text, we set their optimization procedure as For the ������������3-symmetric helical MTIs, the broken Dirac cone appears at the K point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' The first unit cell is optimized with setting ������������ = 6, nonzero topological invariants (2/3,1) and no specific ������������ref, which will auto-update as the mid-frequency of the target gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' For the ������������6-symmetric helical MTIs, the broken Dirac cone appears at the Γ point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' The first unit cell is optimized with setting ������������ = 7, �������������ref (������������), ������������ref (������������)� = (1/2,1), and ������������ref = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' Then the MTI partner is optimized with setting ������������ = 7, �������������ref (������������), ������������ref (������������)� = (0, −1), and ������������ref = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' Appendix D: The supercell’s setup for the edge and the corner states The setups for two example supercells are detailed as For the ������������3-symmetric unit cells in the main text, the script of the truncated supercell is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='1, where it provides an approach to adjust the frequency of the edge states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' This truncation does not break the crystalline symmetry, and the topological edge states would not vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='1(a) and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='1(b), the truncation is set as ������������ = 1/4 × 2������������/√3 , and the edge gap is between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='872-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='1(c) and (d), the eigenvalue spectrum and the crossing waveguide are simulated with the truncation ������������ = 1 × 2������������/√3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' The light-blue areas in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='1 refer to the supercell’s structures in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' For the ������������6-symmetric unit cell, the inner interface between the unit cell pairs can be alternatively truncated as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='2 (a) and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='2(b), the gap of the edge states is found between 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='398-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='425 with the truncation ������������ = 1/2 × ������������.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' The supercell’s script for the eigenvalue spectrum is displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='2(c) with the truncation ������������ = 1 × 2������������/√3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='2(c), except for the inner hexagonal interface, six base medium cylinders with a diameter of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='4������������ are added to adjust the supercell’s corners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' The light-blue areas in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='2 refer to the supercell’s structures in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' (a) (b) (c) (d) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' The supercell’s scripts of the ������������3-symmetric unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' (a) The band of the edge state (light-grey band belongs to the lower interface counterpart), and (b) the script of the ribbon-shaped supercell in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' The supercell’s script (c) for the eigenvalue spectrum and (d) for the simulation of the crossing waveguide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' (a) (b) (c) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' The supercell’s scripts for the ������������6-symmetric unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' (a) The band of the edge states, (b) the script of the ribbon-shaped supercell in (a), and (c) the supercell’s script for the simulation of the crossing waveguide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' Edgestates 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='1 (2A/2πc) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='9 Freq 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='7 X12A 8A T8A 7AEdgestates 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='5 Freq (2A/2πc) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='45 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='35 X12A 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content='4A 8AReference 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' Hasan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' & Kane, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' Space group theory of photonic bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} +page_content=' Phys Rev Lett 121, 263903 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfnQTu/content/2301.03308v1.pdf'} diff --git a/7tAyT4oBgHgl3EQfpvj9/content/tmp_files/2301.00533v1.pdf.txt b/7tAyT4oBgHgl3EQfpvj9/content/tmp_files/2301.00533v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..07d6f1542ada9964ef7b554ec5ef80719f6ff6fb --- /dev/null +++ b/7tAyT4oBgHgl3EQfpvj9/content/tmp_files/2301.00533v1.pdf.txt @@ -0,0 +1,2582 @@ +Analytical comparison between X(3) and X(5) models of the Bohr +Hamiltonian +K.R. Ajulo1; K.J. Oyewumi2 +1,2University of Ilorin, Ilorin, Nigeria. +Abstract +Via the inverse square potential, the solutions of the X(3) model which is a γ-rigid form of +the X(5) critical point symmetry have been achieved. The paper presents X(3), through the +variational technique, as another “window” through which the “pictures” of X(5) −→ SU(3) +symmetry region can be seen. The analytical solutions of the X(3) are compared with the +solutions of the X(5) model. Some new and unique equations connecting the two models in +the: critical order, energy bands, spectra ratios, RL/2, and B(E2) transitional probabilities are +presented. These equations should hold in other potentials with one-parameter such as Kratzer +potential, Davidson potential etc. The spectra ratios and the B(E2) transitional probabilities +are optimized via the optimization procedure. The experimental data of some selected isotopes +are placed accordingly for the theoretical predictions. The deviations from the experiments are +found to be quite small. +Keywords: +Bohr Hamiltonian, X(3), X(5), variation technique, optimization procedure, β- +variable, γ-rigid. +1 +Introduction +X(3) which has been presented in [1-4] is said to be an exactly separable γ-rigid form of the X(5) +critical point symmetry [5]. The X(3) model is defined by the collective coordinate β and two +Euler angles since the γ is assumed to be zero unlike the case of X(5), where γ is varied around +γ0 = 0 value in the harmonic oscillator potential [5]. This implies that, only three variables: β and +θi are involved in the X(3) model. An exact separation of the β variable from the Euler angles is +quite easily achievable. In the Bohr Hamiltonian model [6-9], X(5) critical point symmetry is one +of the two critical point symmetries: it is a phase transition of the first order shape, which were +originally proposed in the works of Iachello [10], while E(5) is the phase transition of the second +order shape [10]. In the present work, the nuclei are taken to be γ-rigid, with the axially symmetric +prolate shape obtained at γ0 = 0. The work presents the usefulness of a one-sided bound inverse +square potential with one parameter. The one-parameter inverse square potential chosen is of the +form +V (β) = +� +� +� +� +� +β0 +β2 , if 0 ≤ β ≤ β0, +∞, if β > β0, +(1) +where β0 is a variation parameter that changes the signatures of the nuclei, as it changes. It +is expected that the solutions should shift forward as the β0 shifts forward and solutions should +shift backward as the β0 shifts backward. A typical inverse square potential is bound on the left +and unbound on the right, and it has a minimum at some positive values of β0 that forces the +particles to infinity as β0 → 0. As a result, the particle’s energy states is one-sided, with energies +escaping through the unbound side. The work is structured as follows: Section 2. presents the +methodology and the solutions of X(3) model via inverse square potential. These solutions are: +the wave functions, the normalization constants and the energy eigenvalues. The B(E2) transition +rates are presented in Section 3. The analytical results, the numerical results, their applications in +certain isotopes are presented and discussed in Section 4. The work is concluded and summarized +in the Section 5. +1E-Mail: 19-68eo001@students.unilorin.edu.ng +2E-Mail: kjoyewumi66@unilorin.edu.ng +1 +arXiv:2301.00533v1 [nucl-th] 2 Jan 2023 + +2 +Methodology of X(3) model with the inverse square potential +In the X(3) model, the Bohr Hamiltonian operator is written as [1,2] +ˆH = − ℏ2 +2B +� +1 +β2 +∂ +∂β β2 ∂ +∂β + +1 +3β2 +� +1 +sin θ +∂ +∂θ sin θ ∂ +∂θ + +1 +sin2 θ +∂2 +∂φ2 +� +− V (β) +� +, +(2) +where the term, +1 +sin θ +∂ +∂θ sin θ ∂ +∂θ + +1 +sin2 θ +∂2 +∂φ2 , +(3) +inside the bracket, represents the angular part of the Laplacian [1,2] . B, β and V (β) are respec- +tively the mass parameter, the collective coordinate and the β-dependent potential. The wave +equation of the Eq.(2) is +ˆHΨ(β, θ, φ) = EΨ(β, θ, φ). +(4) +By the usual method of separation of variable employed in some quantum texts, +Ψ(β, θ, φ) = χ(β)YL,M(θ, φ), +(5) +where YL,M(θ, φ) is the spherical harmonics and χ(β) is the radial part of Eq.(4). The separated +angular part obtained reads [1,2] +− +� +1 +sin θ +∂ +∂θ sin θ ∂ +∂θ + +1 +sin2 θ +∂2 +∂φ2 +� +YL,M(θ, φ) = L(L + 1)YL,M(θ, φ), +(6) +where L is the angular momentum quantum number. +The simplified form of the radial part +equation [1,2] , +� 1 +β2 +d +dβ β2 d +dβ − L(L + 1) +3β2 ++ 2B +ℏ2 [E − V (β)] +� +χ(β) = 0 +(7) +reads +d2 +dβ2 χ(β) + 2 +β +d +dβ χ(β) − L(L + 1) +3β2 +χ(β) − [v(β) − ϵ]χ(β) = 0, +(8) +where ϵ = 2B +ℏ2 E and v(β) = 2B +ℏ2 V (β) are the reduced energy and reduced potential respectively +[5]. +2.1 +Determination of the wave functions +By substituting Eq.(1) for v(β) in Eq.(8) and solving the simplified equation using MAPLE soft- +ware, the eigenfunctions obtained read +χs,ν,L,(β) = β−1/2 �C1,LJν(√ϵβ) + C2,LYν(√ϵβ) +� , +(9) +where C1,L and C2,L are the normalization constants associated with the Bessel functions of the first +kind, Jν, and second kind, Yν, respectively. In the domain of Eq.(1), the critical order associated +with the X(3) model in Eq.(9) is +νX(3) = +� +L +3 (L + 1) + β0 + 1 +4. +(10) +If a boundary condition χs,ν,L(β0) = 0 is considered, then C2,L,nYν(√ϵβ) vanishes and the wave +functions become +χs,ν,L(β) = β−1/2 �C1,LJν(√ϵβ) +� . +(11) +2 + +2.2 +Determination of the energy eigenvalues and the spectral ratio +The procedure for finding the eigenvalues is written in ref. [11]. If the first condition of the listed +procedure is considered, then the acceptable expression for the energy eigenvalues is written as: +Es,L,nβ = ℏ2 +2B k2 +s,ν,nβ, +k2 +s,ν,nβ = ϵs,ν,nβ, +k2 +s,ν,nβ = xnβ,s,ν +β0 +, +(12) +where s = nβ +1, xs,ν,nβ is the s-th zeros of the Bessel function of order ν. The energy eigenvalues +of the β-part in ℏω = 1 unit reads: +ϵs,L,nβ = 2nβ + 1 + νX(3) = 2nβ + 1 + +� +L +3 (L + 1) + β0 + 1 +4 : +nβ = 0, 1, 2, ... +(13) +For the X(3) model, the ground state energy levels are defined with s = 1, the quasi-β1 levels are +defined with s = 2 and the quasi-β2 levels are defined with s = 3. L = 0, 2, 4, 6... There exist no +γ-bands in the X(3) model because, γ0 = 0. +Eq.(13) is the similar to the energy eigenvalues obtained in the β-part of X(5) model [12], the +difference is observed in their critical orders where +νX(5) = +� +L +3 (L + 1) + β0 + 9 +4. +(14) +Since s = nβ + 1, ϵs,L,nβ can be reduced to ϵs,L, then the spectra ratios can be written as +RL/2 = ϵs,L − ϵ1,0 +ϵ1,2 − ϵ1,0 +. +(15) +2.3 +Determination of the normalization constants and the complete wave func- +tions +The normalization condition for the Hamiltonian operator in Eq.(2) is written as [1,2] +� β0 +0 +β2 | χs,ν,L,nβ(β) |2 dβ = 1, +(16) +such that +| χs,ν,L,nβ(β) |2→ 0 +for +β → 0; +| χs,ν,L,nβ(β) |2 β2 → 0 +for +β → ∞. +(17) +If these conditions are satisfied, then +� β0 +0 +β2 | χs,ν,L,nβ(β) |2 dβ < β0. +Using the identity [13-14] +Jν(√ϵβ)Jν(√ϵβ) = +∞ +� +nβ=0 +�1 +2 +√ϵβ +�2ν+2nβ +(2ν + nβ + 1)nβ +nβ![Γ(ν + nβ + 1)]2 +(18) +in Eq.(16), the simplified normalization constants read +C1,L,nβ = +� +���� +� +nβ=0,1,2,3... +(η)nβ +�ks,ν,nβ +2 +�ξ−2 +β(ξ) +0 +nβ! +ξ +� +Γ +�ξ +2 +��2 +� +���� +−1/2 +, +(19) +where +ξ = 2ν + 2nβ + 2, +η = 2ν + nβ + 1 +and +(η)nβ = η(η + 1)(η + 2)...(η + nβ − 1), +(20) +3 + +with (η)0 = 1. Hence, Eq.(11) becomes +χs,ν,L,nβ(β) = +� +���� +� +nβ=0,1,2,3... +(η)nβ +�ks,ν,nβ +2 +�ξ−2 +β(ξ) +0 +nβ! +ξ +� +Γ +�ξ +2 +��2 +� +���� +−1/2 +β−1/2Jν(√ϵβ). +(21) +3 +B(E2) transition rates +The electric quadrupole operator is written as [1,2] +T E2 +µ += tβ +� +D(2) +µ,0(θi) cos γ + 1 +√ +2 +� +D(2) +µ,2(θi) + D(2) +µ,−2(θi) +� +sin γ +� +, +(22) +where D(θi) are the Wigner functions of the Euler angle and t is known as a scale factor. For +γ0 = 0, +T E2 +µ += tβ +�4π +5 Y2µ(θ, φ). +(23) +The B(E2) [1,2,5,15] is written as +B(E2; sL −→ s′L′) = +1 +2sL + 1| +� +s′L′||T E2||sL +� +|2, +(24) += 2s′L′ + 1 +2sL + 1 B(E2; s′L′ −→ sL). +(25) +Eq.(24) or Eq.(25) has been solved in ref. [1] as: +B(E2; sL −→ s′L′) = t2 � +CL′0 +L0,20 +�2 I2 +sL;s′L′, +(26) +where the coefficients, CL′0 +L0,20 are the Clebsch-Gordan coefficients, and +IsL;s′L′ = +� β0 +0 +βχs,ν,L,nβ(β)χs′,ν′,L′,n′ +β(β)β2dβ, +(27) +are the integrals over β. +4 +Numerical results, analytical results, applications and discus- +sion +Some important solutions for the collective model of Eq.(2) are the energy levels, the spectra ratios +and the B(E2) transitions. Their theoretical predictions are important when energy spectra are +assigned to the states for which experimental data are not available. The numerical calculations, +the analytical comparisons and how the search for the experimental realizations of the model was +achieved are discussed accordingly in this section. +Both the X(3) and the X(5) have their critical orders, ν(L, β0), from their Bessel functions which +describes their energy spectra. Firstly, in the comparison of the Eq.(10) and Eq.(14), it can be +deduced from the numerical computation of ν, shown in Table 1., that +νX(3)(β0 = c + 2) = νX(5)(β0 = c) : +c = 0, 1, 2, ... +(28) +In both cases, it increases with increase in the angular momentum, L, and with increase in the +variation parameter, β0. These effects of L and β0 in ν are also seen in the energy values of Eq.(13). +4 + +Table 1: The comparison in the critical order, ν, of the X(5) [12], with the ν of Eq.(10). +ν(L) +L +β0 = 0 +β0 = 2 +β0 = 4 +β0 = 6 +β0 = 102 +β0 = 100 +X(3) +X(5) +X(3) +X(5) +X(3) +X(5) +X(3) +X(5) +X(3) +X(5) +0 +0.500 +1.500 +1.500 +2.062 +2.062 +2.500 +2.500 +2.062 +10.112 +10.112 +2 +1.500 +2.062 +2.062 +2.500 +2.500 +2.872 +2.872 +2.500 +10.210 +10.210 +4 +2.630 +2.986 +2.986 +3.304 +3.304 +3.594 +3.594 +3.304 +10.436 +10.436 +6 +3.775 +4.031 +4.031 +4.272 +4.272 +4.500 +4.500 +4.272 +10.782 +10.782 +8 +4.924 +5.123 +5.123 +5.315 +5.315 +5.500 +5.500 +5.315 +11.236 +11.236 +10 +6.076 +6.238 +6.238 +6.397 +6.397 +6.551 +6.551 +6.397 +11.786 +11.786 +β0 = 1 +β0 = 3 +β0 = 5 +β0 = 7 +β0 = 101 +β0 = 103 +0 +1.118 +1.803 +1.803 +2.291 +2.291 +2.693 +2.693 +3.041 +10.062 +10.259 +2 +1.803 +2.291 +2.291 +2.693 +2.693 +3.041 +3.041 +3.354 +10.161 +10.356 +4 +2.814 +3.149 +3.149 +3.452 +3.452 +3.731 +3.731 +3.990 +10.388 +10.579 +6 +3.905 +4.153 +4.153 +4.387 +4.387 +4.610 +4.610 +4.823 +10.735 +10.920 +8 +5.025 +5.220 +5.220 +5.408 +5.408 +5.590 +5.590 +5.766 +11.191 +11.369 +10 +6.158 +6.318 +6.318 +6.474 +6.474 +6.627 +6.627 +6.776 +11.747 +11.913 +Figure 1: (a) Comparison in the energy levels of the X(3) and X(5) models [15] at β0 = 2 from the gsb up +to the quasi-β2 band. (b): the variation of the critical order, ν, of the X(5) as a function of β0, is compared +with ν of the X(3) at constant angular momenta, L = 0, 2 and L = 4. +5 + +16 +5 +βo= 2 +14 +4.5 +12 +X(3) +B2 +4 +L=4 +3.5 +L=4 +L=0 +rgy +10 +X(3) +X(5) +L=2 +0X(5) +3 +8 +β2 +X(3) +X(5) +X(5) +-β1 +V 2.5 +L=0 +6 +X(5) +β1 +gsb +2 +4 +X(5) +X(3) +gsb +1.5 +2 +X(3) +0 +0.5 +0 +2 +4 +6 +8 +10 +12 +14 +0 +(a) +7 +(b) +0 +2 +4 +8 +10Table 2: Ground state energies, the energies of the quasi-β1 and the quasi-β2 denoted by nβ = +0, s = 1; nβ = 1, s = 2; and nβ = 2, s = 3 respectively for the X(3) and X(5) symmetry [12] in +ℏω = 1 unit. +β0 = 2 +β0 = 3 +β0 = 4 +β0 = 15 +L +nβ = 0; +s = 1 +X(3) +X(5) +X(3) +X(5) +X(3) +X(5) +X(3) +X(5) +0 +2.500 +2.031 +2.803 +2.146 +3.062 +2.250 +4.905 +3.077 +2 +3.062 +2.250 +3.291 +2.346 +3.500 +2.436 +5.153 +3.194 +4 +3.986 +2.652 +4.149 +2.726 +4.304 +2.797 +5.682 +3.445 +6 +5.031 +3.136 +5.153 +3.194 +5.272 +3.250 +6.408 +3.795 +8 +6.123 +3.658 +6.220 +3.704 +6.315 +3.750 +7.265 +4.212 +10 +7.238 +4.198 +7.318 +4.237 +7.397 +4.276 +8.205 +4.671 +12 +8.365 +4.750 +8.433 +4.783 +8.500 +4.816 +9.201 +5.161 +14 +9.500 +5.308 +9.559 +5.337 +9.617 +5.366 +10.233 +5.670 +nβ = 1; +s = 2 +0 +4.500 +4.031 +4.803 +4.146 +5.062 +4.250 +6.905 +5.077 +2 +5.062 +4.250 +5.291 +4.346 +5.500 +4.436 +7.153 +5.194 +4 +5.986 +4.652 +6.149 +4.726 +6.304 +4.797 +7.682 +5.445 +6 +7.031 +5.136 +7.153 +5.194 +7.272 +5.250 +8.408 +5.795 +8 +8.123 +5.658 +8.220 +5.704 +8.315 +5.750 +9.265 +6.211 +10 +9.238 +6.198 +9.318 +6.237 +9.397 +6.276 +10.205 +6.671 +12 +10.365 +6.750 +10.433 +6.783 +10.500 +6.816 +11.201 +7.161 +14 +11.500 +7.308 +11.559 +7.337 +11.617 +7.366 +12.233 +7.670 +nβ = 2; +s = 3 +0 +6.500 +6.031 +6.803 +6.146 +7.062 +6.250 +8.905 +7.077 +2 +7.062 +6.250 +7.291 +6.346 +7.500 +6.436 +9.153 +7.194 +4 +7.986 +6.652 +8.149 +6.726 +8.304 +6.797 +9.682 +7.445 +6 +9.031 +7.136 +9.153 +7.194 +9.272 +7.250 +10.408 +7.795 +8 +10.123 +7.658 +10.220 +7.704 +10.315 +7.750 +11.265 +8.211 +10 +11.238 +8.198 +11.318 +8.237 +11.397 +8.276 +12.205 +8.671 +12 +12.365 +8.750 +12.433 +8.783 +12.500 +8.816 +13.201 +9.161 +14 +13.500 +9.308 +13.559 +9.337 +13.617 +9.366 +14.233 +9.670 +6 + +Figure 2: (a) The plots showing the values of β0 at which energies are minimum. (b) The rate of energy +with respect to β0, showing non stationary property of β0. +Table 3: Comparison of the ground state spectra ratios, defined in Eq.(15), of the inverse square +potential in the X(3) model at different values of the β0, compared with the X(5) [12]. It can be +seen that X(3)(β0 = ∞) ≈ X(5)(β0 = ∞). +Ls,nβ +β0 = 0 +β0 = 0 +β0 = 2 +β0 = 2 +β0 = 4 +β0 = 4 +β0 = ∞ +β0 = ∞ +X(3) +X(5) +X(3) +X(5) +X(3) +X(5) +X(3) +X(5) +gsb +01,0 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +21,0 +1.000 +1.000 +1.000 +1.000 +1.000 +1.000 +1.000 +1.000 +41,0 +2.130 +2.646 +2.646 +2.834 +2.834 +2.938 +3.296 +3.296 +61,0 +3.275 +4.507 +4.507 +5.042 +5.042 +5.372 +6.806 +6.808 +81,0 +4.424 +6.453 +6.453 +7.421 +7.421 +8.508 +11.413 +11.423 +101,0 +5.576 +8.438 +8.438 +9.887 +9.887 +11.881 +16.991 +17.013 +121,0 +6.728 +10.445 +10.445 +12.404 +12.404 +15.686 +23.409 +23.450 +141,0 +7.882 +12.465 +12.465 +14.951 +14.951 +19.740 +30.544 +30.611 +β0 = 1 +β0 = 1 +β0 = 3 +β0 = 3 +β0 = 5 +β0 = 5 +β0 = 15 +β0 = 15 +01,0 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +0.000 +21,0 +1.000 +1.000 +1.000 +1.000 +1.000 +1.000 +1.000 +1.000 +41,0 +2.476 +2.756 +2.756 +2.893 +2.893 +2.946 +3.128 +3.148 +61,0 +4.070 +4.812 +4.812 +5.224 +5.224 +5.529 +6.058 +6.136 +81,0 +5.706 +6.995 +6.995 +7.767 +7.767 +8.638 +9.508 +9.690 +101,0 +7.360 +9.243 +9.243 +10.424 +10.424 +11.915 +13.297 +13.620 +121,0 +9.024 +11.525 +11.525 +13.145 +13.145 +15.854 +17.307 +17.800 +141,0 +10.694 +13.829 +13.829 +15.907 +15.907 +19.899 +21.468 +22.152 +7 + +L=2 +L=4 : +6 +L=2 +L=4 +6 +Po +10 +15 +0 +5 +10 +15 +-0.005- +-1.2 - +-0.010- +-1.4卡 +8v +8v-1.6 +-0.015 +log +SPo +-1.8 - +-0.020- +-2 - +-0.025 +(a) +[b] +-2.2-Table 4: RL/2 ratios, defined in Eq.(15), of the quasi-β1 and quasi-β2 bands of the inverse square +potential in the X(3) model at different values of the β0. +Ls,nβ +β0 = 0 +β0 = 1 +β0 = 2 +β0 = 3 +β0 = 4 +β0 = 15 +β0 = ∞ +quasi-β1 +02,1 +2.000 +2.921 +3.562 +4.094 +4.562 +8.058 +20.124 +22,1 +3.000 +3.921 +4.562 +5.094 +5.562 +9.058 +21.124 +42,1 +4.130 +5.397 +6.208 +6.850 +7.395 +11.187 +23.420 +62,1 +5.275 +6.991 +8.069 +8.906 +9.603 +14.115 +26.929 +82,1 +6.424 +8.626 +10.014 +11.090 +11.982 +17.567 +31.537 +102,1 +7.576 +10.281 +11.999 +13.337 +14.449 +21.356 +37.116 +122,1 +8.728 +11.945 +14.007 +15.619 +16.965 +25.366 +43.534 +142,1 +9.882 +13.615 +16.027 +17.923 +19.513 +29.526 +50.668 +quasi-β2 +03,2 +4.000 +5.842 +7.123 +8.188 +9.123 +16.117 +40.249 +23,2 +5.000 +6.842 +8.123 +9.188 +10.123 +17.117 +41.249 +43,2 +6.130 +8.318 +9.769 +10.944 +11.957 +19.245 +43.545 +63,2 +7.275 +9.912 +11.630 +13.000 +14.165 +22.174 +47.054 +83,2 +8.424 +11.547 +13.576 +15.184 +16.544 +25.625 +51.662 +103,2 +9.576 +13.201 +15.561 +17.431 +19.010 +29.414 +57.240 +123,2 +10.728 +14.866 +17.568 +19.713 +21.527 +33.424 +63.658 +143,2 +11.882 +16.536 +19.589 +22.018 +24.074 +37.584 +70.792 +Figure 3: The ϵs,L +ϵ1,2 +of the X(3). (b): the ϵs,L +ϵ1,2 +of the X(5) both at constant angular momenta, +L = 0...10 are plotted against the variation parameter, β0. +8 + +1.8 +X(3) +1.20- +x(5) +1.6 +L=10. +1.15- +1.4- +L=8 +es,L1.10 +8 +1,2 1.2 +E1,2 +=4 +1.05- +1.0 - +L=2 +O +0.8- +1.00 +=2 +:0 +6 +10 +2 +3 +6 +o +9 +(a) +10 +Po +(b) +PoFigure 4: (a) The comparison in the R4/2 of the X(3) and X(5) for β0 = ∞ and for different values of +the β0,max labelled as X(3)−var and X(5)−var respectively, peculiar to each angular momentum. (b): the +comparison in the R0/2 of the X(3) and X(5) for β0 = ∞ and for different values of the β0,max labelled as +X(3)−var and X(5)−var respectively, peculiar to each angular momentum. +Figure 5: (a) and (b) The visual plots of the potentials correspond to R4/2 and R0/2 respectively. +The values of β0 used correspond to X(3)-var and X(5)-var in the gsb and quasi-β1 bands. +9 + +70 +35 ++X(3)-var +60 +30 +X(5)-var +x(3)-var +50 +25 +(gsb) ++x(5)-var +X(3)- βo = 00 +20 +40 +X(3)-βo = 00 +R +R +15 +→X(5)-Po = 00 +30 +10 +20 +5 +10 +0 +0 +0 +4 +8 +10 +12 +14 +16 +0 +2 +4 +6 +8 +10 +12 +14 +2 +6 +16 +7 +(b) +(a) +7X (3) +x(5) +X (3) + X (5) +F00S +gsb +1-005 +β1 +250- +00 +200 +300- +V(β) 150 +V(β) +200 +100- +100- +50- +(a) +0.2 +03 +(q) +0.1 +to +0.5 +0.1 +0.2 +to +0.5 +β +βFigure 6: (a) and (b) present the RL/2 ratios for the ground state and the quasi-β1 bands of the +X(3) model of inverse square potential respectively, at different values of β0 compared with X(3)- +IW and 162Dy. (c): the RL/2 ratios for the quasi-β2 bands of the X(3) model of inverse square +potential at different values of β0 compared with X(3)-IW [1]. It appears that the gsb solutions of +X(3) at β0 = ∞ lie on the experimental data of 162Dy, which is a typical SU(3) candidate. The +available data on the first exited state lie very close to one another. +Figure 7: (a) and (b) present the RL/2 ratios for the ground state and the quasi-β1 bands of the +X(3) and the X(5) models of inverse square potentials respectively, obtained at different values of +β0,max, labeled X(3)-var and X(5)-var, are compared with the 172−180Os chain. +10 + +55 +35 +50 +O βo= 0 +×βo = 2 +O-βo= 0 +30 +X-βo= 2 +45 +→βo = 3 +→βo = 15 +→βo= 3 +βo= 15 +162 DY +40 +-X(3)-IW +10-βg=8 +25 +_162 Dy +35 +X(3)-IW +(qs) +30 +20 +RL/2 +25 +RL/2 +15 +20 +15 +10 +10 +5 +5 +(b) +0 +(a) 0 +0 +2 +4 +6 +8 +10 +12 +14 +0 +2 +4 +8 +10 +12 +14 +6 +75 +70 +O-βo= 0 +*-βo= 2 +65 +βo= 3 +60 +-βo= 15 +55 +βo= 8 +-X(3)-IW +45 +40 +z/7 +35 +R +30 +25 +20 +15 +10 +5 +(c) +0 +0 +2 +4 +6 +10 +12 +14 +825 +30 +72 +1760s +172 +1760s +25 +20 +180 +Os +1780s +b +Os +20 +β +←x(3)-var +O-X(5)-var +L/2 +-x(3)-var +-x(5)-var +10 +R +R +10 +5 +(a) +(b) 0 +0 +6 +8 +10 +12 +14 +0 +2 +6 +8 +10 +LFigure 8: The Neutron-β0 distribution is employed to show the relative positions of 104−108Ru, +120−126Xe, 184−188Pt and 172−180Os along their common chain. +Figure 9: The B(E2) transition rates of the X(3) normalized to the B(E2 : 21,0 → 01,0) = 100 +units within: (a) the ground state bands at β0 = 0, 1, 2, ∞ and B(E2)-var compared with the +X(3)-IW [1], X(5) experimental data [34] and 158Gd [35], which is a typical SU(3) candidate. +(b): the β1 state bands at β0 = 0, 1, 2, ∞ and B(E2)-var compared with the X(3)-IW [1] and +158Gd. (c): the β2 state bands at β0 = 0, 1, 2, ∞ and B(E2)-var compared with the X(3)-IW [1]. +[Note:-IW denotes infinite well potential.] +11 + +120 +188Pt +110 +186 Pt +184Pt +100 +180Os +178 0s +1760s +90 +N +126 +80 +Xe +124 Xe +122 Xe +70 +108Ru +120 +Xe +60 +104Ru +106Ru +50 +0 +2 +4 +6 +8 +10 +12 +14 +βo1200 +1200 +-βo= 0 +→βo = 1 +-βo= 0 +→βo= 1 +→βo = 2 +×βo=8 +1000 +1000 +→βo = 2 +￥β=8 +-* B(E2)-var +X(3)-IW +米一 +B(E2)-var +- X(3)-IW +800 +800 + X(5)-Exp +600 +600 +400 +2 +2 +400 +E +E +200 +B +200 +(b) +(a) +0 +2 +4 +6 +8 +10 +12 +0 +0 +2 +4 +6 +8 +10 +12 +1400 +βo= 0 +←βo= 1 +1200 +→ β = 2 +￥β= 8 +1000 +* B(E2)-var +-0-. X(3)-IW +800 +600 +2 +400 +200 +B +(c) +0 +0 +2 +4 +6 +8 +10 +12Table 5: The RL/2 ratios, defined in Eq.(15), for the X(3) version of inverse square potential, +labelled X(3)-var, calculated at different values of β0,max, are compared with the X(3)-IW [1]. +[Note: IW denotes infinite well potential]. +Ls,nβ +β0,max +X(3)-var +X(3)-IW +gsb +01,0 +β0 +0.000 +0.000 +21,0 +β0 +1.000 +1.000 +41,0 +0.844 +2.440 +2.440 +61,0 +1.576 +4.244 +4.230 +81,0 +2.033 +6.383 +6.350 +101,0 +2.143 +8.666 +8.780 +121,0 +2.695 +11.421 +11.520 +141,0 +3.643 +14.573 +14.570 +quasi-β1 +02,1 +0.815 +2.703 +2.870 +22,1 +2.101 +4.619 +4.830 +42,1 +3.729 +7.255 +7.370 +62,1 +5.213 +10.327 +10.290 +82,1 +6.098 +13.493 +13.570 +102,1 +6.855 +16.908 +17.180 +122,1 +8.106 +21.009 +21.140 +quasi-β2 +03,2 +2.524 +7.701 +7.650 +23,2 +4.497 +10.553 +10.560 +43,2 +6.523 +14.088 +14.190 +63,2 +8.567 +18.172 +18.220 +83,2 +10.438 +22.613 +22.620 +103,2 +11.932 +25.999 +- +123,2 +13.011 +28.928 +- +12 + +Table 6: The spectra ratios for the X(3) version of inverse square potential are compared with +the experimental data. The values of the β0 and the quality factor, σ, used during the fittings are +recorded. +Ls,nβ +102Mo +102Mo +104Ru +104Ru +106Ru +106Ru +108Ru +108Ru +120Xe +120Xe +122Xe +122Xe +Exp +Theor +Exp +Theor +Exp +Theor +Exp +Theor +Exp +Theor +Exp +gsb +41,0 +2.510 +2.566 +2.480 +2.468 +2.660 +2.662 +2.750 +2.623 +2.470 +2.522 +2.500 +2.571 +61,0 +4.480 +4.483 +4.350 +4.299 +4.800 +4.805 +5.120 +5.102 +4.330 +4.263 +4.430 +4.500 +81,0 +6.810 +6.703 +6.480 +6.488 +7.310 +7.199 +8.020 +7.999 +6.510 +6.361 +6.690 +6.759 +101,0 +9.410 +8.989 +8.690 +8.702 +10.020 +9.920 +11.310 +11.495 +8.900 +8.497 +9.180 +9.216 +121,0 +- +11.498 +- +11.878 +- +12.334 +- +12.879 +- +10.362 +- +11.985 +141,0 +- +13.302 +- +14.522 +- +15.073 +- +15.591 +- +12.640 +- +14.759 +β1 +02,1 +2.350 +3.009 +- +2.569 +3.670 +3.678 +- +4.462 +2.820 +3.000 +3.470 +3.492 +22,1 +3.860 +4.301 +4.230 +4.233 +- +5.127 +- +5.902 +3.950 +4.203 +4.510 +4.608 +42,1 +- +6.289 +5.810 +5.921 +- +7.443 +- +8.285 +5.310 +5.899 +- +6.792 +62,1 +- +8.691 +- +8.009 +- +10.001 +- +11.099 +- +7.958 +- +9.172 +82,1 +- +11.319 +- +11.101 +- +12.519 +- +14.532 +- +10.739 +- +11.829 +β2 +03,2 +- +7.437 +- +6.589 +- +8.388 +- +10.099 +- +6.664 +- +7.722 +23,2 +- +9.000 +- +8.202 +- +10.200 +- +11.811 +- +8.007 +- +9.402 +43,2 +- +11.009 +- +10.341 +- +12.049 +- +12.972 +- +10.229 +- +11.538 +β0 +1.464 +0.963 +- +2.121 +1.830 +1.221 +1.494 +σ +0.569 +0.310 +- +0.422 +0.276 +0.481 +0.295 +124Xe +124Xe +126Xe +126Xe +148Nd +148Nd +184Pt +184Pt +186Pt +186Pt +188Pt +188Pt +Exp +Theor +Exp +Theor +Exp +Theor +Exp +Theor +Exp +Theor +Exp +Theor +gsb +41,0 +2.480 +2.498 +2.420 +2.463 +2.490 +2.500 +2.670 +2.721 +2.560 +2.567 +2.530 +2.890 +61,0 +4.370 +4.418 +4.210 +4.291 +4.240 +4.187 +4.900 +5.001 +4.580 +4.399 +4.460 +4.503 +81,0 +6.580 +6.596 +6.270 +6.325 +6.150 +6.007 +7.550 +7.679 +7.010 +6.768 +6.710 +6.666 +101,0 +8.960 +8.726 +8.640 +8.382 +8.190 +7.986 +10.470 +10.585 +9.700 +8.863 +9.180 +8.969 +121,0 +- +11.998 +- +10.479 +- +9.769 +- +12.807 +- +11.999 +- +11.378 +141,0 +- +15.079 +- +12.681 +- +11.246 +- +15.367 +- +14.875 +- +14.242 +β1 +02,1 +3.580 +3.402 +3.380 +3.201 +3.040 +3.082 +3.020 +3.546 +2.460 +2.798 +3.010 +3.209 +22,1 +4.600 +4.498 +4.320 +4.241 +3.880 +4.006 +5.180 +5.452 +4.170 +4.381 +4.200 +4.397 +42,1 +5.690 +6.289 +5.250 +5.862 +5.320 +5.589 +7.570 +7.943 +6.380 +6.599 +- +6.583 +62,1 +- +8.564 +- +7.828 +7.120 +7.411 +11.040 +11.157 +8.360 +8.581 +- +8.603 +82,1 +- +11.900 +- +10.062 +- +9.752 +- +14.601 +- +12.007 +- +12.100 +β2 +03,2 +- +7.334 +- +6.759 +- +6.249 +- +10.122 +- +7.389 +- +7.603 +23,2 +- +8.888 +- +8.009 +- +7.442 +- +11.900 +- +8.942 +- +9.162 +43,2 +- +10.022 +- +9.287 +- +9.051 +- +12.998 +- +10.208 +- +10.441 +β0 +1.101 +- +0.949 +1.111 +2.639 +1.469 +4.950 +σ +0.279 +- +0.299 +0.347 +0.475 +0.600 +0.729 +13 + +Table 7: The B(E2) transition rates of the X(3) model at β0 = 0, 1, 2, ∞ and its values obtained +at β0,max peculiar to each angular momentum, normalized to the B(E2; 21,0 → 01,0) = 100 units +are compared with the X(3)-IW model [1] and with the experimental data of X(5) [34]. [Note: +-IW denotes infinite well potential.] +L(i) +s,nβ +L(f) +s,nβ +β0 = 0 +β0 = 1 +β0 = 2 +β0 = ∞ +β(i) +0,max → β(f) +0,max +B(E2) − var +X(3)-IW +176Os-Exp +21,0 +01,0 +100.000 +100.000 +100.000 +100.000 +β0 → β0 +100.000 +100.00 +100.00 +41,0 +21,0 +237.513 +190.935 +178.005 +143.992 +0.844 → β0 +189.495 +189.90 +193.00 +61,0 +41,0 +380.702 +286.006 +270.996 +167.292 +1.576 → 0.844 +250.995 +248.90 +267.00 +81,0 +61,0 +523.695 +384.599 +369.090 +185.328 +2.033 → 1.576 +293.038 +291.40 +297.00 +101,0 +81,0 +667.003 +486.036 +469.991 +202.099 +2.143 → 2.033 +324.599 +323.80 +352.50 +121,0 +101,0 +810.954 +587.658 +559.744 +229.986 +2.695 → 2.143 +350.710 +349.50 +- +141,0 +121,0 +954.746 +690.364 +671.484 +253.007 +3.643 → 2.695 +371.992 +370.70 +- +22,1 +02,1 +166.813 +160.292 +152.428 +69.929 +2.011 → 0.815 +78.922 +80.60 +- +42,1 +22,1 +320.619 +260.000 +243.186 +123.888 +3.729 → 2.101 +139.471 +140.10 +- +62,1 +42,1 +470.001 +355.240 +343.376 +156.031 +5.213 → 3.729 +181.730 +182.40 +- +82,1 +62,1 +617.075 +456.792 +439.962 +189.542 +6.098 → 5.213 +213.899 +215.50 +- +102,1 +82,1 +763.927 +557.317 +542.129 +215.763 +6.855 → 6.098 +242.026 +242.40 +- +122,1 +102,1 +899.983 +656.999 +639.677 +233.499 +8.106 → 6.855 +268.543 +265.10 +- +142,1 +122,1 +1009.079 +759.642 +736.660 +251.643 +9.441 → 8.106 +281.320 +- +- +23,2 +03,2 +233.504 +221.942 +209.888 +56.684 +4.497 → 2.524 +72.090 +73.50 +- +43,2 +23,2 +401.982 +327.461 +311.072 +82.831 +6.523 → 4.497 +118.990 +120.50 +- +63,2 +43,2 +559.801 +422.880 +408.564 +116.085 +8.567 → 6.523 +154.892 +154.20 +- +83,2 +63,2 +708.989 +523.436 +512.997 +139.859 +10.438 → 8.567 +183.019 +181.20 +- +103,2 +83,2 +858.095 +624.555 +609.096 +166.646 +11.932 → 10.438 +202.222 +- +- +123,2 +103,2 +1003.933 +727.909 +715.990 +182.910 +13.011 → 11.932 +218.753 +- +- +143,2 +123,2 +1151.239 +832.003 +819.115 +202.421 +14.629 → 13.011 +229.986 +- +- +14 + +Secondly, the exact relationship between the νX(3) and the νX(5) stated in Eq.(28) does not reflect +in the exact comparison of their energy levels. That is, it can be inferred from the results that +ϵX(3)(β0 = c + 2) ̸= ϵX(5)(β0 = c), +(29) +because the total energy of the X(5) contains the γ-part solutions. However, the relation +ϵgs,L = 2 + ϵβ1,L = 4 + ϵβ2,L, +(30) +holds in all the levels for both X(3) and the β-part of X(5): this third remark is shown in the +Table 2. +Another significant remark is such that, the values of ν, for the case of X(5) at L = 2, correspond +to those of X(3), at L = 0. This is shown in Table 1. and the visual comparison is shown with +the lines in Figure 1(b). Analytically, the behaviour of the energies of the X(5) and the X(3) +at constant value of variation parameter, β0, is shown in the Figure 1(a). The critical orders, +ν(L, β0), of the X(5) and that of the X(3), which define their energy levels, are plotted against the +variation parameter, β0, at constant angular momenta and shown in the Figure 1(b): it is shown, +with the numerical values of ν, in the Table 1., that +νX(5)(L = 0) = νX(3)(L = 2) +∀ +β0. +(31) +The derivatives of ν with respect to the β0 are shown in Figures 2(a) and 2(b). The first and the +second derivatives are carried out in order to show the stationary properties of β0 and the values +of β0 at which the energy is minimum. +The variation of the ratio ϵs,L +ϵ1,2 +with respect to the variation parameter, β0, for both X(3) and +X(5) are respectively shown in the Figures 3(a) and 3(b). For all values of β0, its values increase +at L = 0, are constant at L = 2, that is ϵs,L +ϵ1,2 +=1 and decrease at L > 2 . +The ground state bands (gsb) are defined with s = 1; +nβ = 0. The quasi-β1 bands and the +quasi-β2 bands are defined by s = 2; +nβ = 1 and s = 3; +nβ = 2 respectively. The γ bands do +not exist for X(3) model because, γ0 = 0. The increase in the angular momentum, L, at constant +value of β0, increases the energies, in all energy levels. Also, at constant values of the angular +momentum, the increase in the β0 increases the energy levels. The Table 2. shows the numerical +solutions of Eq.(13) obtained for the ground states and the β-bands at β0 = 2, 3, 4 and at β0 = 15. +The Figure 4(a) shows the comparison, in the R4/2, of the X(3) with X(5) at β0 = ∞ and at +β0,max unique to each angular momentum, labelled as X(3)−var and X(5)−var respectively. The +comparison in the R0/2 of the X(3) with X(5), at β0 = ∞ and at different values of the β0,max +peculiar to each angular momentum, labelled as X(3)−var and X(5)−var respectively is shown in +Figure 4(b). +The ‘nature’ of critical point symmetry transitions for different isotopes, constrained to one- +parameter potentials, can be investigated using a variational technique. This technique was used +in ref. [11] to retrieve the U(5) and O(6) ground state bands from the E(5) within the domain +of the one-parameter inverse square potential. The technique has also been used in ref. [12] and +employed in ref. [16] to construct ‘image’ of the X(5) critical symmetry and to construct the Z(5) +critical symmetry respectively. The forward variation of the ‘control parameter’, β0, causes the +nuclei transition from X(5) to SU(3) (i.e. X(5) −→ SU(3) transition symmetry). The nature of +the critical symmetry or the nuclear shape phase region under investigation predicts the directions +of the variation: whether forward variation or backward variation, and also depends on the poten- +tial’s boundary conditions. The rate of change of RL/2(β0) is maximized for each L by using this +approach. As shown in Table 3., each angular momentum is considered and treated separately in +terms of the variation parameter, β0, as the critical values of RL/2 are distinct. Each value of β0 +implies a distinct potential with which the energy is maximized. The method is comparable to +the “normal” variational principle used in some quantum books, in which trial wave functions are +chosen and energy is minimized. +15 + +The comparisons of the ground state spectra ratios, defined in Eq.(15) with the X(5) model [11], +at different values of β0 corresponds to the potentials are displayed in Figures 5(a) and 5(b) and +also shown in Table 3: the visual comparison is shown in Figures 6(a), 6(b) and 6(c). It can +be observed that the solutions of X(3)(β0 = ∞) ≈ X(5)(β0 = ∞). The forward variation of β0 +shifts the solutions to X(3). The solutions leave X(3) and approach SU(3) as β0 tends to ∞. +The available experimental data of 162Dy [17], which is a typical SU(3) candidate are placed for +comparison in Figure 6(a) and Figure 6(b). This is another remark that isotopes which have X(3) +signatures must lie between U(5) −→ SU(3) symmetrical plane. +The two other important relations that can be deduced from the comparison are: +RL/2(gsb) = 2 + RL/2(β1) = 4 + RL/2(β2), +(32) +at β0 = 0 as shown numerically in the Table 3. and Table 4. This is an observable effect or a +signature from Eq.(30) while the effect of Eq.(28) is observed in the spectral ratios of X(3) and +X(5) such that +RX(3) +L/2 (β0 = c + 2) = RX(5) +L/2 (β0 = c) : +c = 0, 1, 2, ... +(33) +In order to obtained the exact solutions of RL/2 ratios rather than vary β0, the technique of +optimizing β0 employed in refs. [11,12,15,16] and others has been used to obtained the solutions +of RL/2 at certain values of the β0 peculiar to the angular momenta. These special values of β0 are +labelled β0,max and they produce exact solutions labelled X(3)-var, shown in Table 5. The values +obtained at different values of β0,max, are compared with X(3)-IW solutions. For all β0,max, 00,0 +and 20,0 levels yield 0.000 and 1.000 respectively. β0,max increases with increase in the angular +momentum and its values are obtained at the points where the increases in β0 become steep. +d +dβ0 +RL/2|β0=max is achieved via a numerical procedure as +d2 +dβ2 +0 +RL/2 vanished. The RL/2 ratios +for the ground state and the quasi-β1 bands of the X(3) and the X(5) models of inverse square +potentials obtained at different values of β0,max, labeled X(3)-var and X(5)-var, are compared with +the experimental data of 172,176,178,180Os [18-21] chain, as shown in Figures 7(a) and 7(b). The +ground state solutions of the X(3) for L = 0 up to L = 10 are in good agreement with 172Os while +those of X(5) are seen lying closer to 176Os than 178Os and 180Os: the generalized comparison is +moderate in the first excited state. This suggests that 172Os is a good candidate for X(3) model +while 176Os shows a signature of X(5) model. +The RL/2 theoretical predictions of the X(3) model are compared with the experimental data of +some selected isotopes: 102Mo [22], 104−108Ru [23-25], 120−126Xe [26-29], 148Nd [30] and 184−188Pt +[31-33] as shown in Table 6. Each energy level is normalized to the particular 20,0 state. The energy +obtained in Eq.(13) is fitted with the experimental energy of each of the isotopes considered. The +equivalent values of the β0 for the isotopes are recorded. The quality factor, σ, used is obtained +from +σ = +��m +i [(Rs,L)Exp +i +− (Rs,L)Theor +i +]2 +m − 1 +, +(34) +where m is the number of available experimental states, (Rs,L)Exp +i +and (Rs,L)Theor +i +represent the +experimental and the theoretical spectral ratios of the ith levels normalized to the ground state +with L = 2, s = 1 and nβ = 0 respectively. +Against the neutron numbers, N, of the chains of the isotopes: 104−108Ru, 120−126Xe, 184−188Pt, +172−180Os, considered for the comparison, the neutron-β0 distribution, showing the relative posi- +tions of the isotopes, is shown in the Figure 8. +The comparison in the ground state, the quasi-β1 bands and the quasi-β2 bands of the B(E2) +transition probabilities at β0 = 0, 1, 2 and β0 = ∞, normalized to the B(E2 : 21,0 → 01,0) = 100 +units with the X(3)-IW [1] and experimental data on X(5) [34] are presented in the Table 7. The +values of β0,max peculiar to each angular momentum, obtained from the optimization of β0, in +16 + +Table 5., are employed to compute the optimized B(E2) transition probabilities, labelled B(E2)- +var. The visuals of these comparisons are shown in the Figures 9(a), 9(b) and 9(c). In order to +show the nature of the solutions along the X(5) −→ SU(3) symmetry region, the experimental +data on the 158Gd [35], which is a typical SU(3) candidate, are placed for comparison in Figures +9(a) and 9(b): the solutions at β0 → ∞ are seen lying close to the 158Gd [35]. The values of the +B(E2) transition probabilities decrease as the variation parameter, β0, increases: they increase as +the angular momentum increases. The forward variation, as the β0 increases, pushes the solutions +to X(5) and the solutions tend to the SU(3) as β0 tends to ∞. +5 +Conclusion +The X(3) solutions of the Bohr Hamiltonian are obtained by solving the radial function of the +Hamiltonian with an inverse square potential with the aid of MAPLE software. Analytically, an +expression for the energy levels is determined from the zeros of the Bessel functions. Through the +use of the variational approach and the optimization procedure, the spectra ratios and the B(E2) +transition probabilities are computed. The analytical solutions of the X(3) model are compared +with the X(5) model of the inverse square potentials. It is worth noting that, X(3) model is +another “window” through which X(5) and SU(3) “pictures” can be seen: X(3) lies between U(5) +and SU(3), hence, X(5) lies between X(3) and SU(3). It has been shown via variational procedure, +that the solutions shift to X(5) from X(3) and approach SU(3) as the variation parameter shifts +forward. +The theoretical predictions on RL/2 and B(E2) with the experimental data for some selected +isotopes are found to be proficient in the gsb and moderate in other levels. This is shown as the +theoretical deviations from the experiments are quite small. +The same manner in which the Davidson potential is employed in ref. [4], the employment of the +one parameter-dependent inverse square potential in the form of Eq.(1), its properties, is efficient +in the variational procedure. Eq.(1) is also a good choice of potential which can be employed for +the description of the nuclei transition at the critical points. For the comparison of X(3) and +X(5) models of Bohr Hamiltonian, with the same formalism employed in this work, it is expected +that Equations (28), (29), (30), (31), (32) and (33) should hold in any one-parameter-dependent +potential domain such as Kratzer potential, Davidson potential and others. +Data availability statement +All the sources of data included in this article for comparison purpose, are cited and referenced +accordingly, in the article. +Funding Information +No funding of any form is received for the course of this work. +References +[1] Bonatsos, D., Lenis, D., Petrellis, D. and Terziev, P.A. and Yigitoglu, I. (2006). +Physics +Letters B, 632, 238-242. doi: http://dx.doi.org/10.1016/j.physletb.2005.10.060 +[2] Budaca, R. (2014). Physics Letters B, 739, 56-61. +doi: http://dx.doi.org/10.1016/j.physletb.2014.10.031 +[3] Alimohammadi, M. and Hassanabadi, H. (2017). 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Data Sheets, 95(2), 387-541. doi: https://doi.org/10.1006/ndsh.2002.0005 +[34] Melon, B. (2011). Investigation of the X(5)-Structure in 176Os Using Absolute Transition +Probabilities. PhD thesis, Universit¨at zu K¨oln. URL: http://kups.ub.uni-koeln.de/id/eprint/4370 +[35] Helmer, R.G. (2004). Nucl. Data Sheets, 101(3), 325-519. doi: https://doi.org/10.1016/j.nds.2004.02.001 +19 + diff --git a/7tAyT4oBgHgl3EQfpvj9/content/tmp_files/load_file.txt b/7tAyT4oBgHgl3EQfpvj9/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b6675c76f5c23fb71d835f9f500b8d833d1f936a --- /dev/null +++ b/7tAyT4oBgHgl3EQfpvj9/content/tmp_files/load_file.txt @@ -0,0 +1,1812 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf,len=1811 +page_content='Analytical comparison between X(3) and X(5) models of the Bohr Hamiltonian K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' Ajulo1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' Oyewumi2 1,2University of Ilorin, Ilorin, Nigeria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' Abstract Via the inverse square potential, the solutions of the X(3) model which is a γ-rigid form of the X(5) critical point symmetry have been achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' The paper presents X(3), through the variational technique, as another “window” through which the “pictures” of X(5) −→ SU(3) symmetry region can be seen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' The analytical solutions of the X(3) are compared with the solutions of the X(5) model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' Some new and unique equations connecting the two models in the: critical order, energy bands, spectra ratios, RL/2, and B(E2) transitional probabilities are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' These equations should hold in other potentials with one-parameter such as Kratzer potential, Davidson potential etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' The spectra ratios and the B(E2) transitional probabilities are optimized via the optimization procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' The experimental data of some selected isotopes are placed accordingly for the theoretical predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' The deviations from the experiments are found to be quite small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' Keywords: Bohr Hamiltonian, X(3), X(5), variation technique, optimization procedure, β- variable, γ-rigid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' 1 Introduction X(3) which has been presented in [1-4] is said to be an exactly separable γ-rigid form of the X(5) critical point symmetry [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' The X(3) model is defined by the collective coordinate β and two Euler angles since the γ is assumed to be zero unlike the case of X(5), where γ is varied around γ0 = 0 value in the harmonic oscillator potential [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' This implies that, only three variables: β and θi are involved in the X(3) model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' An exact separation of the β variable from the Euler angles is quite easily achievable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' In the Bohr Hamiltonian model [6-9], X(5) critical point symmetry is one of the two critical point symmetries: it is a phase transition of the first order shape, which were originally proposed in the works of Iachello [10], while E(5) is the phase transition of the second order shape [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' In the present work, the nuclei are taken to be γ-rigid, with the axially symmetric prolate shape obtained at γ0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' The work presents the usefulness of a one-sided bound inverse square potential with one parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' The one-parameter inverse square potential chosen is of the form V (β) = � � � � � β0 β2 , if 0 ≤ β ≤ β0, ∞, if β > β0, (1) where β0 is a variation parameter that changes the signatures of the nuclei, as it changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' It is expected that the solutions should shift forward as the β0 shifts forward and solutions should shift backward as the β0 shifts backward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' A typical inverse square potential is bound on the left and unbound on the right, and it has a minimum at some positive values of β0 that forces the particles to infinity as β0 → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' As a result, the particle’s energy states is one-sided, with energies escaping through the unbound side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' The work is structured as follows: Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' presents the methodology and the solutions of X(3) model via inverse square potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' These solutions are: the wave functions, the normalization constants and the energy eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' The B(E2) transition rates are presented in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' The analytical results, the numerical results, their applications in certain isotopes are presented and discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' The work is concluded and summarized in the Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' 1E-Mail: 19-68eo001@students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='unilorin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='ng 2E-Mail: kjoyewumi66@unilorin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='ng 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='00533v1 [nucl-th] 2 Jan 2023 2 Methodology of X(3) model with the inverse square potential In the X(3) model, the Bohr Hamiltonian operator is written as [1,2] ˆH = − ℏ2 2B � 1 β2 ∂ ∂β β2 ∂ ∂β + 1 3β2 � 1 sin θ ∂ ∂θ sin θ ∂ ∂θ + 1 sin2 θ ∂2 ∂φ2 � − V (β) � , (2) where the term, 1 sin θ ∂ ∂θ sin θ ∂ ∂θ + 1 sin2 θ ∂2 ∂φ2 , (3) inside the bracket, represents the angular part of the Laplacian [1,2] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' B, β and V (β) are respec- tively the mass parameter, the collective coordinate and the β-dependent potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' The wave equation of the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' (2) is ˆHΨ(β, θ, φ) = EΨ(β, θ, φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' (4) By the usual method of separation of variable employed in some quantum texts, Ψ(β, θ, φ) = χ(β)YL,M(θ, φ), (5) where YL,M(θ, φ) is the spherical harmonics and χ(β) is the radial part of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='(4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' The separated angular part obtained reads [1,2] − � 1 sin θ ∂ ∂θ sin θ ∂ ∂θ + 1 sin2 θ ∂2 ∂φ2 � YL,M(θ, φ) = L(L + 1)YL,M(θ, φ), (6) where L is the angular momentum quantum number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' The simplified form of the radial part equation [1,2] , � 1 β2 d dβ β2 d dβ − L(L + 1) 3β2 + 2B ℏ2 [E − V (β)] � χ(β) = 0 (7) reads d2 dβ2 χ(β) + 2 β d dβ χ(β) − L(L + 1) 3β2 χ(β) − [v(β) − ϵ]χ(β) = 0, (8) where ϵ = 2B ℏ2 E and v(β) = 2B ℏ2 V (β) are the reduced energy and reduced potential respectively [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='1 Determination of the wave functions By substituting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' (1) for v(β) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' (8) and solving the simplified equation using MAPLE soft- ware, the eigenfunctions obtained read χs,ν,L,(β) = β−1/2 �C1,LJν(√ϵβ) + C2,LYν(√ϵβ) � , (9) where C1,L and C2,L are the normalization constants associated with the Bessel functions of the first kind, Jν, and second kind, Yν, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' In the domain of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' (1), the critical order associated with the X(3) model in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' (9) is νX(3) = � L 3 (L + 1) + β0 + 1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' (10) If a boundary condition χs,ν,L(β0) = 0 is considered, then C2,L,nYν(√ϵβ) vanishes and the wave functions become χs,ν,L(β) = β−1/2 �C1,LJν(√ϵβ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' (11) 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='2 Determination of the energy eigenvalues and the spectral ratio The procedure for finding the eigenvalues is written in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' If the first condition of the listed procedure is considered, then the acceptable expression for the energy eigenvalues is written as: Es,L,nβ = ℏ2 2B k2 s,ν,nβ, k2 s,ν,nβ = ϵs,ν,nβ, k2 s,ν,nβ = xnβ,s,ν β0 , (12) where s = nβ +1, xs,ν,nβ is the s-th zeros of the Bessel function of order ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' The energy eigenvalues of the β-part in ℏω = 1 unit reads: ϵs,L,nβ = 2nβ + 1 + νX(3) = 2nβ + 1 + � L 3 (L + 1) + β0 + 1 4 : nβ = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' (13) For the X(3) model, the ground state energy levels are defined with s = 1, the quasi-β1 levels are defined with s = 2 and the quasi-β2 levels are defined with s = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' L = 0, 2, 4, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' There exist no γ-bands in the X(3) model because, γ0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' (13) is the similar to the energy eigenvalues obtained in the β-part of X(5) model [12], the difference is observed in their critical orders where νX(5) = � L 3 (L + 1) + β0 + 9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' (14) Since s = nβ + 1, ϵs,L,nβ can be reduced to ϵs,L, then the spectra ratios can be written as RL/2 = ϵs,L − ϵ1,0 ϵ1,2 − ϵ1,0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' (15) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='3 Determination of the normalization constants and the complete wave func- tions The normalization condition for the Hamiltonian operator in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' (2) is written as [1,2] � β0 0 β2 | χs,ν,L,nβ(β) |2 dβ = 1, (16) such that | χs,ν,L,nβ(β) |2→ 0 for β → 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' | χs,ν,L,nβ(β) |2 β2 → 0 for β → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' (17) If these conditions are satisfied, then � β0 0 β2 | χs,ν,L,nβ(β) |2 dβ < β0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' Using the identity [13-14] Jν(√ϵβ)Jν(√ϵβ) = ∞ � nβ=0 �1 2 √ϵβ �2ν+2nβ (2ν + nβ + 1)nβ nβ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' [Γ(ν + nβ + 1)]2 (18) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' (16), the simplified normalization constants read C1,L,nβ = � ���� � nβ=0,1,2,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' (η)nβ �ks,ν,nβ 2 �ξ−2 β(ξ) 0 nβ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' ξ � Γ �ξ 2 ��2 � ���� −1/2 , (19) where ξ = 2ν + 2nβ + 2, η = 2ν + nβ + 1 and (η)nβ = η(η + 1)(η + 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='(η + nβ − 1), (20) 3 with (η)0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' Hence, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' (11) becomes χs,ν,L,nβ(β) = � ���� � nβ=0,1,2,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' (η)nβ �ks,ν,nβ 2 �ξ−2 β(ξ) 0 nβ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' ξ � Γ �ξ 2 ��2 � ���� −1/2 β−1/2Jν(√ϵβ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' (21) 3 B(E2) transition rates The electric quadrupole operator is written as [1,2] T E2 µ = tβ � D(2) µ,0(θi) cos γ + 1 √ 2 � D(2) µ,2(θi) + D(2) µ,−2(θi) � sin γ � , (22) where D(θi) are the Wigner functions of the Euler angle and t is known as a scale factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' For γ0 = 0, T E2 µ = tβ �4π 5 Y2µ(θ, φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' (23) The B(E2) [1,2,5,15] is written as B(E2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' sL −→ s′L′) = 1 2sL + 1| � s′L′||T E2||sL � |2, (24) = 2s′L′ + 1 2sL + 1 B(E2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' s′L′ −→ sL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' (25) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' (24) or Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' (25) has been solved in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' [1] as: B(E2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' sL −→ s′L′) = t2 � CL′0 L0,20 �2 I2 sL;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='s′L′, (26) where the coefficients, CL′0 L0,20 are the Clebsch-Gordan coefficients, and IsL;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='s′L′ = � β0 0 βχs,ν,L,nβ(β)χs′,ν′,L′,n′ β(β)β2dβ, (27) are the integrals over β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' 4 Numerical results, analytical results, applications and discus- sion Some important solutions for the collective model of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' (2) are the energy levels, the spectra ratios and the B(E2) transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' Their theoretical predictions are important when energy spectra are assigned to the states for which experimental data are not available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' The numerical calculations, the analytical comparisons and how the search for the experimental realizations of the model was achieved are discussed accordingly in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' Both the X(3) and the X(5) have their critical orders, ν(L, β0), from their Bessel functions which describes their energy spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' Firstly, in the comparison of the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' (10) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' (14), it can be deduced from the numerical computation of ν, shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=', that νX(3)(β0 = c + 2) = νX(5)(β0 = c) : c = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' (28) In both cases, it increases with increase in the angular momentum, L, and with increase in the variation parameter, β0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' These effects of L and β0 in ν are also seen in the energy values of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' 4 Table 1: The comparison in the critical order, ν, of the X(5) [12], with the ν of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' ν(L) L β0 = 0 β0 = 2 β0 = 4 β0 = 6 β0 = 102 β0 = 100 X(3) X(5) X(3) X(5) X(3) X(5) X(3) X(5) X(3) X(5) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='500 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='500 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='500 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='062 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='062 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='500 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='500 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='062 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='112 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='112 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='500 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='062 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='062 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='500 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='500 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='872 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='872 2.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='627 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='627 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='776 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='747 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='913 Figure 1: (a) Comparison in the energy levels of the X(3) and X(5) models [15] at β0 = 2 from the gsb up to the quasi-β2 band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' (b): the variation of the critical order, ν, of the X(5) as a function of β0, is compared with ν of the X(3) at constant angular momenta, L = 0, 2 and L = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' 5 16 5 βo= 2 14 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='5 12 X(3) B2 4 L=4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='5 L=4 L=0 rgy 10 X(3) X(5) L=2 0X(5) 3 8 β2 X(3) X(5) X(5) β1 V 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='5 L=0 6 X(5) β1 gsb 2 4 X(5) X(3) gsb 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='5 2 X(3) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='5 0 2 4 6 8 10 12 14 0 (a) 7 (b) 0 2 4 8 10Table 2: Ground state energies, the energies of the quasi-β1 and the quasi-β2 denoted by nβ = 0, s = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' nβ = 1, s = 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' and nβ = 2, s = 3 respectively for the X(3) and X(5) symmetry [12] in ℏω = 1 unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' β0 = 2 β0 = 3 β0 = 4 β0 = 15 L nβ = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' s = 1 X(3) X(5) X(3) X(5) X(3) X(5) X(3) X(5) 0 2.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' Table 3: Comparison of the ground state spectra ratios, defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' (15), of the inverse square potential in the X(3) model at different values of the β0, compared with the X(5) [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' It can be seen that X(3)(β0 = ∞) ≈ X(5)(β0 = ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' Ls,nβ β0 = 0 β0 = 0 β0 = 2 β0 = 2 β0 = 4 β0 = 4 β0 = ∞ β0 = ∞ X(3) X(5) X(3) X(5) X(3) X(5) X(3) X(5) gsb 01,0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} 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+page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='015 log SPo 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='8 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='020- 2 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='025 (a) [b] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='2-Table 4: RL/2 ratios, defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' (15), of the quasi-β1 and quasi-β2 bands of the inverse square potential in the X(3) model at 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='584 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='792 Figure 3: The ϵs,L ϵ1,2 of the X(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' (b): the ϵs,L ϵ1,2 of the X(5) both at constant angular momenta, L = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='10 are plotted against the variation parameter, β0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' 8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='8 X(3) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='20- x(5) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='6 L=10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='15- 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='4- L=8 es,L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='10 8 1,2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='2 E1,2 =4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='05- 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='0 - L=2 O 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='8- 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='00 =2 :0 6 10 2 3 6 o 9 (a) 10 Po (b) PoFigure 4: (a) The comparison in the R4/2 of the X(3) and X(5) for β0 = ∞ and for different values of the β0,max labelled as X(3)−var and X(5)−var respectively, peculiar to each angular momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' (b): the comparison in the R0/2 of the X(3) and X(5) for β0 = ∞ and for different values of the β0,max labelled as X(3)−var and X(5)−var respectively, peculiar to each angular momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' Figure 5: (a) and (b) The visual plots of the potentials correspond to R4/2 and R0/2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' The values of β0 used correspond to X(3)-var and X(5)-var in the gsb and quasi-β1 bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' 9 70 35 +X(3)-var 60 30 X(5)-var x(3)-var 50 25 (gsb) +x(5)-var X(3)- βo = 00 20 40 X(3)-βo = 00 R R 15 →X(5)-Po = 00 30 10 20 5 10 0 0 0 4 8 10 12 14 16 0 2 4 6 8 10 12 14 2 6 16 7 (b) (a) 7X (3) x(5) X (3) X (5) F00S gsb 1-005 β1 250- 00 200 300- V(β) 150 V(β) 200 100- 100- 50- (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='2 03 (q) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='1 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='2 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='5 β βFigure 6: (a) and (b) present the RL/2 ratios for the ground state and the quasi-β1 bands of the X(3) model of inverse square potential respectively, at different values of β0 compared with X(3)- IW and 162Dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' (c): the RL/2 ratios for the quasi-β2 bands of the X(3) model of inverse square potential at different values of β0 compared with X(3)-IW [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' It appears that the gsb solutions of X(3) at β0 = ∞ lie on the experimental data of 162Dy, which is a typical SU(3) candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' The available data on the first exited state lie very close to one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' Figure 7: (a) and (b) present the RL/2 ratios for the ground state and the quasi-β1 bands of the X(3) and the X(5) models of inverse square potentials respectively, obtained at different values of β0,max, labeled X(3)-var and X(5)-var, are compared with the 172−180Os chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='55 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='35 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='O βo= 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='×βo = 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='O-βo= 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='X-βo= 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='45 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='→βo = 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='→βo = 15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='→βo= 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='βo= 15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='162 DY ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='X(3)-IW ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='10-βg=8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='_162 Dy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='35 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='X(3)-IW ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='(qs) ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='LFigure 8: The Neutron-β0 distribution is employed to show the relative positions of 104−108Ru,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' 120−126Xe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' 184−188Pt and 172−180Os along their common chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' Figure 9: The B(E2) transition rates of the X(3) normalized to the B(E2 : 21,0 → 01,0) = 100 units within: (a) the ground state bands at β0 = 0, 1, 2, ∞ and B(E2)-var compared with the X(3)-IW [1], X(5) experimental data [34] and 158Gd [35], which is a typical SU(3) candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' (b): the β1 state bands at β0 = 0, 1, 2, ∞ and B(E2)-var compared with the X(3)-IW [1] and 158Gd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' (c): the β2 state bands at β0 = 0, 1, 2, ∞ and B(E2)-var compared with the X(3)-IW [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' [Note:-IW denotes infinite well potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='] 11 120 188Pt 110 186 Pt 184Pt 100 180Os 178 0s 1760s 90 N 126 80 Xe 124 Xe 122 Xe 70 108Ru 120 Xe 60 104Ru 106Ru 50 0 2 4 6 8 10 12 14 βo1200 1200 βo= 0 →βo = 1 βo= 0 →βo= 1 →βo = 2 ×βo=8 1000 1000 →βo = 2 ￥β=8 -* B(E2)-var X(3)-IW 米一 B(E2)-var X(3)-IW 800 800 X(5)-Exp 600 600 400 2 2 400 E E 200 B 200 (b) (a) 0 2 4 6 8 10 12 0 0 2 4 6 8 10 12 1400 βo= 0 ←βo= 1 1200 → β = 2 ￥β= 8 1000 B(E2)-var 0-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' X(3)-IW 800 600 2 400 200 B (c) 0 0 2 4 6 8 10 12Table 5: The RL/2 ratios, defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' (15), for the X(3) version of inverse square potential, labelled X(3)-var, calculated at different values of β0,max, are compared with the X(3)-IW [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' [Note: IW denotes infinite well potential].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' Ls,nβ β0,max X(3)-var X(3)-IW gsb 01,0 β0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='000 21,0 β0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='000 41,0 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='327 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='290 82,1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='098 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='493 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='570 102,1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='855 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='908 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='180 122,1 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='106 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='009 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='140 quasi-β2 03,2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='524 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='701 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='650 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123,2 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='011 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='928 12 Table 6: The spectra ratios for the X(3) version of inverse square potential are compared with the experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' The values of the β0 and the quality factor, σ, used during the fittings are recorded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' Ls,nβ 102Mo 102Mo 104Ru 104Ru 106Ru 106Ru 108Ru 108Ru 120Xe 120Xe 122Xe 122Xe Exp Theor Exp Theor Exp Theor Exp Theor Exp Theor Exp gsb 41,0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='510 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='566 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='480 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='468 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='660 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='662 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='750 2.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='389 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='603 23,2 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='888 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='009 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='442 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='900 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='942 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='162 43,2 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='022 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='287 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='051 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='998 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='208 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='441 β0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='101 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='949 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='111 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='639 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='469 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='950 σ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='279 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='299 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='347 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='475 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='600 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='729 13 Table 7: The B(E2) transition rates of the X(3) model at β0 = 0, 1, 2, ∞ and its values obtained at β0,max peculiar to each angular momentum, normalized to the B(E2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' 21,0 → 01,0) = 100 units are compared with the X(3)-IW model [1] and with the experimental data of X(5) [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' [Note: IW denotes infinite well potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='] L(i) s,nβ L(f) s,nβ β0 = 0 β0 = 1 β0 = 2 β0 = ∞ β(i) 0,max → β(f) 0,max B(E2) − var X(3)-IW 176Os-Exp 21,0 01,0 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='000 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='000 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='000 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='000 β0 → β0 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='000 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='00 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='00 41,0 21,0 237.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='513 190.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='935 178.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='005 143.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='992 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='844 → β0 189.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='495 189.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='90 193.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='00 61,0 41,0 380.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='702 286.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='006 270.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='996 167.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='292 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='576 → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='844 250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='995 248.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='90 267.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='00 81,0 61,0 523.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='695 384.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='599 369.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='090 185.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='328 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='033 → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='576 293.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='038 291.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='40 297.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='00 101,0 81,0 667.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='003 486.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='036 469.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='991 202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='099 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='143 → 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='033 324.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='599 323.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='80 352.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='50 121,0 101,0 810.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='954 587.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='658 559.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='744 229.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='986 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='695 → 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='143 350.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='710 349.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='50 141,0 121,0 954.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='746 690.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='364 671.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='484 253.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='007 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='643 → 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='695 371.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='992 370.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='70 22,1 02,1 166.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='813 160.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='292 152.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='428 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='929 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='011 → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='815 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='922 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='646 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='932 → 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='438 202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='222 123,2 103,2 1003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='933 727.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='909 715.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='990 182.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='910 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='011 → 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='932 218.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='753 143,2 123,2 1151.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='239 832.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='003 819.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='115 202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='421 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='629 → 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='011 229.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='986 14 Secondly, the exact relationship between the νX(3) and the νX(5) stated in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' (28) does not reflect in the exact comparison of their energy levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' That is, it can be inferred from the results that ϵX(3)(β0 = c + 2) ̸= ϵX(5)(β0 = c), (29) because the total energy of the X(5) contains the γ-part solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' However, the relation ϵgs,L = 2 + ϵβ1,L = 4 + ϵβ2,L, (30) holds in all the levels for both X(3) and the β-part of X(5): this third remark is shown in the Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' Another significant remark is such that, the values of ν, for the case of X(5) at L = 2, correspond to those of X(3), at L = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' This is shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' and the visual comparison is shown with the lines in Figure 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' Analytically, the behaviour of the energies of the X(5) and the X(3) at constant value of variation parameter, β0, is shown in the Figure 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' The critical orders, ν(L, β0), of the X(5) and that of the X(3), which define their energy levels, are plotted against the variation parameter, β0, at constant angular momenta and shown in the Figure 1(b): it is shown, with the numerical values of ν, in the Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=', that νX(5)(L = 0) = νX(3)(L = 2) ∀ β0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' (31) The derivatives of ν with respect to the β0 are shown in Figures 2(a) and 2(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' The first and the second derivatives are carried out in order to show the stationary properties of β0 and the values of β0 at which the energy is minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' The variation of the ratio ϵs,L ϵ1,2 with respect to the variation parameter, β0, for both X(3) and X(5) are respectively shown in the Figures 3(a) and 3(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' For all values of β0, its values increase at L = 0, are constant at L = 2, that is ϵs,L ϵ1,2 =1 and decrease at L > 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' The ground state bands (gsb) are defined with s = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' nβ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' The quasi-β1 bands and the quasi-β2 bands are defined by s = 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' nβ = 1 and s = 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' nβ = 2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' The γ bands do not exist for X(3) model because, γ0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' The increase in the angular momentum, L, at constant value of β0, increases the energies, in all energy levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' Also, at constant values of the angular momentum, the increase in the β0 increases the energy levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' The Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' shows the numerical solutions of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' (13) obtained for the ground states and the β-bands at β0 = 2, 3, 4 and at β0 = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' The Figure 4(a) shows the comparison, in the R4/2, of the X(3) with X(5) at β0 = ∞ and at β0,max unique to each angular momentum, labelled as X(3)−var and X(5)−var respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' The comparison in the R0/2 of the X(3) with X(5), at β0 = ∞ and at different values of the β0,max peculiar to each angular momentum, labelled as X(3)−var and X(5)−var respectively is shown in Figure 4(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' The ‘nature’ of critical point symmetry transitions for different isotopes, constrained to one- parameter potentials, can be investigated using a variational technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' This technique was used in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' [11] to retrieve the U(5) and O(6) ground state bands from the E(5) within the domain of the one-parameter inverse square potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' The technique has also been used in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' [12] and employed in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' [16] to construct ‘image’ of the X(5) critical symmetry and to construct the Z(5) critical symmetry respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' The forward variation of the ‘control parameter’, β0, causes the nuclei transition from X(5) to SU(3) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' X(5) −→ SU(3) transition symmetry).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' The nature of the critical symmetry or the nuclear shape phase region under investigation predicts the directions of the variation: whether forward variation or backward variation, and also depends on the poten- tial’s boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' The rate of change of RL/2(β0) is maximized for each L by using this approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' As shown in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=', each angular momentum is considered and treated separately in terms of the variation parameter, β0, as the critical values of RL/2 are distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' Each value of β0 implies a distinct potential with which the energy is maximized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' The method is comparable to the “normal” variational principle used in some quantum books, in which trial wave functions are chosen and energy is minimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' 15 The comparisons of the ground state spectra ratios, defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' (15) with the X(5) model [11], at different values of β0 corresponds to the potentials are displayed in Figures 5(a) and 5(b) and also shown in Table 3: the visual comparison is shown in Figures 6(a), 6(b) and 6(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' It can be observed that the solutions of X(3)(β0 = ∞) ≈ X(5)(β0 = ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' The forward variation of β0 shifts the solutions to X(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' The solutions leave X(3) and approach SU(3) as β0 tends to ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' The available experimental data of 162Dy [17], which is a typical SU(3) candidate are placed for comparison in Figure 6(a) and Figure 6(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' This is another remark that isotopes which have X(3) signatures must lie between U(5) −→ SU(3) symmetrical plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' The two other important relations that can be deduced from the comparison are: RL/2(gsb) = 2 + RL/2(β1) = 4 + RL/2(β2), (32) at β0 = 0 as shown numerically in the Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' and Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' This is an observable effect or a signature from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' (30) while the effect of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' (28) is observed in the spectral ratios of X(3) and X(5) such that RX(3) L/2 (β0 = c + 2) = RX(5) L/2 (β0 = c) : c = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' (33) In order to obtained the exact solutions of RL/2 ratios rather than vary β0, the technique of optimizing β0 employed in refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' [11,12,15,16] and others has been used to obtained the solutions of RL/2 at certain values of the β0 peculiar to the angular momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' These special values of β0 are labelled β0,max and they produce exact solutions labelled X(3)-var, shown in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' The values obtained at different values of β0,max, are compared with X(3)-IW solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' For all β0,max, 00,0 and 20,0 levels yield 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='000 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='000 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' β0,max increases with increase in the angular momentum and its values are obtained at the points where the increases in β0 become steep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' d dβ0 RL/2|β0=max is achieved via a numerical procedure as d2 dβ2 0 RL/2 vanished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' The RL/2 ratios for the ground state and the quasi-β1 bands of the X(3) and the X(5) models of inverse square potentials obtained at different values of β0,max, labeled X(3)-var and X(5)-var, are compared with the experimental data of 172,176,178,180Os [18-21] chain, as shown in Figures 7(a) and 7(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' The ground state solutions of the X(3) for L = 0 up to L = 10 are in good agreement with 172Os while those of X(5) are seen lying closer to 176Os than 178Os and 180Os: the generalized comparison is moderate in the first excited state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' This suggests that 172Os is a good candidate for X(3) model while 176Os shows a signature of X(5) model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' The RL/2 theoretical predictions of the X(3) model are compared with the experimental data of some selected isotopes: 102Mo [22], 104−108Ru [23-25], 120−126Xe [26-29], 148Nd [30] and 184−188Pt [31-33] as shown in Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' Each energy level is normalized to the particular 20,0 state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' The energy obtained in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' (13) is fitted with the experimental energy of each of the isotopes considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' The equivalent values of the β0 for the isotopes are recorded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' The quality factor, σ, used is obtained from σ = ��m i [(Rs,L)Exp i − (Rs,L)Theor i ]2 m − 1 , (34) where m is the number of available experimental states, (Rs,L)Exp i and (Rs,L)Theor i represent the experimental and the theoretical spectral ratios of the ith levels normalized to the ground state with L = 2, s = 1 and nβ = 0 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' Against the neutron numbers, N, of the chains of the isotopes: 104−108Ru, 120−126Xe, 184−188Pt, 172−180Os, considered for the comparison, the neutron-β0 distribution, showing the relative posi- tions of the isotopes, is shown in the Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' The comparison in the ground state, the quasi-β1 bands and the quasi-β2 bands of the B(E2) transition probabilities at β0 = 0, 1, 2 and β0 = ∞, normalized to the B(E2 : 21,0 → 01,0) = 100 units with the X(3)-IW [1] and experimental data on X(5) [34] are presented in the Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' The values of β0,max peculiar to each angular momentum, obtained from the optimization of β0, in 16 Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=', are employed to compute the optimized B(E2) transition probabilities, labelled B(E2)- var.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' The visuals of these comparisons are shown in the Figures 9(a), 9(b) and 9(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' In order to show the nature of the solutions along the X(5) −→ SU(3) symmetry region, the experimental data on the 158Gd [35], which is a typical SU(3) candidate, are placed for comparison in Figures 9(a) and 9(b): the solutions at β0 → ∞ are seen lying close to the 158Gd [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' The values of the B(E2) transition probabilities decrease as the variation parameter, β0, increases: they increase as the angular momentum increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' The forward variation, as the β0 increases, pushes the solutions to X(5) and the solutions tend to the SU(3) as β0 tends to ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' 5 Conclusion The X(3) solutions of the Bohr Hamiltonian are obtained by solving the radial function of the Hamiltonian with an inverse square potential with the aid of MAPLE software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' Analytically, an expression for the energy levels is determined from the zeros of the Bessel functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' Through the use of the variational approach and the optimization procedure, the spectra ratios and the B(E2) transition probabilities are computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' The analytical solutions of the X(3) model are compared with the X(5) model of the inverse square potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' It is worth noting that, X(3) model is another “window” through which X(5) and SU(3) “pictures” can be seen: X(3) lies between U(5) and SU(3), hence, X(5) lies between X(3) and SU(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' It has been shown via variational procedure, that the solutions shift to X(5) from X(3) and approach SU(3) as the variation parameter shifts forward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' The theoretical predictions on RL/2 and B(E2) with the experimental data for some selected isotopes are found to be proficient in the gsb and moderate in other levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' This is shown as the theoretical deviations from the experiments are quite small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' The same manner in which the Davidson potential is employed in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' [4], the employment of the one parameter-dependent inverse square potential in the form of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' (1), its properties, is efficient in the variational procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' (1) is also a good choice of potential which can be employed for the description of the nuclei transition at the critical points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' For the comparison of X(3) and X(5) models of Bohr Hamiltonian, with the same formalism employed in this work, it is expected that Equations (28), (29), (30), (31), (32) and (33) should hold in any one-parameter-dependent potential domain such as Kratzer potential, Davidson potential and others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' Data availability statement All the sources of data included in this article for comparison purpose, are cited and referenced accordingly, in the article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' Funding Information No funding of any form is received for the course of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' References [1] Bonatsos, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=', Lenis, D.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' Plus, 136(500).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' doi: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='1140/epjp/s13360-021-01451-7 [12] Ajulo, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=', Oyewumi, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=', Oyun, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' and Ajibade, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' Plus 137(90).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' doi: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='1140/epjp/s13360-021-02276-0 [13] Abramowitz, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' and Stegun, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' (1965).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' Handbook of Mathematical Functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' Dover, New York.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' [14] Gradshteyn, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' and Ryzhik, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' (1980).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' Table of Integral Series and Products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' Academic, New York.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' [15] Bonatsos, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=', Lenis, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=', Minkov, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=', Raychev, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' and Terziev, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' Physics Letter B, 584, 1-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' doi: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='physletb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='018 [16] Ajulo, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' and Oyewumi, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' Physica Scripta, 137(90).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' doi: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='1088/1402- 4896/ac76ed [17] Helmer, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' (1985).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' Data Sheets, 44(4), 661-781.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' doi: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='1016/S0090- 3752(85)80054-5 [18] Greenwood, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' (1975).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' Data Sheets, 15, 497.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' doi: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='1016/S0090- 3752(87)80023-6 [19] Horen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' Harmatz, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' (1976).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' Data Sheets, 19, 383.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' doi: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='1016/S0090- 3752(76)80067-1 [20] Browne, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' (1988).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' Data Sheets, 54, 199.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' doi: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='1016/S0090- 3752(88)80132-7 [21] Wu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' and Niu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' Data Sheets, 100(4), 483-705.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' doi: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='1006/ndsh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='0018 [22] DE Frenne, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' Data Sheets, 110(8), 1745-1915.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' doi: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='nds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='002 18 [23] Blachot, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' Data Sheets, 108(10), 2035-2172.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' doi: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='nds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='001 [24] DE Frenne, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' and Negret, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' Data Sheets, 109(4), 943-1102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' doi: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='nds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='002 [25] Blachot, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' Data Sheets, 91(2), 135-296.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' doi: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='1006/ndsh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='0017 [26] Kitao, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=', Tendow, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' and Hashizume, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' Data Sheets, 96(2), 241-390.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' doi: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='1006/ndsh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='0012 [27] Tamura, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' Data Sheets, 108(3), 455-632.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' doi: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='nds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='001 [28] Katakura, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' and Wu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' Data Sheets, 109(7), 1655-1877.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' doi: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='nds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='001 [29] Limura, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=', Katakura, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' and Ohya, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' Data Sheets, 180, 1-413.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' doi: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='nds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='001 [30] Nica, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' Data Sheets 117, 1-229.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' doi: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='nds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='001 [31] Baglin, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' Data Sheets, 111(2), 275-523.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' doi: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='nds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='001 [32] Baglin, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' Data Sheets, 99(9), 1-196.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' doi: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='1006/ndsh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='0007 [33] Singh, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' Data Sheets, 95(2), 387-541.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content=' doi: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='1006/ndsh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAyT4oBgHgl3EQfpvj9/content/2301.00533v1.pdf'} +page_content='0005 [34] Melon, B.' metadata={'source': 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0000000000000000000000000000000000000000..5cf2540250f8ac4efbd6d7a5b55546e4bdd62551 --- /dev/null +++ b/8NE2T4oBgHgl3EQflQeg/content/tmp_files/2301.03987v1.pdf.txt @@ -0,0 +1,1376 @@ +API Entity and Relation Joint Extraction from Text via +Dynamic Prompt-tuned Language Model +QING HUANG, Jiangxi Normal University, School of Computer Information Engineering, China +YANBANG SUN∗, Jiangxi Normal University, School of Computer Information Engineering, China +ZHENCHANG XING, CSIRO’s Data61 & Australian National University, College of Engineering and +Computer Science, Australia +MIN YU†, Jiangxi Normal University, School of Computer Information Engineering, China +XIWEI XU, CSIRO’s Data61, Australia +QINGHUA LU, CSIRO’s Data61, Australia +Extraction of Application Programming Interfaces (APIs) and their semantic relations from unstructured +text (e.g., Stack Overflow) is a fundamental work for software engineering tasks (e.g., API recommendation). +However, existing approaches are rule-based and sequence-labeling based. They must manually enumerate the +rules or label data for a wide range of sentence patterns, which involves a significant amount of labor overhead +and is exacerbated by morphological and common-word ambiguity. In contrast to matching or labeling API +entities and relations, this paper formulates heterogeneous API extraction and API relation extraction task as +a sequence-to-sequence generation task, and proposes AERJE, an API entity-relation joint extraction model +based on the large pre-trained language model. After training on a small number of ambiguous but correctly +labeled data, AERJE builds a multi-task architecture that extracts API entities and relations from unstructured +text using dynamic prompts. We systematically evaluate AERJE on a set of long and ambiguous sentences +from Stack Overflow. The experimental results show that AERJE achieves high accuracy and discrimination +ability in API entity-relation joint extraction, even with zero or few-shot fine-tuning. +Additional Key Words and Phrases: API Entity, API Relation, Joint Extraction, Dynamic Prompt +ACM Reference Format: +Qing Huang, Yanbang Sun, Zhenchang Xing, Min Yu, Xiwei Xu, and Qinghua Lu. 2023. API Entity and +Relation Joint Extraction from Text via Dynamic Prompt-tuned Language Model. 1, 1 (January 2023), 20 pages. +https://doi.org/10.1145/nnnnnnn.nnnnnnn +1 +INTRODUCTION +Application Programming Interfaces (APIs) are important software engineering artifacts that can +be frequently found in a wide range of natural language texts, from official API references and +tutorials to informal online forums. Meanwhile, API relations are also embedded in these texts. +For example, the text “To manipulate data you actually need executeUpdate() rather than execute- +Query()” in the Stack Overflow (SO) post 1 describes the Function-Replace relation [1] between +executeUpdate() and executeQuery(). This API relation reveals that we should replace executeQuery() +with executeUpdate() to solve the question in the post, i.e., “why cannot issue data manipulation +statements with executeQuery()”. API entity and relation extraction from unstructured texts is +∗Y. Sun and Q. Huang are co-first authors. +†M. Yu is the corresponding author. +1https://stackoverflow.com/questions/1905607 +Authors’ addresses: Qing Huang, Jiangxi Normal University, School of Computer Information Engineering, Nanchang, +Jiangxi, China, qh@whu.edu.cn; Yanbang Sun, Jiangxi Normal University, School of Computer Information Engineering, +Nanchang, Jiangxi, China, ybsun@jxnu.edu.cn; Zhenchang Xing, CSIRO’s Data61 & Australian National University, College +of Engineering and Computer Science, Canberra, Australia, zhenchang.xing@data61.csiro.au; Min Yu, Jiangxi Normal +University, School of Computer Information Engineering, Nanchang, Jiangxi, China, myu@jxnu.edu.cn; Xiwei Xu, CSIRO’s +Data61, Sydney, Australia, xiwei.xu@data61.csiro.au; Qinghua Lu, CSIRO’s Data61, Sydney, Australia, qinghua.lu@data61. +csiro.au. +, Vol. 1, No. 1, Article . Publication date: January 2023. +arXiv:2301.03987v1 [cs.SE] 10 Jan 2023 + +2 +Qing Huang, Yanbang Sun, Zhenchang Xing, Min Yu, Xiwei Xu, and Qinghua Lu +Table 1. Three types of ambiguities for API entities and relations. +PostID +Sentence +#47871272 +You need to override remove() in your iterator. +#14200489 +This code is invalid since l.remove() is called during iteration over l. +#60017952 +You may be calling iterator.remove more than once. +#34682267 +By default, printWriter calls flush in println, whereas it doesn’t do this in print. +#703396 +If the idea is to :::: +print integer stored as doubles... +#322715 +linkedlist and arraylist are two different implementations of the list interface. +#33405095 +nextline() will read the entire line, but next() will only read the next word. +#355089 +StringBuffer is synchronized, StringBuilder is not. +Note: API mention is tagged with an underline; common word is tagged with a wavy line. +fundamental for efficiently accessing and applying API knowledge to various software engineering +tasks. Once extracted, these entities and relations can be organized into structured knowledge +(particularly in the form of knowledge graphs) to support a variety of software engineering tasks +such as API linking [2, 3], API recommendation [4, 5], and API comparison [6]. +There are currently two main types of approaches for extracting API entities from unstructured +text. The first is a rule-based approach such as language-convention based regular expressions [7, 8], +island parsing [9, 10] and heuristic rule matching [1, 6, 11]. Because it is impossible to manually +enumerate the rules that adapt to all sentence patterns, it suffers from rule design overhead. The +second is a sequence labeling based approach such as CRF [2, 12] and Bi-LSTM-CRF [13]. Because +it is impossible to manually label entities for a large amount of sentences, it suffers from data +labeling overhead. Compared with API entity extraction, relation extraction from software text is +rather primitive, which relies on either API syntax (e.g., a class declares a method) [6], special-tag +annotated relations (e.g., “see also” keyword and hyperlink-based method) [14], or some ad-hoc +relation phrases (e.g., “differ in” and “be similar to”) [1]. Same as rule-based entity extraction, these +relation extraction methods suffer from rule-design overhead. We refer to rule design overhead +and data labeling overhead as labor overhead in this work. +This labor overhead is exacerbated by three types of ambiguities, which necessitate the manual +design of more rules or the labeling of more data to distinguish ambiguous sentences. Morphological +ambiguity, which includes abbreviations, synonyms, and misspellings, is one type of ambiguity [12]. +It is common in informal discussions, because people rarely write full API names that exactly match +the API names in the library [15]. Three sentences in the first row of Table 1, for example, shows +three morphological variations of API java.util.iterator.remove(), that either omit some prefixes +and special symbols (e.g., package names, class names, and “()”), or are preceded by user-defined +variable names. The second type is common-word ambiguity between common words and API +mentions [12], which occurs because people frequently write API method names without proper +punctuation, parentheses, and uppercase letters. For example, as shown in the second row of Table 1, +even though the word print appears in two sentences, print in the first sentence refers to the API +java.io.printwriter.print(), whereas print in the second one is only a verb. The final type is expression +ambiguity of API semantic relations, which is caused by changes in sentence patterns. In general, +the same API relation can be expressed in multiple sentence patterns. For example, as shown in the +third raw of Table 1, three sentence patterns are used in the three sentences, all of which express +the Behavioral-Difference relation between API entities. They are “API1 and API2 are different”, +“API1 does one thing, but API2 does the other thing”, and “API1 is (adjective), API2 is not”. +, Vol. 1, No. 1, Article . Publication date: January 2023. + +API Entity and Relation Joint Extraction from Text via Dynamic Prompt-tuned Language Model +3 +To alleviate the labor overhead, we devise a novel idea of extracting API entities and relations +using a large pre-trained language model (LLM). LLM stores a large amount of prior knowledge +and can serve as a neural knowledge base of real-world entities and relations [16]. In addition, LLM +can provide better model initialization [17] and strong learning capbility. By fine-tuning a LLM +with a small set of domain-specific training data, we can prompote the LLM to identify as many +API entities and relations as possible. In order to make LLM be more discriminative, the training +data should contain sufficient morphological and common-word ambiguity, and the API entities +and relations should be labeled correctly. To reduce manual labeling, we devise morphology and +verb-based data augmentation strategies to generate more ambiguous data but correctly labeled +sentences for the LLM fine-tuning. +Existing work [18] separates API entity extraction and relation extraction as two tasks, leaving +relation extraction heavily reliant on entity extraction results, which leads to error propagation [19]. +Instead, we consider API entity extraction and relation extraction as correlated tasks and adopt +a unified model for joint entity and relation extraction, inspired by the recent work on universal +information extraction (UIE [20]). However, the LLM (i.e., T5 [21]) in UIE often fails with complex +sentences, particularly long and ambiguous sentences containing API entities and various relations, +because it only uses one static prompt to recognize all types of API relations. To tackle this issue, we +design a dynamic prompt generator, inspired by the input-dependent prompt tuning method [22], +that generates dynamic prompts for a small number of potentially relevant relations at inference +time based on the actual input sentences, rather than relying on the same static prompt for all +inputs. When confronted with complex sentences, the more relation types to recognize, the more +noise it suffers from, the more difficult it is for LLM to understand to what extent a complex +sentence contains certain relations. As our dynamic prompt reduces the number of relation types +to recognize and mitigate noise interference, it improves the extraction accuracy of API relations. +In this paper, we propose a API Entity-Relation Joint Extraction framework, called AERJE. It +consists of a dynamic prompt generator and a joint entity-relation extractor. The kernel of the +prompt generator is a BERT-based text classifier that is used to classify the input text. Each class +represents an API relation and the prompt generator generates dynamic prompts based on the +top-N possible API relations. The generated dynamic prompt and the input sentence are fed into +the joint entity-relation extractor to extract the API entities and relations contained in the text. In +our current implementation, the joint entity-relation extractor is Transformer-base LLM (T5). The +prompt generator and the entity-relation extractor are fine-tuned in an end-to-end manner. +No model, to the best of our knowledge, can simultaneously extract both API entities and +relations. AERJE, on the other hand, achieves an F1 score of 96.51% for API entity extraction, which +is approximately 6% higher than the state-of-the-art API entity recognition model ARCLIN [13] +and 7% higher than APIReal [2], and an F1 score of 81.20% for API relation extraction. Then, we +evaluate the impact of intrinsic factors (two data augmentation strategies and the number of API +relations in the dynamic prompts) on performance. Our experiments find that data augmentation +helps to improve AERJE’s discriminative capability for API entities and relations, and the dynamic +prompts with four API relations can significantly improve AERJE’s extraction accuracy. Finally, +we assess AERJE’s generalization and ability to extract API entities and relations in low-resource +scenarios (i.e., less than 0.8% fine-tuning data) and find that, even under low resource conditions, +our AERJE still has strong extraction ability, outperforming APIReal [2] and ARCLIN [13]. +The main contributions of this paper are as follows: +• Conceptually, we are the first to formulate heterogeneous API extraction and API relation +extraction tasks as a uniform sequence-to-sequence generation task, and propose AERJE, an +API entity-relation joint extraction framework based on pre-trained LLMs. +, Vol. 1, No. 1, Article . Publication date: January 2023. + +4 +Qing Huang, Yanbang Sun, Zhenchang Xing, Min Yu, Xiwei Xu, and Qinghua Lu +In fact, collections.sort() has +already been migrated to +list.sort(). +You better using getint() +instead of get(). +You can use +double.parsedouble() to +convert a string to a double. +Input Sentences +Bert +Dynamic +prompt + + + ( + ( API: getint() + ( function replace: get() ) + ) + ( API: get() ) + ) + ( + ( API: collection.sort() ) + ( API: list.sort() ) + ) +[spot] API [asso] function replace [asso] efficiency comparison [asso] type conversion +[text] You better using getint() instead of get(). +[spot] API [asso] function similarity [asso] logic constraint [asso] type conversion +[text] In fact, collections.sort() has already been migrated to list.sort(). +[spot] API [asso] logic constraint [asso] type conversion [asso] function similarity +[text] You can use double.parsedouble() to convert a string to a double. +Joint Entity-Relation Extractor +Linear +P(Relation) +top-3 Relations +a +b +c +Ⅰ +Ⅱ +CLS +E +... +E +E +D +... +D +D +Latent Vector +... +b +c +a +b +c +a +Ⅲ +Dynamic Prompt Generator +T5 + + ( + ( API: string + ( type conversion: double ) + ) + ( API: double ) + ( API: double.parsedouble() ) + ) +Structured Extraction Language +...... +...... +...... +Fig. 1. Overall Framework of AERJE. The Dynamic prompt’s bold font represents the semantic relation to be +extracted from the input sentence. II.b lacks a bold font because I.b contains no semantic relation. +• We devise two data augmentation strategies in order to obtain more ambiguous but correctly +labeled sentences. Learning such sentences enables AERJE to be more discriminative for API +entities and relations. +• Unlike the single task model, we build a multi-task architecture that encodes the structures +of entity and relation extraction into a unified structure language for extracting API entities +and relations simultaneously. +• Instead of using a single static prompt with all types of API relations for all sentences, we +design a dynamic prompt based on relation classification, which reduces the number of +relation types to recognize, eliminates noise interference, and lowers the difficulty of relation +extraction. +• We systematically evaluate the AERJE’s intrinsic factors, performance, generalization, and +few-shot learning capabilities. It is the first approach to extract API entities and relations +simultaneously, and it achieves superior performance than independent API extraction and +API relation extraction. Our data package can be found here2, the code will be released after +the paper is accepted. +2 +APPROACH +We formulate heterogeneous API extraction and API relation extraction tasks as a uniform sequence- +to-sequence generation task, and propose a novel model AERJE to accomplish it. As shown in +Fig. 1, AERJE consists of a dynamic prompt generator and a joint entity-relation extractor. The +dynamic prompt generator generates dynamic prompts based on the input texts (one at a time). +The input text is then appended to the prompt to form a whole input that is fed into the joint +entity-relation extractor, which generates a structured extraction language sequence with API +entities and relations. +2.1 +Dynamic Prompt Generator +This section describes how to build a prompt that unifies heterogeneous API extraction and API +relation extraction tasks, followed by a discussion of how to design dynamic prompt to improve +AERJE’s API relation extraction performance. +2.1.1 +Prompt Construction for Multi-tasking. In order to extract both API entities and API relations +from an input text, the prompt consists of API entity type, API relation type, and the input text, +which are labeled by [spot], [asso] and [text], respectively. For example, “[spot] API [asso] function +2https://anonymous.4open.science/r/AERJE-6DBF/README.md +, Vol. 1, No. 1, Article . Publication date: January 2023. + +API Entity and Relation Joint Extraction from Text via Dynamic Prompt-tuned Language Model +5 +replace [asso] efficiency comparison [text] You better using getint() instead of get()” represents +an API entity type “API”, two relation types “function replace” and “efficiency comparison”, and +an input text “You better using getint() instead of get()”. In this work, we consider a generic API +entity type “API” and seven relation types defined in [1], including “function similarity”, “behavior +difference”, “logic constraint”, “type conversion”, “function collaboration”, “efficiency comparison”, +“function replace”. Note that more fine-grained API entity types can be used, such as “class”, +“method”, “field” [23], but we leave it as our future work. +2.1.2 +Dynamic Prompt Generation. As stated in Section 1, the more relation types there are, the +harder it is for T5 to determine which types of relation the API entities in the input text belong to, +especially when the sentence is long and ambiguous. If we adopt the static prompt that includes +all seven relation types, the relation extraction performance of the model will decrease (cf. RQ3). +As a result, we design a dynamic prompt generation method to make the content of the prompt +more accurate and instructive for the complex input text. The dynamic prompts, as shown in II.a of +Fig.1, contain only the top-N relations and provide better guidance to the subsequent T5-supported +joint entity-relation extractor. Here, the prompt generator is implemented as a text classifier which +predicts the API relations present in the input text. We use a BERT-based classifier because the +pre-training task (i.e., Next Sentence Prediction) of BERT [24] is consistent with our task, both +of which are classification tasks. Given a sentence containing API entities (see I.a of Fig. 1), the +BERT-based classifier outputs the probability that the sentence belongs to each semantic relation; +the top-3 relations are then chosen as candidate relations. Finally, entity type, candidate relations, +and input sentence are connected by labels (i.e., [spot], [asso], [text]) to generate the dynamic +prompt (see II.a of Fig. 1). +Note that the BERT-based classifier in our current implementation aims to narrow the scope and +provide candidate relations, and it cannot replace the API relations extractor. When the candidate +relations classified by the classifier do not fit these entities in the sentence, the extractor does not +force a relation to be selected from the incorrect candidate relations, but instead assumes that no +relation exists between these entities. For example, given a sentence with no relations between API +entities (see I.b of Fig.1), the dynamic prompt generator generates a dynamic prompt (see II.b of +Fig.1). Based on such a dynamic prompt, the subsequent extractor will not extract relations from +the sentence as none of the candidate relation types is applicable to the input sentence. +To summarize, too many candidate relations may reduce the extractor’s ability to recognize +them, while too few candidate relations may cause the extractor to miss the correct relations. As a +result, we should investigate the appropriate number of candidate relations (cf. RQ3). +2.2 +API Joint Entity-Relation Extractor +We adopt a structured extraction language (SEL) [20] to encode the structures of entity extraction +and relation extraction into a unified representation, so that heterogeneous API extraction and +API relation extraction tasks can be modeled uniformly within a sequence-to-sequence generation +framework. The first sequence refers to the dynamic prompt, while the second sequence refers to +the SEL sequence. +2.2.1 +Structured Extraction Language. SEL sequence is proposed to encode different information +extraction structures via the hierarchical spotting-associating structure. Fig. 2.a shows its universal +format. “Spot Name: Info Span” denotes various entity type and the object of a specific entity type; +“Asso Name: Info Span” denotes various relation types and the associated object of a specific relation +type. Fig. 2.b shows the concrete SEL sequence in our work. “API: getint()” represents that “getint()” +is an API entity; “function replace: get()” represents that the relation between “getint()” and “get()” +, Vol. 1, No. 1, Article . Publication date: January 2023. + +6 +Qing Huang, Yanbang Sun, Zhenchang Xing, Min Yu, Xiwei Xu, and Qinghua Lu + ( + ( Spot Name: Info Span + （ Asso Name: Info Span ) + ) + ( Spot Name: Info Span ) + ) + ( + ( API: getint() + （ function replace: get() ) + ) + ( API: get() ) + ) +a +b +Fig. 2. Specialization of Universal Structured Extraction Language. +is “function replace”. From this concrete SEL sequence, we can extract API entities and relations +simultaneously as it unifies the structure of API entities and relations. +2.2.2 +SEL Sequence Generation. We implement our API joint entity-relation extractor as the +sequence-to-sequence generation framework: +� +𝑦1, . . . ,𝑦|𝑦| +� += JE( +� +𝑝1, . . . , 𝑝 |𝑝 | +� +) +(1) +where JE is a Transformer-based LLM, +� +𝑝1, . . . , 𝑝 |𝑝 | +� is the dynamic prompt, +� +𝑦1, . . . ,𝑦|𝑦| +� is the +linearized SEL sequence that contains the API entities and relations to be extracted. In this frame- +work, we feed the dynamic prompt into the LLM (as shown in Fig. 1.II), and the LLM generates the +SEL sequence (as shown in Fig. 1.III), from which we can obtain API entities and relations. The +dynamic prompt to the JE can also be written in the format described in Section 2.1.1: +� +𝑝1, . . . , 𝑝 |𝑝 | +� +=[[ spot ], . . . [ spot ] . . . , +[ asso ], . . . , [ asso ] . . . , +[ text ],𝑥1,𝑥2, . . . ,𝑥 |𝑥 | +� +(2) +where 𝑥 = +� +𝑥1, . . . ,𝑥 |𝑥 | +� +denotes the input text. +To better illustrate the framework’s internal mechanics, an encoder-decoder-style architecture is +introduced. Given the dynamic prompt 𝑝, JE computes the hidden representation H = +� +p1, . . . , p|𝑝 | +� +of each token: +H = Encoder �𝑝1, . . . , 𝑝 |𝑝 | +� +(3) +where Encoder(·) is a Transformer encoder. Then JE decodes the prompt into a SEL sequence in an +auto-regressive style. At the step 𝑖 of decoding, JE generates the 𝑖-th token 𝑦𝑖 in the SEL sequence +and the decoder state h𝑑 +𝑖 as following: +𝑦𝑖, h𝑑 +𝑖 = Decoder +�� +H; h𝑑 +1, . . . , h𝑑 +𝑖−1 +�� +(4) +Decoder(·) is a Transformer decoder that predicts the conditional probability 𝑝 (𝑦𝑖 | 𝑦 <𝑖, 𝑝) of +token 𝑦𝑖 until the end symbol is output. +2.3 +Model Training +This section describes data collection and augmentation, model training, which includes training a +BERT-based classifier and fine-tuning a Transformer-based LLM. +, Vol. 1, No. 1, Article . Publication date: January 2023. + +API Entity and Relation Joint Extraction from Text via Dynamic Prompt-tuned Language Model +7 +2.3.1 +Data Collection. Given that the relation types we consider are all from a knowledge graph +of Java APIs [1], we randomly chose 5,000 Java-tagged posts from the Stack Overflow data dump 3. +Each post is accompanied by its answers and post tags (such as “java”, “arrays”, “java.lang”). We +choose the most voted answers from the posts to ensure the quality of the training data, but we +exclude code snippets and all HTML tags because the focus of our study is informal text. All the +answers are then splitted into sentences using spaCy 4, yielding 28,140 sentences. Every sentence is +accompanied by multiple category tags from the post to which it belongs. Then, for each sentence, +we parse it into tokens using the software-specific tokenizer [12] which preserves the integrity of +an API mention. iterator.remove(), for example, is treated as a single token. Finally, we crawl all +APIs in JDK 1.8 5, and use these APIs to filter out the sentences containing API entities, as inspired +by a previous study [13], based on the following criteria: +• Because of the large number of morphological ambiguities, a token may be an API entity if it +partially matches any of the crawled APIs (e.g., remove() and java.util.Iterator.remove()). +• Since API mentions usually end with “()”, the token is treated as an API entity if it contains +“()”. +• API mentions typically include “.” to indicate a function call (e.g., iterator.remove(), or l.remove()); +thus, if token contains “.”, we consider it to be an API entity . +After filtering, we obtain 9,111 sentences that may contain API entities. However, this is rough +sentence filtering. In order to do accurate sentence filtering, We invite 12 master students (all +with more than five years Java experience) to examine the API entities and annotate the semantic +relations between APIs in order to further verify whether these sentences contain API entities and +the seven types of API relations we aim to extract. We train the annotators prior to annotation to +ensure that they can recognize these API relations in the text. After training, the annotators were +divided into six groups, with two students from each group annotating the same content. After the +annotation, we assign two authors to deal with the annotation results’ conflicts, and the Cohen’s +Kappa [25] coefficient is 0.859 (i.e., almost perfect agreement). As a result, we get a total of 2917 +sentences, with 2471 containing only entities and 446 containing both entities and relations. +2.3.2 +Data Augmentation. To improve the AERJE’s ability to recognize API entities and relations +from long and ambiguous sentences, we devise two data augmentation strategies to obtain more +ambiguous sentences for model training. +Morphology based Mutation. Inspired by [13], we change the form of each API entity in the +sentence. Specifically, we replace the API entity itself with the final piece of its fully qualified name. +For example, iterator.remove() is replaced with remove() or remove. +Verb based Mutation. We use spaCy to locate the verbs on which each API entity relies, and +then replace those verbs with synonyms, as Liu et al. [26] do to obtain similar question titles. As +shown in the seventh sentence of Table 1, we replace “read” with “load”. However, because spaCy +may not obtain the correct API entity, we must identify the dependency between the API entity’s +subtoken and the verb to ensure the mutation quality. For example, there is a dependency between +“nextline” and “read”, so we can reliably mutate “read” with synonyms. +Our data augmentation strategy does not include sentence pattern mutation [26], which uses +different sentence patterns to present the same API relation between the same API entities. Unlike +the morphology-based and verb-based mutation, this mutation is not reliable in software text +which demands stricter semantics than general text. The sentence pattern mutation could result in +sentence structure reconstruction, which would likely change the sentence semantics, contaminate +3Retrieved June 6, 2022 from https://archive.org/download/stackexchange/ +4https://spacy.io +5https://docs.oracle.com/javase/8/docs/api +, Vol. 1, No. 1, Article . Publication date: January 2023. + +8 +Qing Huang, Yanbang Sun, Zhenchang Xing, Min Yu, Xiwei Xu, and Qinghua Lu +the training data, and compromise AERJE training. For example, the original sentence “StringBuffer +is synchronized, StringBuilder is not” may be mutated into “StringBuffer and StringBuilder differ in +synchronized”. The original sentence indicates that StringBuffer is synchronized and StringBuilder +is asynchronous, but the mutated sentence does not specify who is synchronous or asynchronous. +We obtain 2,334 sentences as the initial training set and 583 sentences as the initial test set in an +8:2 ratio from the 2,917 sentences collected. The number of sentences after applying the two data +augmentation strategies to the initial training and test sets is 10,678 and 2,686, referred to as the +final training set and the final test set, respectively. This final training set is used to fine-tune the +LLM-based extractor, and the final test set is used to test the fine-tuned extractor. Here, we split +the sentences into training and testing sets and then mutated them. This ensures that the sentence +before and after the mutation is in the same set, preventing the leaking of training data into the test +set (e.g., one sentence in the training set and its mutation in the test set). Furthermore, we obtain +1,639 sentences with both entities and relations as the classifier training set from the final training +set. Similarly, we obtain 387 sentences with both entities and relations as the classifier test set from +the final test set. +2.3.3 +BERT-based Relation Classifier Training. We choose BERT [24] as a relation classifier because +its pre-training task (i.e., Next Sentence Prediction) is consistent with our task, both of which are +classification tasks. However, the implementation of relation classifier is not limited to BERT, we +can also use TextCNN [27] and FastText [28]. In our current implementation, we use the BERT-base +classifier to classify each input sentence into N relation types. Based on the N relation types, +dynamic prompt generator generates the corresponding dynamic prompt. +A mask language model (BERT) [24] and a linear layer comprise the classifier. Due to the seven +API relation types, the linear layer’s output dimension is set to 7. We obtain the latent vector from +the CLS token when we enter the sentence into BERT. The latent vector obtained from the CLS token +characterizes the sentence features better than other positions, resulting in better classification +performance. The latent vector is then fed into the linear layer, which produces a vector with seven +dimensions, each corresponding to a relation type. Finally, the classifier is trained on the classifier +training set. In back propagation, we use the cross-loss entropy to calculate the classifier’s loss and +adjust the BERT and linear layer parameters. The loss function is formulated as follows, where +𝑧 = [𝑧0, . . . ,𝑧𝐶−1] represents the linear layer’s output result, and C represents the sentence’s label. +Loss(𝑧,𝑐) = −𝑧[𝑐] + log +�𝐶−1 +∑︁ +𝑗=0 +exp(𝑧[𝑗]) +� +(5) +2.3.4 +LLM-based Extractor Fine-tuning. We use the pre-trained T5-v1.1-large model [21] as the +LLM in our current implementation because T5’s training objective aligns perfectly with our +formulation of the API entity and relation extraction task as a sequence to sequence generation +task. Furthermore, studies [29, 30] confirm that T5 is capable of capturing rich text information and +demonstrate its effectiveness in a variety of downstream NLP tasks. Our approach is not limited to +T5, but can use any Transformer-based LLM. +In order to fine-tune T5, we convert each labeled sentence in the final training set into a SEL +sequence (y), then feed it into the dynamic prompt generator to obtain its dynamic prompt (p), and +finally construct the labeled corpus: De = {(𝑝,𝑦)}. On the labeled corpus, we fine-tune T5 for 50 +epoch with batch size 10 using the Adam optimizer with a learning rate of 1e-4, linear scheduling +with a warming up proportion of 6%, and the teacher-forcing cross-entropy loss: +LFT = +∑︁ +(𝑝,𝑦) ∈De +− log 𝑃 (𝑦 | 𝑝;𝜃𝑒,𝜃𝑑) +(6) +, Vol. 1, No. 1, Article . Publication date: January 2023. + +API Entity and Relation Joint Extraction from Text via Dynamic Prompt-tuned Language Model +9 +where 𝜃𝑒 and 𝜃𝑑 are the parameter of encoder and decoder, respectively. +3 +EXPERIMENTAL SETUP +This section starts with five questions about AERJE’s performance, followed by a description of the +experimental setup, which includes the dataset, baseline, and evaluation metrics. +3.1 +Rearch question +• RQ1: Effectiveness of Data Augmentation +• RQ2: Optimal Num. of Relation Types for Dynamic Prompt +• RQ3: Joint Extraction Performance of AERJE +• RQ4: Generalization Ability of AERJE +• RQ5: AERJE’s Performance in Low-Resource Scenario +3.2 +Dataset +As described in section 2.3.2, there are three groups of data sets. The first group refers to the +sentences collected initially, some of which contain only entities and others contain both entities +and relations. +• The initial training set consists 2,334 sentences, of which 362 contain both entities and +relations. +• The initial test set consists 583 sentences, of which 84 contain both entities and relations. +The second group refers to the sentences after applying the two data augmentation strategies, +some of which contain only entities and others contain both entities and relations. +• The final training set with a total of 10,678 sentences, 1639 of which contain both entities +and relations. +• The final test set with a total of 2,686 sentences,387 of which contain both entities and +relations. +The third group refers to the sentences containing both entities and relations in the final training +and testing sets. +• The classifier training set with a total of 1,639 sentences. +• The classifier test set with a total of 387 sentences. +3.3 +Baselines +Our AERJE is capable of API entity-relation joint extraction. However, to the best of our knowledge, +no previous work has focused on extracting both API entities and relations from unstructured texts +at the same time. As a result, we can only compare AERJE with the existing work in the respective +fields of API entity extraction and API relation extraction. +For API entity extraction, there are rule-based methods (such as regular Expressions [7, 8]), +heuristic rule matching methods [1, 6, 11], and sequence-labeling based methods (such as AR- +CLIN [13] using BI-LSTM as encoder and CRF as decoder, APIReal [2] using only CRF). Since the +performance of the first two classes of methods is not as good as that of the last class of methods [13], +we choose ARCLIN, APIReal as baselines. We obtain them source code from Github 6 7, and label +the API entities and non-entities in the sentences with the “BIO” tag (i.e., “B”: the beginning of +API entity segment, “I”: the inside of API entity segment, “O”: non-entity). Then we create pairs +of original sentences and labeled sentences to train these two baselines. Finally, we the trained +models on the final test set, from which we obtain API entities based on the “BIO” tag. +6https://github.com/YintongHuo/ARCLIN +7https://github.com/baolingfeng/APIExing +, Vol. 1, No. 1, Article . Publication date: January 2023. + +10 +Qing Huang, Yanbang Sun, Zhenchang Xing, Min Yu, Xiwei Xu, and Qinghua Lu +For API relation extraction, there are only rule matching methods that rely on API syntax [6], +special-tag annotated relations [14], or some ad-hoc relation phrases [1]. It is very difficult to +re-implement these methods due to the rule-design overhead. Furthermore, it is impractical to apply +these methods as we assume plain texts without any special annotations. Instead, we implement a +variant 𝐴𝐸𝑅𝐽𝐸𝑤/𝑜𝐷𝑃𝐺 (𝐷𝑃𝐺 means the dynamic prompt generator), which uses a static prompt +with all 7 relation types to evaluate the performance of the full-version of AERJE. +In addition, we also implement two other variants of AERJE as our baseline. One is 𝐴𝐸𝑅𝐽𝐸𝑠𝑖𝑛𝑔𝑙𝑒, +which separate API entity and relation extraction into two independent tasks. For API entity +extraction, the prompt contains only “[spot] API”. For API relation extraction, the prompt contains +only “[asso] relation type”. 𝐴𝐸𝑅𝐽𝐸𝑠𝑖𝑛𝑔𝑙𝑒 still uses dynamic prompt in relation extraction. After +relation extraction, we merge the extracted entities and relations as the final results of 𝐴𝐸𝑅𝐽𝐸𝑠𝑖𝑛𝑔𝑙𝑒. +e.g., the extracted entities getint(), get() and relation function replace are merged as (API: getint() +(function replace: get())). We compare 𝐴𝐸𝑅𝐽𝐸𝑠𝑖𝑛𝑔𝑙𝑒 with AERJE to understand the effectiveness of +joint entity-relation extraction. Meanwhile, the entity extraction results of 𝐴𝐸𝑅𝐽𝐸𝑠𝑖𝑛𝑔𝑙𝑒 is equivalent +to fine-tuning pre-trained model for entity extraction, and its final results is equivalent to fine- +tuning pre-trained model for relation extraction. Therefore, 𝐴𝐸𝑅𝐽𝐸𝑠𝑖𝑛𝑔𝑙𝑒 also reflects the capability +of fine-tuning pre-trained model for entity and relation extraction separately. Another variant is +𝐴𝐸𝑅𝐽𝐸𝑏𝑎𝑠𝑒, which replace T5-v.1.1-large in AERJE with a smaller model backbone, i.e., T5-v1.1-base. +We use it to explore the impact of large pre-trained language models on AERJE performance. +All variants use the same hyper-parameters as AERJE and remain constant across experimental +scenarios. Note that SEL used in AERJE has been demonstrated to be effective in the extraction +task [20]. As such, we do not to verify the effectiveness of SEL in AERJE. +3.4 +Evaluation Metrics +We use Precision, Recall, and F1 score as metrics to evaluate the performance of AERJE and baseline +models on our test set. Precision means what percentage of API entities and relations extracted +are correct, recall means what percentage of the real API entities and relations are extracted, and +F1 score is the harmonic mean of precision and recall. It is important to note that the relation +is only correct if the relation type and corresponding entities are both correct. In context of our +work, we are not concerned with the top-N relation classification accuracy. As long as the top-N +includes relevant relation types, the extractor does not care about the order of these relation types. +Furthermore, a sentence may have 2 or more relations, which renders the top-1 accuracy irrelevant. +Finally, as the extractor has the capability of ruling out irrelevant relation types in the prompt, it is +also not necessary to evaluate the classification precision and recall at N. +4 +EXPERIMENTAL RESULTS +This section delves into five research questions to evaluate and discuss the AERJE’s performance. +4.1 +RQ1: Effectiveness of Data Augmentation +4.1.1 +Motivation. To reduce manual labeling effort and improve model training, we devise two data +augmentation strategies. We want to investigate if ambiguous but correctly annotated sentences +obtained through two data augmentation strategies could improve AERJE’s discriminative capability +for extracting API entities and relations, in order to demonstrate the effectiveness of two data +augmentation strategies. +4.1.2 +Methodology. We set up two scenarios 𝐴𝐸𝑅𝐽𝐸𝑤/𝑜𝐷𝐴 and 𝐴𝐸𝑅𝐽𝐸𝑤𝐷𝐴 (𝐷𝐴 means the data +augmentation). 𝐴𝐸𝑅𝐽𝐸𝑤/𝑜𝐷𝐴 is trained on the initial training set, while 𝐴𝐸𝑅𝐽𝐸𝑤𝐷𝐴 is trained on +, Vol. 1, No. 1, Article . Publication date: January 2023. + +API Entity and Relation Joint Extraction from Text via Dynamic Prompt-tuned Language Model +11 +Table 2. Impact of data augmentation strategy on AERJE +Strategy +Entity +Relation +P +R +F1 +P +R +F1 +𝐴𝐸𝑅𝐽𝐸𝑤𝐷𝐴 +97.57 +95.48 +96.51 +86.54 +76.48 +81.20 +𝐴𝐸𝑅𝐽𝐸𝑤/𝑜𝐷𝐴 +95.11 +92.19 +93.63 +77.71 +75.66 +76.67 +the final training set. Both 𝐴𝐸𝑅𝐽𝐸𝑤/𝑜𝐷𝐴 and 𝐴𝐸𝑅𝐽𝐸𝑤𝐷𝐴are tested on the same final test set. This +setting allows us to compare the effectiveness of data augmentation. +4.1.3 +Result. Table 2 shows the experimental results. In terms of API entity extraction, 𝐴𝐸𝑅𝐽𝐸𝑤𝐷𝐴 +has precision, recall, and F1-scores of 97.57%, 95.48%, and 96.51%, while 𝐴𝐸𝑅𝐽𝐸𝑤/𝑜𝐷𝐴 has precision, +recall, and F1-scores of 95.11%, 92.19%, and 93.63%. 𝐴𝐸𝑅𝐽𝐸𝑤𝐷𝐴’s precision, recall, and F1-score are +all higher than 𝐴𝐸𝑅𝐽𝐸𝑤/𝑜𝐷𝐴’s, with 𝐴𝐸𝑅𝐽𝐸𝑤𝐷𝐴’s recall and F1-score being about 3% higher. +In terms of API relation extraction,𝐴𝐸𝑅𝐽𝐸𝑤𝐷𝐴 has precision, recall, and F1-score of 86.54%, 76.48%, +and 81.20%, while 𝐴𝐸𝑅𝐽𝐸𝑤/𝑜𝐷𝐴 has precision, recall, and F1-score of 77.71%, 75.66%, and 76.67%. +𝐴𝐸𝑅𝐽𝐸𝑤𝐷𝐴’s precision, recall, and F1-score are all higher than 𝐴𝐸𝑅𝐽𝐸𝑤/𝑜𝐷𝐴’s. The precision of +𝐴𝐸𝑅𝐽𝐸𝑤𝐷𝐴 is 8.83% higher than that of 𝐴𝐸𝑅𝐽𝐸𝑤/𝑜𝐷𝐴, and the F1-score of 𝐴𝐸𝑅𝐽𝐸𝑤𝐷𝐴 is 4.53% higher +than that of 𝐴𝐸𝑅𝐽𝐸𝑤/𝑜𝐷𝐴. This demonstrates that fine-tuning 𝐴𝐸𝑅𝐽𝐸𝑤𝐷𝐴 using a large number +of ambiguous sentences with API relations benefits 𝐴𝐸𝑅𝐽𝐸𝑤𝐷𝐴 to distinguish between relations +and non-relations, as well as between correct and incorrect relations. In contrast, 𝐴𝐸𝑅𝐽𝐸𝑤/𝑜𝐷𝐴 has +not been fine-tuned on ambiguous sentences and thus does not perform as well as 𝐴𝐸𝑅𝐽𝐸𝑤𝐷𝐴. For +example, an ambiguous sentence “you want to read up on processbuilder to launch the exe file +and then waitfor() to wait until the process is complete”. 𝐴𝐸𝑅𝐽𝐸𝑤𝐷𝐴 correctly extracts two API +entities, ProcBuilder and waitfor(), as well as the “logic constraint” relation between them, from the +sentence. In contrast, 𝐴𝐸𝑅𝐽𝐸𝑤/𝑜𝐷𝐴 only extracts one API waitfor() from the sentence. This shows +AERJE’s capability to extract API entities and relations from ambiguous sentences can be improved +by fine-tuning with the augmentated data. +AERJE’s discriminative capability for API entities and relations can be improved by fine-tuning +it with ambiguous but correctly labeled sentences obtained through the data augmentation +strategies. +4.2 +RQ2: Optimal Num. of Relation Types for Dynamic Prompt +4.2.1 +Motivation. As described in section 2.1.2, given an input sentence, the dynamic prompt +generator employs the BERT-based classifier to predict a set of candidate relation types, which are +then included in the dynamic prompt to guide the subsequent joint entity-relation extractor. In this +RQ, we would like to investigate how many candidate relation types (i.e., top-N classifier results) +can provide the most effective guidance to the extractor. +4.2.2 +Methodology. We exhaust all cases of N values (from 1 to 6) in the dynamic prompt generator, +then fine-tune AERJE on the same final training set and test it on the same final test set to select the +most appropriate N value based on experimental results. We do not test N=7 because it is essentially +the static prompt with all seven relation types (i.e., 𝐴𝐸𝑅𝐽𝐸𝑤/𝑜𝐷𝑃𝐺 studied in RQ3). +4.2.3 +Result. As shown in Table 3, changing the N value has small effect on entity extraction +because N represents the number of relation types in the dynamic prompt which does not directly +affect entity extraction. At N=3, AERJE achieves the marginally best F1-score 96.51% for API entity +extraction. +, Vol. 1, No. 1, Article . Publication date: January 2023. + +12 +Qing Huang, Yanbang Sun, Zhenchang Xing, Min Yu, Xiwei Xu, and Qinghua Lu +Table 3. Model results for different values of N +top-N +Entity +Relation +P +R +F1 +P +R +F1 +1 +96.88 +94.23 +95.54 +75.92 +71.92 +73.87 +2 +97.04 +95.18 +96.10 +77.75 +73.80 +75.72 +3 +97.57 +95.48 +96.51 +86.54 +76.48 +81.20 +4 +97.84 +94.39 +96.08 +83.51 +73.22 +78.03 +5 +96.72 +94.39 +95.54 +77.90 +73.30 +75.53 +6 +96.44 +94.75 +95.59 +75.35 +72.61 +73.95 +For relation extraction, changing the N value has larger effect on both precision and recall. As +N increases, both precision and recall improve until N=3. When N=3, the precision, recall and +F1-score of AERJE reaches the highest 86.54%, 76.48% and 81.20%, respectively. This means that +the correct API relation type is most likely covered in the top-3 candidate relations predicted +by the classifier. When N is less than 3, however, the F1-score of AERJE in relation extraction +decreases because the top-N candidate relations may miss the correct relation type. Here is an +example: “A TreeMap has the same limitation (as does a HashMap, which also breaks when the +hashcode of its elements changes after insertion)”. When N=2, classifier predicts two relations +between TreeMap and Hashmap, including “behavior difference” and “logic constraint” , but ignores +the “function similarity” relation. This ignored relation is at the third relation predicted by the +classifier. However, when N is greater than 3, the F1-score of AERJE in relation extraction decreases +because the dynamic prompt may contain some incorrect relation types, which may mislead the +extractor. This misleading effect has bigger impact on precision than on recall. +The optimal number of relation types for dynamic prompt should be set to 3. This not only +ensures that the majority of the correct relation types appear in the dynamic prompts, but it +also prevents the dynamic prompts from containing too many noise relation types which may +make the model sacrifice precision for recall. +4.3 +RQ3: Joint Extraction Performance of AERJE +4.3.1 +Motivation. We would like to evaluate AERJE’s performance in API entity and relation joint +extraction, compared with the state-of-the-art methods for API entity extraction and API relation +extraction. Note that only our AERJE can achieve joint API entity and relation extraction. +4.3.2 +Methodology. AERJE is compared to APIReal and ARCLIN for API entity extraction, and +three variant models (i.e., 𝐴𝐸𝑅𝐽𝐸𝑤/𝑜𝐷𝑃𝐺, 𝐴𝐸𝑅𝐽𝐸𝑠𝑖𝑛𝑔𝑙𝑒, and 𝐴𝐸𝑅𝐽𝐸𝑏𝑎𝑠𝑒) for both API entity and +relation extraction. Note that the entity and relation extraction results by 𝐴𝐸𝑅𝐽𝐸𝑠𝑖𝑛𝑔𝑙𝑒 represents +the capability of fine-tuning the pre-trained model for the two tasks separately. All models are +trained and tested on the same final training and test sets. Details on configuration can be found in +Section 3.3. +4.3.3 +Result. Table 4 shows the evaluation result of AERJE and five baselines on final test sets. We +see that AERJE’s F1-score is higher 7.5% than APIReal’s F1-score and 5.7% than ARCLIN’s F1-score +on API entity extraction. Compared with the three variant models, AERJE’s F1-score for API entity +extraction is only slightly lower (0.18%) than the best performer (i.e., 96.69% by 𝐴𝐸𝑅𝐽𝐸𝑠𝑖𝑛𝑔𝑙𝑒), but +AERJE’s F1-score for relation extraction is 6.83% higher than that of the second best performer +(74.37% by 𝐴𝐸𝑅𝐽𝐸𝑠𝑖𝑛𝑔𝑙𝑒). +, Vol. 1, No. 1, Article . Publication date: January 2023. + +API Entity and Relation Joint Extraction from Text via Dynamic Prompt-tuned Language Model +13 +Table 4. Comparison of Overall Performance +Model +Entity +Relation +P +R +F1 +P +R +F1 +APIReal +89.13 +88.90 +89.01 +- +- +- +ARCLIN +94.76 +87.17 +90.81 +- +- +- +AERJE +97.57 +95.48 +96.51 +86.54 +76.48 +81.20 +𝐴𝐸𝑅𝐽𝐸𝑠𝑖𝑛𝑔𝑙𝑒 +98.03 +95.38 +96.69 +82.83 +67.47 +74.37 +𝐴𝐸𝑅𝐽𝐸𝑤/𝑜𝐷𝑃𝐺 +97.52 +95.78 +96.64 +75.38 +70.62 +72.92 +𝐴𝐸𝑅𝐽𝐸𝑏𝑎𝑠𝑒 +96.39 +95.15 +95.77 +75.97 +70.28 +73.01 +For APIReal and ARCLIN performance on API entity extraction, both AERJE and it variant +models outperform them largely. This superior performance is due to the backbone large pre- +trained language models (T5) in AERJE. During the pre-training, T5 learns linguistic and semantic +knowledge in text and has powerful abilities in word and sentence representations. Through fine- +tuning, the semantic knowledge packed in the T5 can be transferred to the downstream tasks and +benefit API entity extraction. +The amount of knowledge in the T5 also affects the AERJE’s performance on API entity and +relation extraction. Compared with AERJE, the F1-score of 𝐴𝐸𝑅𝐽𝐸𝑏𝑎𝑠𝑒 is reduced by 0.74% and 8.19% +in API entity extraction and API relation extraction, respectively. The decrease of 𝐴𝐸𝑅𝐽𝐸𝑏𝑎𝑠𝑒’s F1 +score on API entity extraction is very small compared with the decrease on API relation extraction. +It is because the number of sentences containing API entities in the final training set is 6 times +more than the number of sentences containing both API entity and relation (i.e., 10,678 vs 1,639). +Sufficient fine-tuning data for API entity extraction allows the basic T5 model to achieve the +equivalent performance on API entity extraction as the large T5. In contrast, the relation extraction +is more complex than the entity extraction, and the amount of fine-tuning data is smaller. In such +case, the basic T5 cannot compete with the large T5. +For 𝐴𝐸𝑅𝐽𝐸𝑠𝑖𝑛𝑔𝑙𝑒 and AERJE, they achieve almost the same entity extraction performance. How- +ever, in terms of API relation extraction, AERJE’s F1-score (81.20%), precision (86.54%) and recall +(76.48%) are much higher than 𝐴𝐸𝑅𝐽𝐸𝑠𝑖𝑛𝑔𝑙𝑒’s F1-score (74.37%), precision (82.83%) and recall (67.47%), +respectively. This suggests that fine-tuning pre-trained model for API entity extraction individually +or jointly with API relation extraction does not affect the quality of API entity extraction. But +joint entity and relation extraction is much more effective for the relation extraction task than +fine-tuning the model just for the relation extraction. +For 𝐴𝐸𝑅𝐽𝐸𝑤/𝑜𝐷𝑃𝐺 and AERJE, they also achieve almost the same entity extraction performance. +This is due to the fact that dynamic prompt only affects the relation type, not the entity type. In +terms of API relation extraction, AERJE’s precision (86.54%), recall (76.48%) and F1-score (81.20%) are +higher than 𝐴𝐸𝑅𝐽𝐸𝑤/𝑜𝐷𝑃𝐺’s precision (75.38%), recall (70.62%) and F1-score (72.92%), respectively. +This is because 𝐴𝐸𝑅𝐽𝐸𝑤/𝑜𝐷𝑃𝐺 uses the same static prompt that includes all seven relation types +for all input sentences. The more types of relations there are in the prompt, the more noise the +prompt is, and the more difficult it is for AERJE to identify and extract the correct relations in the +input sentence. In contrast, AERJE’s use of dynamic prompt reduces the number of relation types +to recognize, improving its ability to extract API relations. +Standing on the shoulder of large pre-trained language model (T5), AERJE outperforms traditional +sequence labeling models for API entity extraction. Dynamic prompt has no impact on API +entity extraction, but can largely boost the performance of API relation extraction. Fine-tuning +the pre-trained model jointly is much more effective than fine-tuning the model just for one +task, which makes joint entity-relation extraction more accurate on both tasks than separate +entity and relation extraction. +, Vol. 1, No. 1, Article . Publication date: January 2023. + +14 +Qing Huang, Yanbang Sun, Zhenchang Xing, Min Yu, Xiwei Xu, and Qinghua Lu +4.4 +RQ4: Generalization Ability of AERJE +4.4.1 +Motivation. Each API comes with its own API package, which often have different forms. +Furthermore, as APIs from different packages support diverse functionalities, the texts in which +they appear may be different in content and linguistic properties. It is impossible for AERJE to see +all API packages during fine-tuning. In this RQ, we want to investigate if AERJE can recognize +APIs and their relations from the API packages that it does not see during fine-tuning. +4.4.2 +Methodology. In order to collect as much data from different packages as possible, we +combine the final training set and the final test set into a new data set with a total of 13,364 +sentences. Every sentence, as stated in Section 2.3.1, is accompanied by multiple post tags, some of +which show the relationship between the sentence and the API package. For example, the tag “io” is +associated with the package name “java.io”. Therefore, we filter out sentences with package names +by matching each tag of a sentence to any JDK 1.8 package name. Here is a partial match, which +means it matches a portion of the package name, for example, “swing” can match “javax.swing”. +And then we pool the package names that appear with the sentences and select the three package +names that appear the most frequently (i.e., javax.swing, java.io, and java.util). Finally, we gather +1651 sentences whose tags match these three package names. +To ensure the correctness of the sentences obtained through approximate match, we invite six +students (who have previously participated in annotation) and divide them into three groups to +annotate sentences from three different packages. Two students in each group annotate the same +sentences. They independently determine whether the API entities in each sentence are from the +specific package (i.e., java.io, java.util, javax.swing). Here is an example “you can use lines() method +in BufferedRead” for java.io package. The sentence is annotated as True, since the API entities +line() and BufferedRead only correspond to java.io. Instead, if any API entity in the sentence do +not belong to specific package, the sentence is annotated as False. Then we assign an author to +handle conflicts between the group members. Finally, we obtain 999 sentences that strictly matched +these packages names. Cohen’s Kappa [25] coefficient is 0.795 (i.e., substantial agreement). The +data details for each package are as follows: +• The java.io dataset has 235 sentences, 51 of which contain both entities and relations. 12 of +the 51 sentences are non-augmented sentences. +• The javax.swing dataset has 435 sentences, 76 of which contain both entities and relations. +14 of the 76 sentences are non-augmented sentences. +• The java.util dataset has 329 sentences, 68 of which contain both entities and relations. 18 of +the 68 sentences are non-augmented sentences. +Our AERJE and baseline models are all trained on one of the three datasets and tested on the +two others. As AERJE outperforms its variants. We don’t consider these variants here. +4.4.3 +Result. Table 5 shows the results that reflect each model’s generalization ability. For API +entity extraction, AERJE’s F1-score achieves 95.05%, when trained on the java.util dataset, far +exceeding APIReal’s F1-score (61.27%) and ARCLIN’s F1-score (58.50%). We attribute this to the +underlying LLM on which AERJE is built. As Qiu et al. [17] show, LLM provides better model +initialization, which usually leads to better generalization performance on the target tasks. Similar +observations can be made for training the models on the java.io dataset and the javax.swing dataset. +Generally, ARCLIN and APIReal may perform well on either precision or recall, but not both and +thus poor F1-score. In contrast, AERJE is very stable with much better precision and recall and +with only small fluctuations in F1-scores across the experiments. +For API relation extraction, AERJE’s F1-score is 40.98%, 79.99% and 68.48% when trained on +the java.io, java.util and javax.swing datasets, respectively. In the across-package training-testing +, Vol. 1, No. 1, Article . Publication date: January 2023. + +API Entity and Relation Joint Extraction from Text via Dynamic Prompt-tuned Language Model +15 +Table 5. Comparison of Generalization Ability +Model +java.io +java.util +javax.swing +Entity +Relation +Entity +Relation +Entity +Relation +P +R +F1 +P +R +F1 +P +R +F1 +P +R +F1 +P +R +F1 +P +R +F1 +APIReal +85.02 +36.97 +51.53 +- +- +- +98.11 +44.54 +61.27 +- +- +- +98.99 +25.62 +40.70 +- +- +- +ARCLIN +95.93 +70.84 +81.50 +- +- +- +98.55 +41.60 +58.50 +- +- +- +98.64 +56.57 +71.90 +- +- +- +AERJE +92.00 +89.35 +90.66 +45.87 +37.03 +40.98 +95.17 +94.93 +95.05 +78.68 +81.35 +79.99 +93.89 +89.91 +91.86 +96.92 +52.94 +68.48 +Table 6. Experimental results in a low-resource scenario +Model +1-Shot +5-Shot +10-Shot +Entity +Relation +Entity +Relation +Entity +Relation +P +R +F1 +P +R +F1 +P +R +F1 +P +R +F1 +P +R +F1 +P +R +F1 +APIReal +80.30 +15.92 +26.57 +- +- +- +86.17 +60.67 +71.20 +- +- +- +83.94 +68.59 +75.49 +- +- +- +ARCLIN +55.07 +62.38 +58.50 +- +- +- +74.64 +72.91 +73.76 +- +- +- +83.58 +75.52 +79.34 +- +- +- +AERJE +72.76 +85.06 +78.43 +9.34 +44.49 +15.44 +79.09 +91.62 +84.90 +31.68 +65.96 +42.80 +82.47 +93.74 +87.74 +35.00 +72.94 +47.30 +setting, the performance of AERJE degrades, compared with the non-across-package setting (see +Table 4). However, when trained on the java.util dataset, AERJE’s F1-score (79.99%) is only about +1% less than non-across-package setting (81.20%). This suggests that AERJE is capable of dealing +with the data drift across different packages. In addition, different across-package training-testing +settings also bring different results. When using java.util for training AERJE, its F1-score is about +39% higher than the F1-score of AERJE trained on java.io. First, due to java.io having fewer sentences +with relations than java.util (51 vs 68). Second, java.io data has fewer non-augmented sentences +with relations than java.util (12 vs 18), which makes java.io data less diverse than java.util. +Our AERJE has a strong generalization ability in face of the data drift across different API +packages. This ability comes from the generalization ability of the underlying LLM. +4.5 +RQ5: AERJE’s Performance in Low-Resource Scenario +4.5.1 +Motivation. Labor overhead means that the data available for training is limited. In this RQ, +we want to investigate how well AERJE perform when trained with the extremely small amount of +training data. +4.5.2 +Methodology. We conduct a K-shot experiment, where K can be 1, 5, or 10. To begin the +K-shot experiment, we randomly select K sentences from the final training set for each relation +type. Then we choose K sentences at random from the final training set that contain only entities +but no relations. This yields a training set containing 8*k sentences. Finally, we train our AERJE +and baseline models on this training set and test them on the final test set. Note that, to avoid the +influence of random sampling, we repeat each K-shot experiment ten times with different samples. +4.5.3 +Result. For API entity extraction, Table 6 shows the performance of each model in three +low-resource scenarios (i.e., 1-shot, 5-shot, and 10-shot) where AERJE significantly outperforms +APIReal and ARCLIN. Especially, in the 1-shot scenario, AERJE’s F1-score is 78.43%, which is +significantly higher than APIReal’s (26.57%) and ARCLIN’s (58.50%). Compared to APIReal and +ARCLIN, the LLM-based AERJE has a large amount of prior knowledge from the LLM pre-training. +As the fine-tuning shot increases, the accuracy of AERJE improves fast, especially on F1-score, +reaching the F1-score 84.90% at 5-shot and 87.74% at 10-shot. +For API relation extraction, in the 1-shot scenario, AERJE does not perform well, but it still +magically achieves the recall 44.49%. However, with only 4 more examples (at 5-shot), the F1-score +of AERJE significantly increases from below 16% at 1-shot to about 43% at 5-shot. This suggests +that the underlying LLM can quickly adapt to the API relation extraction task that it does not see +, Vol. 1, No. 1, Article . Publication date: January 2023. + +16 +Qing Huang, Yanbang Sun, Zhenchang Xing, Min Yu, Xiwei Xu, and Qinghua Lu +during pre-training with only a few examples. In the few-shot setting, we see that precision is +much more difficult to improve than recall. It could be due to the ambiguities of relations, i.e. the +same type of relation can be expressed in very different forms (as shown in table 1), while different +types of relations may be expressed in the similar forms (e.g., “API1 be ADJ to API2” represents +function similarity or function opposite relation). With only a few examples of each type of relation, +it makes learning to distinguish between them more difficult. Furthermore, AERJE’s F1-score for +entity extraction is 78.43% at one-shot, while its F1-score for relation extraction is only 15.44%. The +primary cause for this is that the training set of one-shot contains almost all API entity ambiguity +types but only a few API relation ambiguity types. As a result, relation extraction is more difficult +than entity extraction. +AERJE can quickly adapt the underlying LLM to the API entity and relation extraction tasks with +only a small number of fine-tuning data. Prior knowledge in LLM enables this quick adaptation. +Relation extraction is much harder than entity extraction in the few-shot setting. +5 +DISCUSSION +The major threat to internal validity is the manual labeling of training and testing datasets. Incorrect +human labels could harm modeling training and testing. To mitigate this threat, we invited two +students to annotate the same content and assigned an author to resolve disagreements in the +labeling results. However, even humans can’t always tell if a token references an API, especially +when it comes to common nouns that reference basic computing concepts, such as policy and time, +which can be either basic noun concepts or APIs (java.security.policy class, java.time package). We +take a conservative strategy, i.e., common nouns as API entities, unless both annotators agree. +The threat to external validity is three-fold. The first external threat is that we only collect data +on Stack Overflow. Although our model performed well on the SO data set, we intend further to +validate its generalization performance in the other data sources (e.g., Java Tutorial8, SitePoint9, +and Reddit10). The second external threat is that AERJE has only been tested on Java packages. We +chose Java because previous work [1, 6, 11] has demonstrated how difficult it is to extract these +API entities and relations from it. In the future, we plan to expand AERJE to other programming +languages (such as Python and C#). The third external threat stems from two AERJE components: +the BERT-based classifier and the T5-based extractor. There are numerous alternative models for +both components of the model. TextCNN [27] and FastText [28] can be used to build the classifier. +It is possible to use BART [31] and GPT-3 [32] to implement the extractor. In the future, we will +compare two AERJE components with alternative models to determine the best performing model. +6 +RELATED WORK +API entity and relation extraction is a fundamental work in software engineering. It is useful +in the construction of knowledge graphs; extracted structured API knowledge can help with +many software engineering tasks such as API linking [2, 3, 8, 33], API misuse detection [11], API +recommendation [4, 5], and API comparison [6]. This section describes the methods for extracting +API entities and relations from unstructured text. +Bacchelli [7, 34] and Dagenais [3] detect class and method mentions in developer emails, docu- +mentation and forum posts using regular expressions of distinct orthographic features. Ren [11], +Huang [1], and Liu [6] extract entities from SO posts using the HTML tag. Bacchelli et +al. [10] extract coarse-grained structured code fragments from natural language text with island +8http://www.java2s.com +9https://www.sitepoint.com/ +10https://www.reddit.com +, Vol. 1, No. 1, Article . Publication date: January 2023. + +API Entity and Relation Joint Extraction from Text via Dynamic Prompt-tuned Language Model +17 +parsing. Huang [1], and Liu [6] extract semantic relations between entities based on syntactic +patterns. However, their API entity and relation extraction method from natural language text +relies on unique orthographic features of APIs, and suffer from the rule design overhead. +To mitigate the overhead of rule design, researchers extract API entities using machine learning +methods. Ye et al. [2] propose APIReal, which uses CRF to identify API entities. They label the +API entities and non-entities in the sentence with the “BIO” tag and form the pair of the labeled +sequence and the sequence. They then train CRF on these pairs, and use the trained CRF to label +the input text, from which they obtain API entities with the “BI” or “B” tag. Huo et al. [13], on +the other hand, propose ARCLIN, which uses BI-LSTM as encoder and CRF as decoder to identify +API entities, rather than just CRF. However, these methods suffer from data labeling overhead +because preparing a large number of high-quality training data for these sequence labeling models +is unrealistic. +To solve the two overhead issues mentioned above, researchers use LLM to extract entities. Li +et al. [35] use BERT and Yan et al. [36] use XLNet [37] to extract entities in the natural language +domain. These models, however, are limited to a single natural language processing task, i.e., the +entity extraction only. In order to realize joint extraction of multiple tasks, researchers propose LLM- +based unified architectural models, such as UIE [20] and OpenUE [38]. In particular, UIE proposes +SEL to encode different information extraction structures via the hierarchical spotting-associating +structure. Motivated by this, we consider adapting UIE to the joint API entity-relation extraction. +However, UIE is not good at dealing with complex sentences, particularly long and ambiguous +sentences containing API entities and various relations, because UIE has only one static prompt to +identify all types of API relations. As a result, when confronted with ambiguous sentences, the +more relation types to recognize, the more noise interference, and the lower the UIE recognition +rate. In contrast, we propose LLM-based AERJE, which extracts API entities and relations from +unstructured complex sentences at the same time. Different from UIE, our dynamic prompt design +could generate a small number of potentially relevant relations for input text to eliminate noise +interference and lessens the difficulty of API relation extraction. +7 +CONCLUSION AND FUTURE WORK +In this paper, we are the first to formulate heterogeneous API extraction and API relation extraction +task as a sequence-to-sequence task, and proposes AERJE to extract API entities and relations +from unstructured text simultaneously using pre-trained LLM and dynamic prompt learning. The +systematic evaluation of AERJE is conducted on a set of long and ambiguous sentences from Stack +Overflow. The experimental results show that AERJE’s ability to extract API entities and relations +can be activated with a small amount of data, allowing it to accurately identify API entities and +relations from complex text that the model has never seen during fine-tuning. In the future, we +will carry out the plans mentioned in the discussion and apply AERJE to any software engineering +task supported by API entity and relation extraction, such as API linking, API search, and API +recommendation. +ACKNOWLEDGMENTS +The work is partly supported by the National Nature Science Foundation of China under Grant +(Nos.62262031, 61902162), the Nature Science Foundation of Jiangxi Province (20202BAB202015), +the Central Guided Local Science and Technology Development Special Project (20222ZDH04090), +the Graduate Innovative Special Fund Projects of Jiangxi Province (YC2021-S308, YC2022-S258). +, Vol. 1, No. 1, Article . Publication date: January 2023. + +18 +Qing Huang, Yanbang Sun, Zhenchang Xing, Min Yu, Xiwei Xu, and Qinghua Lu +REFERENCES +[1] Qing Huang, Zhiqiang Yuan, Zhenchang Xing, Zhengkang Zuo, Changjing Wang, and Xin Xia. 1+1>2: Programming +know-what and know-how knowledge fusion, semantic enrichment and coherent application. 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Named entity recognition by using xlnet-bilstm-crf. Neural Process. Lett., +53:3339–3356, 2021. +[37] Zhilin Yang, Zihang Dai, Yiming Yang, Jaime G. Carbonell, Ruslan Salakhutdinov, and Quoc V. Le. Xlnet: Generalized +autoregressive pretraining for language understanding. In NeurIPS, 2019. +[38] Ningyu Zhang, Shumin Deng, Zhen Bi, Haiyang Yu, Jiacheng Yang, Mosha Chen, Fei Huang, Wei Zhang, and Huajun +Chen. Openue: An open toolkit of universal extraction from text. In EMNLP, 2020. +, Vol. 1, No. 1, Article . Publication date: January 2023. + +20 +Qing Huang, Yanbang Sun, Zhenchang Xing, Min Yu, Xiwei Xu, and Qinghua Lu +QING HUANG received the M.S degree in computer application and +technology from Nanchang University, in 2009, and the PH.D. degree in +computer software and theory from Wuhan University, in 2018. He is +currently an Assistant Professor with the School of Computer and Informa- +tion Engineering, Jiangxi Normal University, China. His research interests +include information security, software engineering and knowledge graph. +Yanbang Sun is a second-year master student at the School of Computer +and Information Engineering, Jiangxi Normal University, China. His re- +search interests include software engineering and knowledge graph. +Zhenchang Xing is a Senior Research Scientist with Data61, CSIRO, +Eveleigh, NSW, Australia. In addition, he is an Associate Professor in +the Research School of Computer Science, Australian National University. +Previously, he was an Assistant Professor in the School of Computer Sci- +ence and Engineering, Nanyang Technological University, Singapore, from +2012-2016. His main research areas are software engineering, applied data +analytics, and human-computer interaction. +MIN YU is a Professor in Communication, Electronic Engineering, and +Computer Science at Jiangxi Normal University, was a visiting scholar at +the University of California, Irvine, the USA, and interested in Distributed +computing, Wireless Sensor Network, and Indoor Positioning. +Xiwei Xu is a Senior Research Scientist with Architecture& Analytics +Platforms Team, Data61, CSIRO. She is also a Conjoint Lecturer with UNSW. +She started working on blockchain since 2015. Her main research interest +is software architecture. She also does research in the areas of service +computing, business process, and cloud computing and dependability. +Qinghua Lu is a Senior Research Scientist with Data61, CSIRO, Eveleigh, +NSW, Australia. She has published more than 100 academic papers in +international journals and conferences. Her research interests include the +software architecture, blockchain, software engineering for AI, and AI +ethics. +, Vol. 1, No. 1, Article . Publication date: January 2023. + diff --git a/8NE2T4oBgHgl3EQflQeg/content/tmp_files/load_file.txt b/8NE2T4oBgHgl3EQflQeg/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..de21bfec82d2188ea756c0bbbdcd7b370f0d1329 --- /dev/null +++ b/8NE2T4oBgHgl3EQflQeg/content/tmp_files/load_file.txt @@ -0,0 +1,1268 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf,len=1267 +page_content='API Entity and Relation Joint Extraction from Text via Dynamic Prompt-tuned Language Model QING HUANG,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Jiangxi Normal University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' School of Computer Information Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' China YANBANG SUN∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Jiangxi Normal University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' School of Computer Information Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' China ZHENCHANG XING,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' CSIRO’s Data61 & Australian National University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' College of Engineering and Computer Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Australia MIN YU†,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Jiangxi Normal University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' School of Computer Information Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' China XIWEI XU,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' CSIRO’s Data61,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Australia QINGHUA LU,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' CSIRO’s Data61,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Australia Extraction of Application Programming Interfaces (APIs) and their semantic relations from unstructured text (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=', Stack Overflow) is a fundamental work for software engineering tasks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=', API recommendation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' However, existing approaches are rule-based and sequence-labeling based.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' They must manually enumerate the rules or label data for a wide range of sentence patterns, which involves a significant amount of labor overhead and is exacerbated by morphological and common-word ambiguity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' In contrast to matching or labeling API entities and relations, this paper formulates heterogeneous API extraction and API relation extraction task as a sequence-to-sequence generation task, and proposes AERJE, an API entity-relation joint extraction model based on the large pre-trained language model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' After training on a small number of ambiguous but correctly labeled data, AERJE builds a multi-task architecture that extracts API entities and relations from unstructured text using dynamic prompts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' We systematically evaluate AERJE on a set of long and ambiguous sentences from Stack Overflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' The experimental results show that AERJE achieves high accuracy and discrimination ability in API entity-relation joint extraction, even with zero or few-shot fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Additional Key Words and Phrases: API Entity, API Relation, Joint Extraction, Dynamic Prompt ACM Reference Format: Qing Huang, Yanbang Sun, Zhenchang Xing, Min Yu, Xiwei Xu, and Qinghua Lu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' API Entity and Relation Joint Extraction from Text via Dynamic Prompt-tuned Language Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' 1, 1 (January 2023), 20 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='1145/nnnnnnn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='nnnnnnn 1 INTRODUCTION Application Programming Interfaces (APIs) are important software engineering artifacts that can be frequently found in a wide range of natural language texts, from official API references and tutorials to informal online forums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Meanwhile, API relations are also embedded in these texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' For example, the text “To manipulate data you actually need executeUpdate() rather than execute- Query()” in the Stack Overflow (SO) post 1 describes the Function-Replace relation [1] between executeUpdate() and executeQuery().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' This API relation reveals that we should replace executeQuery() with executeUpdate() to solve the question in the post, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=', “why cannot issue data manipulation statements with executeQuery()”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' API entity and relation extraction from unstructured texts is ∗Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Sun and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Huang are co-first authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' †M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Yu is the corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' 1https://stackoverflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='com/questions/1905607 Authors’ addresses: Qing Huang, Jiangxi Normal University, School of Computer Information Engineering, Nanchang, Jiangxi, China, qh@whu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Yanbang Sun, Jiangxi Normal University, School of Computer Information Engineering, Nanchang, Jiangxi, China, ybsun@jxnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Zhenchang Xing, CSIRO’s Data61 & Australian National University, College of Engineering and Computer Science, Canberra, Australia, zhenchang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='xing@data61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='csiro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='au;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Min Yu, Jiangxi Normal University, School of Computer Information Engineering, Nanchang, Jiangxi, China, myu@jxnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Xiwei Xu, CSIRO’s Data61, Sydney, Australia, xiwei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='xu@data61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='csiro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='au;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Qinghua Lu, CSIRO’s Data61, Sydney, Australia, qinghua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='lu@data61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' csiro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' , Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' 1, Article .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Publication date: January 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='03987v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='SE] 10 Jan 2023 2 Qing Huang, Yanbang Sun, Zhenchang Xing, Min Yu, Xiwei Xu, and Qinghua Lu Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Three types of ambiguities for API entities and relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' PostID Sentence #47871272 You need to override remove() in your iterator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' #14200489 This code is invalid since l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='remove() is called during iteration over l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' #60017952 You may be calling iterator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='remove more than once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' #34682267 By default, printWriter calls flush in println, whereas it doesn’t do this in print.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' #703396 If the idea is to :::: print integer stored as doubles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' #322715 linkedlist and arraylist are two different implementations of the list interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' #33405095 nextline() will read the entire line, but next() will only read the next word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' #355089 StringBuffer is synchronized, StringBuilder is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Note: API mention is tagged with an underline;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' common word is tagged with a wavy line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' fundamental for efficiently accessing and applying API knowledge to various software engineering tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Once extracted, these entities and relations can be organized into structured knowledge (particularly in the form of knowledge graphs) to support a variety of software engineering tasks such as API linking [2, 3], API recommendation [4, 5], and API comparison [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' There are currently two main types of approaches for extracting API entities from unstructured text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' The first is a rule-based approach such as language-convention based regular expressions [7, 8], island parsing [9, 10] and heuristic rule matching [1, 6, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Because it is impossible to manually enumerate the rules that adapt to all sentence patterns, it suffers from rule design overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' The second is a sequence labeling based approach such as CRF [2, 12] and Bi-LSTM-CRF [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Because it is impossible to manually label entities for a large amount of sentences, it suffers from data labeling overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Compared with API entity extraction, relation extraction from software text is rather primitive, which relies on either API syntax (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=', a class declares a method) [6], special-tag annotated relations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=', “see also” keyword and hyperlink-based method) [14], or some ad-hoc relation phrases (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=', “differ in” and “be similar to”) [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Same as rule-based entity extraction, these relation extraction methods suffer from rule-design overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' We refer to rule design overhead and data labeling overhead as labor overhead in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' This labor overhead is exacerbated by three types of ambiguities, which necessitate the manual design of more rules or the labeling of more data to distinguish ambiguous sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Morphological ambiguity, which includes abbreviations, synonyms, and misspellings, is one type of ambiguity [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' It is common in informal discussions, because people rarely write full API names that exactly match the API names in the library [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Three sentences in the first row of Table 1, for example, shows three morphological variations of API java.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='util.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='iterator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='remove(), that either omit some prefixes and special symbols (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=', package names, class names, and “()”), or are preceded by user-defined variable names.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' The second type is common-word ambiguity between common words and API mentions [12], which occurs because people frequently write API method names without proper punctuation, parentheses, and uppercase letters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' For example, as shown in the second row of Table 1, even though the word print appears in two sentences, print in the first sentence refers to the API java.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='io.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='printwriter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='print(), whereas print in the second one is only a verb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' The final type is expression ambiguity of API semantic relations, which is caused by changes in sentence patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' In general, the same API relation can be expressed in multiple sentence patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' For example, as shown in the third raw of Table 1, three sentence patterns are used in the three sentences, all of which express the Behavioral-Difference relation between API entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' They are “API1 and API2 are different”, “API1 does one thing, but API2 does the other thing”, and “API1 is (adjective), API2 is not”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' , Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' 1, Article .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Publication date: January 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' API Entity and Relation Joint Extraction from Text via Dynamic Prompt-tuned Language Model 3 To alleviate the labor overhead, we devise a novel idea of extracting API entities and relations using a large pre-trained language model (LLM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' LLM stores a large amount of prior knowledge and can serve as a neural knowledge base of real-world entities and relations [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' In addition, LLM can provide better model initialization [17] and strong learning capbility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' By fine-tuning a LLM with a small set of domain-specific training data, we can prompote the LLM to identify as many API entities and relations as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' In order to make LLM be more discriminative, the training data should contain sufficient morphological and common-word ambiguity, and the API entities and relations should be labeled correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' To reduce manual labeling, we devise morphology and verb-based data augmentation strategies to generate more ambiguous data but correctly labeled sentences for the LLM fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Existing work [18] separates API entity extraction and relation extraction as two tasks, leaving relation extraction heavily reliant on entity extraction results, which leads to error propagation [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Instead, we consider API entity extraction and relation extraction as correlated tasks and adopt a unified model for joint entity and relation extraction, inspired by the recent work on universal information extraction (UIE [20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' However, the LLM (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=', T5 [21]) in UIE often fails with complex sentences, particularly long and ambiguous sentences containing API entities and various relations, because it only uses one static prompt to recognize all types of API relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' To tackle this issue, we design a dynamic prompt generator, inspired by the input-dependent prompt tuning method [22], that generates dynamic prompts for a small number of potentially relevant relations at inference time based on the actual input sentences, rather than relying on the same static prompt for all inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' When confronted with complex sentences, the more relation types to recognize, the more noise it suffers from, the more difficult it is for LLM to understand to what extent a complex sentence contains certain relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' As our dynamic prompt reduces the number of relation types to recognize and mitigate noise interference, it improves the extraction accuracy of API relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' In this paper, we propose a API Entity-Relation Joint Extraction framework, called AERJE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' It consists of a dynamic prompt generator and a joint entity-relation extractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' The kernel of the prompt generator is a BERT-based text classifier that is used to classify the input text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Each class represents an API relation and the prompt generator generates dynamic prompts based on the top-N possible API relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' The generated dynamic prompt and the input sentence are fed into the joint entity-relation extractor to extract the API entities and relations contained in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' In our current implementation, the joint entity-relation extractor is Transformer-base LLM (T5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' The prompt generator and the entity-relation extractor are fine-tuned in an end-to-end manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' No model, to the best of our knowledge, can simultaneously extract both API entities and relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' AERJE, on the other hand, achieves an F1 score of 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='51% for API entity extraction, which is approximately 6% higher than the state-of-the-art API entity recognition model ARCLIN [13] and 7% higher than APIReal [2], and an F1 score of 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='20% for API relation extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Then, we evaluate the impact of intrinsic factors (two data augmentation strategies and the number of API relations in the dynamic prompts) on performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Our experiments find that data augmentation helps to improve AERJE’s discriminative capability for API entities and relations, and the dynamic prompts with four API relations can significantly improve AERJE’s extraction accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Finally, we assess AERJE’s generalization and ability to extract API entities and relations in low-resource scenarios (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=', less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='8% fine-tuning data) and find that, even under low resource conditions, our AERJE still has strong extraction ability, outperforming APIReal [2] and ARCLIN [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' The main contributions of this paper are as follows: Conceptually, we are the first to formulate heterogeneous API extraction and API relation extraction tasks as a uniform sequence-to-sequence generation task, and propose AERJE, an API entity-relation joint extraction framework based on pre-trained LLMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' , Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' 1, Article .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Publication date: January 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' 4 Qing Huang, Yanbang Sun, Zhenchang Xing, Min Yu, Xiwei Xu, and Qinghua Lu In fact, collections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='sort() has already been migrated to list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='sort().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' You better using getint() instead of get().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' You can use double.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='parsedouble() to convert a string to a double.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Input Sentences Bert Dynamic prompt ( ( API: getint() ( function replace: get() ) ) ( API: get() ) ) ( ( API: collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='sort() ) ( API: list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='sort() ) ) [spot] API [asso] function replace [asso] efficiency comparison [asso] type conversion [text] You better using getint() instead of get().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' [spot] API [asso] function similarity [asso] logic constraint [asso] type conversion [text] In fact, collections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='sort() has already been migrated to list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='sort().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' [spot] API [asso] logic constraint [asso] type conversion [asso] function similarity [text] You can use double.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='parsedouble() to convert a string to a double.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Joint Entity-Relation Extractor Linear P(Relation) top-3 Relations a b c Ⅰ Ⅱ CLS E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' E E D .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' D D Latent Vector .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' b c a b c a Ⅲ Dynamic Prompt Generator T5 ( ( API: string ( type conversion: double ) ) ( API: double ) ( API: double.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='parsedouble() ) ) Structured Extraction Language .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='. .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='. .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='. Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Overall Framework of AERJE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' The Dynamic prompt’s bold font represents the semantic relation to be extracted from the input sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='b lacks a bold font because I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='b contains no semantic relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' We devise two data augmentation strategies in order to obtain more ambiguous but correctly labeled sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Learning such sentences enables AERJE to be more discriminative for API entities and relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Unlike the single task model, we build a multi-task architecture that encodes the structures of entity and relation extraction into a unified structure language for extracting API entities and relations simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Instead of using a single static prompt with all types of API relations for all sentences, we design a dynamic prompt based on relation classification, which reduces the number of relation types to recognize, eliminates noise interference, and lowers the difficulty of relation extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' We systematically evaluate the AERJE’s intrinsic factors, performance, generalization, and few-shot learning capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' It is the first approach to extract API entities and relations simultaneously, and it achieves superior performance than independent API extraction and API relation extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Our data package can be found here2, the code will be released after the paper is accepted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' 2 APPROACH We formulate heterogeneous API extraction and API relation extraction tasks as a uniform sequence- to-sequence generation task, and propose a novel model AERJE to accomplish it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' 1, AERJE consists of a dynamic prompt generator and a joint entity-relation extractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' The dynamic prompt generator generates dynamic prompts based on the input texts (one at a time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' The input text is then appended to the prompt to form a whole input that is fed into the joint entity-relation extractor, which generates a structured extraction language sequence with API entities and relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='1 Dynamic Prompt Generator This section describes how to build a prompt that unifies heterogeneous API extraction and API relation extraction tasks, followed by a discussion of how to design dynamic prompt to improve AERJE’s API relation extraction performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='1 Prompt Construction for Multi-tasking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' In order to extract both API entities and API relations from an input text, the prompt consists of API entity type, API relation type, and the input text, which are labeled by [spot], [asso] and [text], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' For example, “[spot] API [asso] function 2https://anonymous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='4open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='science/r/AERJE-6DBF/README.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='md , Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' 1, Article .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Publication date: January 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' API Entity and Relation Joint Extraction from Text via Dynamic Prompt-tuned Language Model 5 replace [asso] efficiency comparison [text] You better using getint() instead of get()” represents an API entity type “API”, two relation types “function replace” and “efficiency comparison”, and an input text “You better using getint() instead of get()”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' In this work, we consider a generic API entity type “API” and seven relation types defined in [1], including “function similarity”, “behavior difference”, “logic constraint”, “type conversion”, “function collaboration”, “efficiency comparison”, “function replace”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Note that more fine-grained API entity types can be used, such as “class”, “method”, “field” [23], but we leave it as our future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='2 Dynamic Prompt Generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' As stated in Section 1, the more relation types there are, the harder it is for T5 to determine which types of relation the API entities in the input text belong to, especially when the sentence is long and ambiguous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' If we adopt the static prompt that includes all seven relation types, the relation extraction performance of the model will decrease (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' RQ3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' As a result, we design a dynamic prompt generation method to make the content of the prompt more accurate and instructive for the complex input text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' The dynamic prompts, as shown in II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='a of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='1, contain only the top-N relations and provide better guidance to the subsequent T5-supported joint entity-relation extractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Here, the prompt generator is implemented as a text classifier which predicts the API relations present in the input text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' We use a BERT-based classifier because the pre-training task (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=', Next Sentence Prediction) of BERT [24] is consistent with our task, both of which are classification tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Given a sentence containing API entities (see I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='a of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' 1), the BERT-based classifier outputs the probability that the sentence belongs to each semantic relation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' the top-3 relations are then chosen as candidate relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Finally, entity type, candidate relations, and input sentence are connected by labels (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=', [spot], [asso], [text]) to generate the dynamic prompt (see II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='a of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Note that the BERT-based classifier in our current implementation aims to narrow the scope and provide candidate relations, and it cannot replace the API relations extractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' When the candidate relations classified by the classifier do not fit these entities in the sentence, the extractor does not force a relation to be selected from the incorrect candidate relations, but instead assumes that no relation exists between these entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' For example, given a sentence with no relations between API entities (see I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='b of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='1), the dynamic prompt generator generates a dynamic prompt (see II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='b of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Based on such a dynamic prompt, the subsequent extractor will not extract relations from the sentence as none of the candidate relation types is applicable to the input sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' To summarize, too many candidate relations may reduce the extractor’s ability to recognize them, while too few candidate relations may cause the extractor to miss the correct relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' As a result, we should investigate the appropriate number of candidate relations (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' RQ3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='2 API Joint Entity-Relation Extractor We adopt a structured extraction language (SEL) [20] to encode the structures of entity extraction and relation extraction into a unified representation, so that heterogeneous API extraction and API relation extraction tasks can be modeled uniformly within a sequence-to-sequence generation framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' The first sequence refers to the dynamic prompt, while the second sequence refers to the SEL sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='1 Structured Extraction Language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' SEL sequence is proposed to encode different information extraction structures via the hierarchical spotting-associating structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='a shows its universal format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' “Spot Name: Info Span” denotes various entity type and the object of a specific entity type;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' “Asso Name: Info Span” denotes various relation types and the associated object of a specific relation type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='b shows the concrete SEL sequence in our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' “API: getint()” represents that “getint()” is an API entity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' “function replace: get()” represents that the relation between “getint()” and “get()” , Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' 1, Article .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Publication date: January 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' 6 Qing Huang, Yanbang Sun, Zhenchang Xing, Min Yu, Xiwei Xu, and Qinghua Lu ( ( Spot Name: Info Span （ Asso Name: Info Span ) ) ( Spot Name: Info Span ) ) ( ( API: getint() （ function replace: get() ) ) ( API: get() ) ) a b Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Specialization of Universal Structured Extraction Language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' is “function replace”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' From this concrete SEL sequence, we can extract API entities and relations simultaneously as it unifies the structure of API entities and relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='2 SEL Sequence Generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' We implement our API joint entity-relation extractor as the sequence-to-sequence generation framework: � 𝑦1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' ,𝑦|𝑦| � = JE( � 𝑝1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' , 𝑝 |𝑝 | � ) (1) where JE is a Transformer-based LLM, � 𝑝1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' , 𝑝 |𝑝 | � is the dynamic prompt, � 𝑦1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' ,𝑦|𝑦| � is the linearized SEL sequence that contains the API entities and relations to be extracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' In this frame- work, we feed the dynamic prompt into the LLM (as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='II), and the LLM generates the SEL sequence (as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='III), from which we can obtain API entities and relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' The dynamic prompt to the JE can also be written in the format described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='1: � 𝑝1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' , 𝑝 |𝑝 | � =[[ spot ], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' [ spot ] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' , [ asso ], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' , [ asso ] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' , [ text ],𝑥1,𝑥2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' ,𝑥 |𝑥 | � (2) where 𝑥 = � 𝑥1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' ,𝑥 |𝑥 | � denotes the input text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' To better illustrate the framework’s internal mechanics, an encoder-decoder-style architecture is introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Given the dynamic prompt 𝑝, JE computes the hidden representation H = � p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' , p|𝑝 | � of each token: H = Encoder �𝑝1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' , 𝑝 |𝑝 | � (3) where Encoder(·) is a Transformer encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Then JE decodes the prompt into a SEL sequence in an auto-regressive style.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' At the step 𝑖 of decoding, JE generates the 𝑖-th token 𝑦𝑖 in the SEL sequence and the decoder state h𝑑 𝑖 as following: 𝑦𝑖, h𝑑 𝑖 = Decoder �� H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' h𝑑 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' , h𝑑 𝑖−1 �� (4) Decoder(·) is a Transformer decoder that predicts the conditional probability 𝑝 (𝑦𝑖 | 𝑦 <𝑖, 𝑝) of token 𝑦𝑖 until the end symbol is output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='3 Model Training This section describes data collection and augmentation, model training, which includes training a BERT-based classifier and fine-tuning a Transformer-based LLM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' , Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' 1, Article .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Publication date: January 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' API Entity and Relation Joint Extraction from Text via Dynamic Prompt-tuned Language Model 7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='1 Data Collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Given that the relation types we consider are all from a knowledge graph of Java APIs [1], we randomly chose 5,000 Java-tagged posts from the Stack Overflow data dump 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Each post is accompanied by its answers and post tags (such as “java”, “arrays”, “java.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='lang”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' We choose the most voted answers from the posts to ensure the quality of the training data, but we exclude code snippets and all HTML tags because the focus of our study is informal text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' All the answers are then splitted into sentences using spaCy 4, yielding 28,140 sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Every sentence is accompanied by multiple category tags from the post to which it belongs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Then, for each sentence, we parse it into tokens using the software-specific tokenizer [12] which preserves the integrity of an API mention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' iterator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='remove(), for example, is treated as a single token.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Finally, we crawl all APIs in JDK 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='8 5, and use these APIs to filter out the sentences containing API entities, as inspired by a previous study [13], based on the following criteria: Because of the large number of morphological ambiguities, a token may be an API entity if it partially matches any of the crawled APIs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=', remove() and java.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='util.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='Iterator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='remove()).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Since API mentions usually end with “()”, the token is treated as an API entity if it contains “()”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' API mentions typically include “.” to indicate a function call (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=', iterator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='remove(), or l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='remove());' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' thus, if token contains “.”, we consider it to be an API entity .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' After filtering, we obtain 9,111 sentences that may contain API entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' However, this is rough sentence filtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' In order to do accurate sentence filtering, We invite 12 master students (all with more than five years Java experience) to examine the API entities and annotate the semantic relations between APIs in order to further verify whether these sentences contain API entities and the seven types of API relations we aim to extract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' We train the annotators prior to annotation to ensure that they can recognize these API relations in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' After training, the annotators were divided into six groups, with two students from each group annotating the same content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' After the annotation, we assign two authors to deal with the annotation results’ conflicts, and the Cohen’s Kappa [25] coefficient is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='859 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=', almost perfect agreement).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' As a result, we get a total of 2917 sentences, with 2471 containing only entities and 446 containing both entities and relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='2 Data Augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' To improve the AERJE’s ability to recognize API entities and relations from long and ambiguous sentences, we devise two data augmentation strategies to obtain more ambiguous sentences for model training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Morphology based Mutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Inspired by [13], we change the form of each API entity in the sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Specifically, we replace the API entity itself with the final piece of its fully qualified name.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' For example, iterator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='remove() is replaced with remove() or remove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Verb based Mutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' We use spaCy to locate the verbs on which each API entity relies, and then replace those verbs with synonyms, as Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' [26] do to obtain similar question titles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' As shown in the seventh sentence of Table 1, we replace “read” with “load”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' However, because spaCy may not obtain the correct API entity, we must identify the dependency between the API entity’s subtoken and the verb to ensure the mutation quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' For example, there is a dependency between “nextline” and “read”, so we can reliably mutate “read” with synonyms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Our data augmentation strategy does not include sentence pattern mutation [26], which uses different sentence patterns to present the same API relation between the same API entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Unlike the morphology-based and verb-based mutation, this mutation is not reliable in software text which demands stricter semantics than general text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' The sentence pattern mutation could result in sentence structure reconstruction, which would likely change the sentence semantics, contaminate 3Retrieved June 6, 2022 from https://archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='org/download/stackexchange/ 4https://spacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='io 5https://docs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='oracle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='com/javase/8/docs/api , Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' 1, Article .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Publication date: January 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' 8 Qing Huang, Yanbang Sun, Zhenchang Xing, Min Yu, Xiwei Xu, and Qinghua Lu the training data, and compromise AERJE training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' For example, the original sentence “StringBuffer is synchronized, StringBuilder is not” may be mutated into “StringBuffer and StringBuilder differ in synchronized”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' The original sentence indicates that StringBuffer is synchronized and StringBuilder is asynchronous, but the mutated sentence does not specify who is synchronous or asynchronous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' We obtain 2,334 sentences as the initial training set and 583 sentences as the initial test set in an 8:2 ratio from the 2,917 sentences collected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' The number of sentences after applying the two data augmentation strategies to the initial training and test sets is 10,678 and 2,686, referred to as the final training set and the final test set, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' This final training set is used to fine-tune the LLM-based extractor, and the final test set is used to test the fine-tuned extractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Here, we split the sentences into training and testing sets and then mutated them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' This ensures that the sentence before and after the mutation is in the same set, preventing the leaking of training data into the test set (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=', one sentence in the training set and its mutation in the test set).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Furthermore, we obtain 1,639 sentences with both entities and relations as the classifier training set from the final training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Similarly, we obtain 387 sentences with both entities and relations as the classifier test set from the final test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='3 BERT-based Relation Classifier Training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' We choose BERT [24] as a relation classifier because its pre-training task (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=', Next Sentence Prediction) is consistent with our task, both of which are classification tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' However, the implementation of relation classifier is not limited to BERT, we can also use TextCNN [27] and FastText [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' In our current implementation, we use the BERT-base classifier to classify each input sentence into N relation types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Based on the N relation types, dynamic prompt generator generates the corresponding dynamic prompt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' A mask language model (BERT) [24] and a linear layer comprise the classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Due to the seven API relation types, the linear layer’s output dimension is set to 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' We obtain the latent vector from the CLS token when we enter the sentence into BERT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' The latent vector obtained from the CLS token characterizes the sentence features better than other positions, resulting in better classification performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' The latent vector is then fed into the linear layer, which produces a vector with seven dimensions, each corresponding to a relation type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Finally, the classifier is trained on the classifier training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' In back propagation, we use the cross-loss entropy to calculate the classifier’s loss and adjust the BERT and linear layer parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' The loss function is formulated as follows, where 𝑧 = [𝑧0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' ,𝑧𝐶−1] represents the linear layer’s output result, and C represents the sentence’s label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Loss(𝑧,𝑐) = −𝑧[𝑐] + log �𝐶−1 ∑︁ 𝑗=0 exp(𝑧[𝑗]) � (5) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='4 LLM-based Extractor Fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' We use the pre-trained T5-v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='1-large model [21] as the LLM in our current implementation because T5’s training objective aligns perfectly with our formulation of the API entity and relation extraction task as a sequence to sequence generation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Furthermore, studies [29, 30] confirm that T5 is capable of capturing rich text information and demonstrate its effectiveness in a variety of downstream NLP tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Our approach is not limited to T5, but can use any Transformer-based LLM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' In order to fine-tune T5, we convert each labeled sentence in the final training set into a SEL sequence (y), then feed it into the dynamic prompt generator to obtain its dynamic prompt (p), and finally construct the labeled corpus: De = {(𝑝,𝑦)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' On the labeled corpus, we fine-tune T5 for 50 epoch with batch size 10 using the Adam optimizer with a learning rate of 1e-4, linear scheduling with a warming up proportion of 6%, and the teacher-forcing cross-entropy loss: LFT = ∑︁ (𝑝,𝑦) ∈De − log 𝑃 (𝑦 | 𝑝;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='𝜃𝑒,𝜃𝑑) (6) , Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' 1, Article .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Publication date: January 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' API Entity and Relation Joint Extraction from Text via Dynamic Prompt-tuned Language Model 9 where 𝜃𝑒 and 𝜃𝑑 are the parameter of encoder and decoder, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' 3 EXPERIMENTAL SETUP This section starts with five questions about AERJE’s performance, followed by a description of the experimental setup, which includes the dataset, baseline, and evaluation metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='1 Rearch question RQ1: Effectiveness of Data Augmentation RQ2: Optimal Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' of Relation Types for Dynamic Prompt RQ3: Joint Extraction Performance of AERJE RQ4: Generalization Ability of AERJE RQ5: AERJE’s Performance in Low-Resource Scenario 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='2 Dataset As described in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='2, there are three groups of data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' The first group refers to the sentences collected initially, some of which contain only entities and others contain both entities and relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' The initial training set consists 2,334 sentences, of which 362 contain both entities and relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' The initial test set consists 583 sentences, of which 84 contain both entities and relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' The second group refers to the sentences after applying the two data augmentation strategies, some of which contain only entities and others contain both entities and relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' The final training set with a total of 10,678 sentences, 1639 of which contain both entities and relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' The final test set with a total of 2,686 sentences,387 of which contain both entities and relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' The third group refers to the sentences containing both entities and relations in the final training and testing sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' The classifier training set with a total of 1,639 sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' The classifier test set with a total of 387 sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='3 Baselines Our AERJE is capable of API entity-relation joint extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' However, to the best of our knowledge, no previous work has focused on extracting both API entities and relations from unstructured texts at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' As a result, we can only compare AERJE with the existing work in the respective fields of API entity extraction and API relation extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' For API entity extraction, there are rule-based methods (such as regular Expressions [7, 8]), heuristic rule matching methods [1, 6, 11], and sequence-labeling based methods (such as AR- CLIN [13] using BI-LSTM as encoder and CRF as decoder, APIReal [2] using only CRF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Since the performance of the first two classes of methods is not as good as that of the last class of methods [13], we choose ARCLIN, APIReal as baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' We obtain them source code from Github 6 7, and label the API entities and non-entities in the sentences with the “BIO” tag (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=', “B”: the beginning of API entity segment, “I”: the inside of API entity segment, “O”: non-entity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Then we create pairs of original sentences and labeled sentences to train these two baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Finally, we the trained models on the final test set, from which we obtain API entities based on the “BIO” tag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' 6https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='com/YintongHuo/ARCLIN 7https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='com/baolingfeng/APIExing , Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' 1, Article .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Publication date: January 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' 10 Qing Huang, Yanbang Sun, Zhenchang Xing, Min Yu, Xiwei Xu, and Qinghua Lu For API relation extraction, there are only rule matching methods that rely on API syntax [6], special-tag annotated relations [14], or some ad-hoc relation phrases [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' It is very difficult to re-implement these methods due to the rule-design overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Furthermore, it is impractical to apply these methods as we assume plain texts without any special annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Instead, we implement a variant 𝐴𝐸𝑅𝐽𝐸𝑤/𝑜𝐷𝑃𝐺 (𝐷𝑃𝐺 means the dynamic prompt generator), which uses a static prompt with all 7 relation types to evaluate the performance of the full-version of AERJE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' In addition, we also implement two other variants of AERJE as our baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' One is 𝐴𝐸𝑅𝐽𝐸𝑠𝑖𝑛𝑔𝑙𝑒, which separate API entity and relation extraction into two independent tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' For API entity extraction, the prompt contains only “[spot] API”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' For API relation extraction, the prompt contains only “[asso] relation type”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' 𝐴𝐸𝑅𝐽𝐸𝑠𝑖𝑛𝑔𝑙𝑒 still uses dynamic prompt in relation extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' After relation extraction, we merge the extracted entities and relations as the final results of 𝐴𝐸𝑅𝐽𝐸𝑠𝑖𝑛𝑔𝑙𝑒.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=', the extracted entities getint(), get() and relation function replace are merged as (API: getint() (function replace: get())).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' We compare 𝐴𝐸𝑅𝐽𝐸𝑠𝑖𝑛𝑔𝑙𝑒 with AERJE to understand the effectiveness of joint entity-relation extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Meanwhile, the entity extraction results of 𝐴𝐸𝑅𝐽𝐸𝑠𝑖𝑛𝑔𝑙𝑒 is equivalent to fine-tuning pre-trained model for entity extraction, and its final results is equivalent to fine- tuning pre-trained model for relation extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Therefore, 𝐴𝐸𝑅𝐽𝐸𝑠𝑖𝑛𝑔𝑙𝑒 also reflects the capability of fine-tuning pre-trained model for entity and relation extraction separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Another variant is 𝐴𝐸𝑅𝐽𝐸𝑏𝑎𝑠𝑒, which replace T5-v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='1-large in AERJE with a smaller model backbone, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=', T5-v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='1-base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' We use it to explore the impact of large pre-trained language models on AERJE performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' All variants use the same hyper-parameters as AERJE and remain constant across experimental scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Note that SEL used in AERJE has been demonstrated to be effective in the extraction task [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' As such, we do not to verify the effectiveness of SEL in AERJE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='4 Evaluation Metrics We use Precision, Recall, and F1 score as metrics to evaluate the performance of AERJE and baseline models on our test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Precision means what percentage of API entities and relations extracted are correct, recall means what percentage of the real API entities and relations are extracted, and F1 score is the harmonic mean of precision and recall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' It is important to note that the relation is only correct if the relation type and corresponding entities are both correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' In context of our work, we are not concerned with the top-N relation classification accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' As long as the top-N includes relevant relation types, the extractor does not care about the order of these relation types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Furthermore, a sentence may have 2 or more relations, which renders the top-1 accuracy irrelevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Finally, as the extractor has the capability of ruling out irrelevant relation types in the prompt, it is also not necessary to evaluate the classification precision and recall at N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' 4 EXPERIMENTAL RESULTS This section delves into five research questions to evaluate and discuss the AERJE’s performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='1 RQ1: Effectiveness of Data Augmentation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='1 Motivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' To reduce manual labeling effort and improve model training, we devise two data augmentation strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' We want to investigate if ambiguous but correctly annotated sentences obtained through two data augmentation strategies could improve AERJE’s discriminative capability for extracting API entities and relations, in order to demonstrate the effectiveness of two data augmentation strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='2 Methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' We set up two scenarios 𝐴𝐸𝑅𝐽𝐸𝑤/𝑜𝐷𝐴 and 𝐴𝐸𝑅𝐽𝐸𝑤𝐷𝐴 (𝐷𝐴 means the data augmentation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' 𝐴𝐸𝑅𝐽𝐸𝑤/𝑜𝐷𝐴 is trained on the initial training set, while 𝐴𝐸𝑅𝐽𝐸𝑤𝐷𝐴 is trained on , Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' 1, Article .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Publication date: January 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' API Entity and Relation Joint Extraction from Text via Dynamic Prompt-tuned Language Model 11 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Impact of data augmentation strategy on AERJE Strategy Entity Relation P R F1 P R F1 𝐴𝐸𝑅𝐽𝐸𝑤𝐷𝐴 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='57 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='48 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='51 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='54 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='48 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='20 𝐴𝐸𝑅𝐽𝐸𝑤/𝑜𝐷𝐴 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='11 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='19 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='63 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='71 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='66 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='67 the final training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Both 𝐴𝐸𝑅𝐽𝐸𝑤/𝑜𝐷𝐴 and 𝐴𝐸𝑅𝐽𝐸𝑤𝐷𝐴are tested on the same final test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' This setting allows us to compare the effectiveness of data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='3 Result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Table 2 shows the experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' In terms of API entity extraction, 𝐴𝐸𝑅𝐽𝐸𝑤𝐷𝐴 has precision, recall, and F1-scores of 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='57%, 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='48%, and 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='51%, while 𝐴𝐸𝑅𝐽𝐸𝑤/𝑜𝐷𝐴 has precision, recall, and F1-scores of 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='11%, 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='19%, and 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='63%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' 𝐴𝐸𝑅𝐽𝐸𝑤𝐷𝐴’s precision, recall, and F1-score are all higher than 𝐴𝐸𝑅𝐽𝐸𝑤/𝑜𝐷𝐴’s, with 𝐴𝐸𝑅𝐽𝐸𝑤𝐷𝐴’s recall and F1-score being about 3% higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' In terms of API relation extraction,𝐴𝐸𝑅𝐽𝐸𝑤𝐷𝐴 has precision, recall, and F1-score of 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='54%, 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='48%, and 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='20%, while 𝐴𝐸𝑅𝐽𝐸𝑤/𝑜𝐷𝐴 has precision, recall, and F1-score of 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='71%, 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='66%, and 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='67%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' 𝐴𝐸𝑅𝐽𝐸𝑤𝐷𝐴’s precision, recall, and F1-score are all higher than 𝐴𝐸𝑅𝐽𝐸𝑤/𝑜𝐷𝐴’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' The precision of 𝐴𝐸𝑅𝐽𝐸𝑤𝐷𝐴 is 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='83% higher than that of 𝐴𝐸𝑅𝐽𝐸𝑤/𝑜𝐷𝐴, and the F1-score of 𝐴𝐸𝑅𝐽𝐸𝑤𝐷𝐴 is 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='53% higher than that of 𝐴𝐸𝑅𝐽𝐸𝑤/𝑜𝐷𝐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' This demonstrates that fine-tuning 𝐴𝐸𝑅𝐽𝐸𝑤𝐷𝐴 using a large number of ambiguous sentences with API relations benefits 𝐴𝐸𝑅𝐽𝐸𝑤𝐷𝐴 to distinguish between relations and non-relations, as well as between correct and incorrect relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' In contrast, 𝐴𝐸𝑅𝐽𝐸𝑤/𝑜𝐷𝐴 has not been fine-tuned on ambiguous sentences and thus does not perform as well as 𝐴𝐸𝑅𝐽𝐸𝑤𝐷𝐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' For example, an ambiguous sentence “you want to read up on processbuilder to launch the exe file and then waitfor() to wait until the process is complete”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' 𝐴𝐸𝑅𝐽𝐸𝑤𝐷𝐴 correctly extracts two API entities, ProcBuilder and waitfor(), as well as the “logic constraint” relation between them, from the sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' In contrast, 𝐴𝐸𝑅𝐽𝐸𝑤/𝑜𝐷𝐴 only extracts one API waitfor() from the sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' This shows AERJE’s capability to extract API entities and relations from ambiguous sentences can be improved by fine-tuning with the augmentated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' AERJE’s discriminative capability for API entities and relations can be improved by fine-tuning it with ambiguous but correctly labeled sentences obtained through the data augmentation strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='2 RQ2: Optimal Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' of Relation Types for Dynamic Prompt 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='1 Motivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' As described in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='2, given an input sentence, the dynamic prompt generator employs the BERT-based classifier to predict a set of candidate relation types, which are then included in the dynamic prompt to guide the subsequent joint entity-relation extractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' In this RQ, we would like to investigate how many candidate relation types (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=', top-N classifier results) can provide the most effective guidance to the extractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='2 Methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' We exhaust all cases of N values (from 1 to 6) in the dynamic prompt generator, then fine-tune AERJE on the same final training set and test it on the same final test set to select the most appropriate N value based on experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' We do not test N=7 because it is essentially the static prompt with all seven relation types (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=', 𝐴𝐸𝑅𝐽𝐸𝑤/𝑜𝐷𝑃𝐺 studied in RQ3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='3 Result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' As shown in Table 3, changing the N value has small effect on entity extraction because N represents the number of relation types in the dynamic prompt which does not directly affect entity extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' At N=3, AERJE achieves the marginally best F1-score 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='51% for API entity extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' , Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' 1, Article .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Publication date: January 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' 12 Qing Huang, Yanbang Sun, Zhenchang Xing, Min Yu, Xiwei Xu, and Qinghua Lu Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Model results for different values of N top-N Entity Relation P R F1 P R F1 1 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='88 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='23 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='54 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='92 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='92 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='87 2 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='04 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='18 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='10 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='75 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='80 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='72 3 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='57 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='48 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='51 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='54 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='48 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='20 4 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='84 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='39 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='08 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='51 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='22 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='03 5 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='72 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='39 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='54 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='90 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='30 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='53 6 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='44 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='75 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='59 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='35 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='61 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='95 For relation extraction, changing the N value has larger effect on both precision and recall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' As N increases, both precision and recall improve until N=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' When N=3, the precision, recall and F1-score of AERJE reaches the highest 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='54%, 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='48% and 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='20%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' This means that the correct API relation type is most likely covered in the top-3 candidate relations predicted by the classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' When N is less than 3, however, the F1-score of AERJE in relation extraction decreases because the top-N candidate relations may miss the correct relation type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Here is an example: “A TreeMap has the same limitation (as does a HashMap, which also breaks when the hashcode of its elements changes after insertion)”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' When N=2, classifier predicts two relations between TreeMap and Hashmap, including “behavior difference” and “logic constraint” , but ignores the “function similarity” relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' This ignored relation is at the third relation predicted by the classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' However, when N is greater than 3, the F1-score of AERJE in relation extraction decreases because the dynamic prompt may contain some incorrect relation types, which may mislead the extractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' This misleading effect has bigger impact on precision than on recall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' The optimal number of relation types for dynamic prompt should be set to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' This not only ensures that the majority of the correct relation types appear in the dynamic prompts, but it also prevents the dynamic prompts from containing too many noise relation types which may make the model sacrifice precision for recall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='3 RQ3: Joint Extraction Performance of AERJE 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='1 Motivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' We would like to evaluate AERJE’s performance in API entity and relation joint extraction, compared with the state-of-the-art methods for API entity extraction and API relation extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Note that only our AERJE can achieve joint API entity and relation extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='2 Methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' AERJE is compared to APIReal and ARCLIN for API entity extraction, and three variant models (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=', 𝐴𝐸𝑅𝐽𝐸𝑤/𝑜𝐷𝑃𝐺, 𝐴𝐸𝑅𝐽𝐸𝑠𝑖𝑛𝑔𝑙𝑒, and 𝐴𝐸𝑅𝐽𝐸𝑏𝑎𝑠𝑒) for both API entity and relation extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Note that the entity and relation extraction results by 𝐴𝐸𝑅𝐽𝐸𝑠𝑖𝑛𝑔𝑙𝑒 represents the capability of fine-tuning the pre-trained model for the two tasks separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' All models are trained and tested on the same final training and test sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Details on configuration can be found in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='3 Result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Table 4 shows the evaluation result of AERJE and five baselines on final test sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' We see that AERJE’s F1-score is higher 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='5% than APIReal’s F1-score and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='7% than ARCLIN’s F1-score on API entity extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Compared with the three variant models, AERJE’s F1-score for API entity extraction is only slightly lower (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='18%) than the best performer (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=', 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='69% by 𝐴𝐸𝑅𝐽𝐸𝑠𝑖𝑛𝑔𝑙𝑒), but AERJE’s F1-score for relation extraction is 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='83% higher than that of the second best performer (74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='37% by 𝐴𝐸𝑅𝐽𝐸𝑠𝑖𝑛𝑔𝑙𝑒).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' , Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' 1, Article .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Publication date: January 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' API Entity and Relation Joint Extraction from Text via Dynamic Prompt-tuned Language Model 13 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Comparison of Overall Performance Model Entity Relation P R F1 P R F1 APIReal 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='13 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='90 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='01 ARCLIN 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='76 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='17 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='81 AERJE 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='57 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='48 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='51 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='54 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='48 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='20 𝐴𝐸𝑅𝐽𝐸𝑠𝑖𝑛𝑔𝑙𝑒 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='03 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='38 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='69 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='83 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='47 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='37 𝐴𝐸𝑅𝐽𝐸𝑤/𝑜𝐷𝑃𝐺 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='52 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='78 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='64 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='38 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='62 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='92 𝐴𝐸𝑅𝐽𝐸𝑏𝑎𝑠𝑒 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='39 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='15 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='77 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='97 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='28 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='01 For APIReal and ARCLIN performance on API entity extraction, both AERJE and it variant models outperform them largely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' This superior performance is due to the backbone large pre- trained language models (T5) in AERJE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' During the pre-training, T5 learns linguistic and semantic knowledge in text and has powerful abilities in word and sentence representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Through fine- tuning, the semantic knowledge packed in the T5 can be transferred to the downstream tasks and benefit API entity extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' The amount of knowledge in the T5 also affects the AERJE’s performance on API entity and relation extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Compared with AERJE, the F1-score of 𝐴𝐸𝑅𝐽𝐸𝑏𝑎𝑠𝑒 is reduced by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='74% and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='19% in API entity extraction and API relation extraction, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' The decrease of 𝐴𝐸𝑅𝐽𝐸𝑏𝑎𝑠𝑒’s F1 score on API entity extraction is very small compared with the decrease on API relation extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' It is because the number of sentences containing API entities in the final training set is 6 times more than the number of sentences containing both API entity and relation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=', 10,678 vs 1,639).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Sufficient fine-tuning data for API entity extraction allows the basic T5 model to achieve the equivalent performance on API entity extraction as the large T5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' In contrast, the relation extraction is more complex than the entity extraction, and the amount of fine-tuning data is smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' In such case, the basic T5 cannot compete with the large T5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' For 𝐴𝐸𝑅𝐽𝐸𝑠𝑖𝑛𝑔𝑙𝑒 and AERJE, they achieve almost the same entity extraction performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' How- ever, in terms of API relation extraction, AERJE’s F1-score (81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='20%), precision (86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='54%) and recall (76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='48%) are much higher than 𝐴𝐸𝑅𝐽𝐸𝑠𝑖𝑛𝑔𝑙𝑒’s F1-score (74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='37%), precision (82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='83%) and recall (67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='47%), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' This suggests that fine-tuning pre-trained model for API entity extraction individually or jointly with API relation extraction does not affect the quality of API entity extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' But joint entity and relation extraction is much more effective for the relation extraction task than fine-tuning the model just for the relation extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' For 𝐴𝐸𝑅𝐽𝐸𝑤/𝑜𝐷𝑃𝐺 and AERJE, they also achieve almost the same entity extraction performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' This is due to the fact that dynamic prompt only affects the relation type, not the entity type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' In terms of API relation extraction, AERJE’s precision (86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='54%), recall (76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='48%) and F1-score (81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='20%) are higher than 𝐴𝐸𝑅𝐽𝐸𝑤/𝑜𝐷𝑃𝐺’s precision (75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='38%), recall (70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='62%) and F1-score (72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='92%), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' This is because 𝐴𝐸𝑅𝐽𝐸𝑤/𝑜𝐷𝑃𝐺 uses the same static prompt that includes all seven relation types for all input sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' The more types of relations there are in the prompt, the more noise the prompt is, and the more difficult it is for AERJE to identify and extract the correct relations in the input sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' In contrast, AERJE’s use of dynamic prompt reduces the number of relation types to recognize, improving its ability to extract API relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Standing on the shoulder of large pre-trained language model (T5), AERJE outperforms traditional sequence labeling models for API entity extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Dynamic prompt has no impact on API entity extraction, but can largely boost the performance of API relation extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Fine-tuning the pre-trained model jointly is much more effective than fine-tuning the model just for one task, which makes joint entity-relation extraction more accurate on both tasks than separate entity and relation extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' , Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' 1, Article .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Publication date: January 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' 14 Qing Huang, Yanbang Sun, Zhenchang Xing, Min Yu, Xiwei Xu, and Qinghua Lu 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='4 RQ4: Generalization Ability of AERJE 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='1 Motivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Each API comes with its own API package, which often have different forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Furthermore, as APIs from different packages support diverse functionalities, the texts in which they appear may be different in content and linguistic properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' It is impossible for AERJE to see all API packages during fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' In this RQ, we want to investigate if AERJE can recognize APIs and their relations from the API packages that it does not see during fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='2 Methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' In order to collect as much data from different packages as possible, we combine the final training set and the final test set into a new data set with a total of 13,364 sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Every sentence, as stated in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='1, is accompanied by multiple post tags, some of which show the relationship between the sentence and the API package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' For example, the tag “io” is associated with the package name “java.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='io”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Therefore, we filter out sentences with package names by matching each tag of a sentence to any JDK 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='8 package name.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Here is a partial match, which means it matches a portion of the package name, for example, “swing” can match “javax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='swing”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' And then we pool the package names that appear with the sentences and select the three package names that appear the most frequently (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=', javax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='swing, java.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='io, and java.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='util).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Finally, we gather 1651 sentences whose tags match these three package names.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' To ensure the correctness of the sentences obtained through approximate match, we invite six students (who have previously participated in annotation) and divide them into three groups to annotate sentences from three different packages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Two students in each group annotate the same sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' They independently determine whether the API entities in each sentence are from the specific package (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=', java.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='io, java.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='util, javax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='swing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Here is an example “you can use lines() method in BufferedRead” for java.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='io package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' The sentence is annotated as True, since the API entities line() and BufferedRead only correspond to java.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='io.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Instead, if any API entity in the sentence do not belong to specific package, the sentence is annotated as False.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Then we assign an author to handle conflicts between the group members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Finally, we obtain 999 sentences that strictly matched these packages names.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Cohen’s Kappa [25] coefficient is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='795 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=', substantial agreement).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' The data details for each package are as follows: The java.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='io dataset has 235 sentences, 51 of which contain both entities and relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' 12 of the 51 sentences are non-augmented sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' The javax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='swing dataset has 435 sentences, 76 of which contain both entities and relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' 14 of the 76 sentences are non-augmented sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' The java.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='util dataset has 329 sentences, 68 of which contain both entities and relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' 18 of the 68 sentences are non-augmented sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Our AERJE and baseline models are all trained on one of the three datasets and tested on the two others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' As AERJE outperforms its variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' We don’t consider these variants here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='3 Result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Table 5 shows the results that reflect each model’s generalization ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' For API entity extraction, AERJE’s F1-score achieves 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='05%, when trained on the java.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='util dataset, far exceeding APIReal’s F1-score (61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='27%) and ARCLIN’s F1-score (58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='50%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' We attribute this to the underlying LLM on which AERJE is built.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' As Qiu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' [17] show, LLM provides better model initialization, which usually leads to better generalization performance on the target tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Similar observations can be made for training the models on the java.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='io dataset and the javax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='swing dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Generally, ARCLIN and APIReal may perform well on either precision or recall, but not both and thus poor F1-score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' In contrast, AERJE is very stable with much better precision and recall and with only small fluctuations in F1-scores across the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' For API relation extraction, AERJE’s F1-score is 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='98%, 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='99% and 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='48% when trained on the java.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='io, java.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='util and javax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='swing datasets, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' In the across-package training-testing , Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' 1, Article .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Publication date: January 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' API Entity and Relation Joint Extraction from Text via Dynamic Prompt-tuned Language Model 15 Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Comparison of Generalization Ability Model java.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='io java.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='util javax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='swing Entity Relation Entity Relation Entity Relation P R F1 P R F1 P R F1 P R F1 P R F1 P R F1 APIReal 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='02 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='97 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='53 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='11 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='54 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='27 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='99 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='62 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='70 ARCLIN 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='93 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='84 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='50 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='55 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='60 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='50 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='64 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='57 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='90 AERJE 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='00 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='35 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='66 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='87 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='03 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='98 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='17 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='93 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='05 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='68 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='35 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='99 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='89 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='91 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='86 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='92 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='94 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='48 Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Experimental results in a low-resource scenario Model 1-Shot 5-Shot 10-Shot Entity Relation Entity Relation Entity Relation P R F1 P R F1 P R F1 P R F1 P R F1 P R F1 APIReal 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='30 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='92 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='57 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='17 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='67 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='20 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='94 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='59 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='49 ARCLIN 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='07 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='38 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='50 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='64 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='91 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='76 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='58 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='52 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='34 AERJE 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='76 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='06 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='43 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='34 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='49 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='44 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='09 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='62 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='90 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='68 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='96 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='80 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='47 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='74 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='74 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='00 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='94 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='30 setting, the performance of AERJE degrades, compared with the non-across-package setting (see Table 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' However, when trained on the java.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='util dataset, AERJE’s F1-score (79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='99%) is only about 1% less than non-across-package setting (81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='20%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' This suggests that AERJE is capable of dealing with the data drift across different packages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' In addition, different across-package training-testing settings also bring different results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' When using java.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='util for training AERJE, its F1-score is about 39% higher than the F1-score of AERJE trained on java.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='io.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' First, due to java.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='io having fewer sentences with relations than java.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='util (51 vs 68).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Second, java.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='io data has fewer non-augmented sentences with relations than java.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='util (12 vs 18), which makes java.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='io data less diverse than java.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='util.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Our AERJE has a strong generalization ability in face of the data drift across different API packages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' This ability comes from the generalization ability of the underlying LLM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='5 RQ5: AERJE’s Performance in Low-Resource Scenario 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='1 Motivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Labor overhead means that the data available for training is limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' In this RQ, we want to investigate how well AERJE perform when trained with the extremely small amount of training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='2 Methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' We conduct a K-shot experiment, where K can be 1, 5, or 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' To begin the K-shot experiment, we randomly select K sentences from the final training set for each relation type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Then we choose K sentences at random from the final training set that contain only entities but no relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' This yields a training set containing 8*k sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Finally, we train our AERJE and baseline models on this training set and test them on the final test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Note that, to avoid the influence of random sampling, we repeat each K-shot experiment ten times with different samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='3 Result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' For API entity extraction, Table 6 shows the performance of each model in three low-resource scenarios (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=', 1-shot, 5-shot, and 10-shot) where AERJE significantly outperforms APIReal and ARCLIN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Especially, in the 1-shot scenario, AERJE’s F1-score is 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='43%, which is significantly higher than APIReal’s (26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='57%) and ARCLIN’s (58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='50%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Compared to APIReal and ARCLIN, the LLM-based AERJE has a large amount of prior knowledge from the LLM pre-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' As the fine-tuning shot increases, the accuracy of AERJE improves fast, especially on F1-score, reaching the F1-score 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='90% at 5-shot and 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='74% at 10-shot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' For API relation extraction, in the 1-shot scenario, AERJE does not perform well, but it still magically achieves the recall 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='49%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' However, with only 4 more examples (at 5-shot), the F1-score of AERJE significantly increases from below 16% at 1-shot to about 43% at 5-shot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' This suggests that the underlying LLM can quickly adapt to the API relation extraction task that it does not see , Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' 1, Article .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Publication date: January 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' 16 Qing Huang, Yanbang Sun, Zhenchang Xing, Min Yu, Xiwei Xu, and Qinghua Lu during pre-training with only a few examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' In the few-shot setting, we see that precision is much more difficult to improve than recall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' It could be due to the ambiguities of relations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' the same type of relation can be expressed in very different forms (as shown in table 1), while different types of relations may be expressed in the similar forms (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=', “API1 be ADJ to API2” represents function similarity or function opposite relation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' With only a few examples of each type of relation, it makes learning to distinguish between them more difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Furthermore, AERJE’s F1-score for entity extraction is 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='43% at one-shot, while its F1-score for relation extraction is only 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='44%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' The primary cause for this is that the training set of one-shot contains almost all API entity ambiguity types but only a few API relation ambiguity types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' As a result, relation extraction is more difficult than entity extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' AERJE can quickly adapt the underlying LLM to the API entity and relation extraction tasks with only a small number of fine-tuning data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Prior knowledge in LLM enables this quick adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Relation extraction is much harder than entity extraction in the few-shot setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' 5 DISCUSSION The major threat to internal validity is the manual labeling of training and testing datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Incorrect human labels could harm modeling training and testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' To mitigate this threat, we invited two students to annotate the same content and assigned an author to resolve disagreements in the labeling results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' However, even humans can’t always tell if a token references an API, especially when it comes to common nouns that reference basic computing concepts, such as policy and time, which can be either basic noun concepts or APIs (java.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='policy class, java.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='time package).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' We take a conservative strategy, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=', common nouns as API entities, unless both annotators agree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' The threat to external validity is three-fold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' The first external threat is that we only collect data on Stack Overflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Although our model performed well on the SO data set, we intend further to validate its generalization performance in the other data sources (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=', Java Tutorial8, SitePoint9, and Reddit10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' The second external threat is that AERJE has only been tested on Java packages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' We chose Java because previous work [1, 6, 11] has demonstrated how difficult it is to extract these API entities and relations from it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' In the future, we plan to expand AERJE to other programming languages (such as Python and C#).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' The third external threat stems from two AERJE components: the BERT-based classifier and the T5-based extractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' There are numerous alternative models for both components of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' TextCNN [27] and FastText [28] can be used to build the classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' It is possible to use BART [31] and GPT-3 [32] to implement the extractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' In the future, we will compare two AERJE components with alternative models to determine the best performing model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' 6 RELATED WORK API entity and relation extraction is a fundamental work in software engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' It is useful in the construction of knowledge graphs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' extracted structured API knowledge can help with many software engineering tasks such as API linking [2, 3, 8, 33], API misuse detection [11], API recommendation [4, 5], and API comparison [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' This section describes the methods for extracting API entities and relations from unstructured text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Bacchelli [7, 34] and Dagenais [3] detect class and method mentions in developer emails, docu- mentation and forum posts using regular expressions of distinct orthographic features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Ren [11], Huang [1], and Liu [6] extract entities from SO posts using the HTML tag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Bacchelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' [10] extract coarse-grained structured code fragments from natural language text with island 8http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='java2s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='com 9https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='sitepoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='com/ 10https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='reddit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='com , Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' 1, Article .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Publication date: January 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' API Entity and Relation Joint Extraction from Text via Dynamic Prompt-tuned Language Model 17 parsing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Huang [1], and Liu [6] extract semantic relations between entities based on syntactic patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' However, their API entity and relation extraction method from natural language text relies on unique orthographic features of APIs, and suffer from the rule design overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' To mitigate the overhead of rule design, researchers extract API entities using machine learning methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Ye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' [2] propose APIReal, which uses CRF to identify API entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' They label the API entities and non-entities in the sentence with the “BIO” tag and form the pair of the labeled sequence and the sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' They then train CRF on these pairs, and use the trained CRF to label the input text, from which they obtain API entities with the “BI” or “B” tag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Huo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' [13], on the other hand, propose ARCLIN, which uses BI-LSTM as encoder and CRF as decoder to identify API entities, rather than just CRF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' However, these methods suffer from data labeling overhead because preparing a large number of high-quality training data for these sequence labeling models is unrealistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' To solve the two overhead issues mentioned above, researchers use LLM to extract entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' [35] use BERT and Yan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' [36] use XLNet [37] to extract entities in the natural language domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' These models, however, are limited to a single natural language processing task, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=', the entity extraction only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' In order to realize joint extraction of multiple tasks, researchers propose LLM- based unified architectural models, such as UIE [20] and OpenUE [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' In particular, UIE proposes SEL to encode different information extraction structures via the hierarchical spotting-associating structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Motivated by this, we consider adapting UIE to the joint API entity-relation extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' However, UIE is not good at dealing with complex sentences, particularly long and ambiguous sentences containing API entities and various relations, because UIE has only one static prompt to identify all types of API relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' As a result, when confronted with ambiguous sentences, the more relation types to recognize, the more noise interference, and the lower the UIE recognition rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' In contrast, we propose LLM-based AERJE, which extracts API entities and relations from unstructured complex sentences at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Different from UIE, our dynamic prompt design could generate a small number of potentially relevant relations for input text to eliminate noise interference and lessens the difficulty of API relation extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' 7 CONCLUSION AND FUTURE WORK In this paper, we are the first to formulate heterogeneous API extraction and API relation extraction task as a sequence-to-sequence task, and proposes AERJE to extract API entities and relations from unstructured text simultaneously using pre-trained LLM and dynamic prompt learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' The systematic evaluation of AERJE is conducted on a set of long and ambiguous sentences from Stack Overflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' The experimental results show that AERJE’s ability to extract API entities and relations can be activated with a small amount of data, allowing it to accurately identify API entities and relations from complex text that the model has never seen during fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' In the future, we will carry out the plans mentioned in the discussion and apply AERJE to any software engineering task supported by API entity and relation extraction, such as 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' 20 Qing Huang, Yanbang Sun, Zhenchang Xing, Min Yu, Xiwei Xu, and Qinghua Lu QING HUANG received the M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='S degree in computer application and technology from Nanchang University, in 2009, and the PH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' degree in computer software and theory from Wuhan University, in 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' He is currently an Assistant Professor with the School of Computer and Informa- tion Engineering, Jiangxi Normal University, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' His research interests include information security, software engineering and knowledge graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Yanbang Sun is a second-year master student at the School of Computer and Information Engineering, Jiangxi Normal University, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' His re- search interests include software engineering and knowledge graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Zhenchang Xing is a Senior Research Scientist with Data61, CSIRO, Eveleigh, NSW, Australia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' In addition, he is an Associate Professor in the Research School of Computer Science, Australian National University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Previously, he was an Assistant Professor in the School of Computer Sci- ence and Engineering, Nanyang Technological University, Singapore, from 2012-2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' His main research areas are software engineering, applied data analytics, and human-computer interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' MIN YU is a Professor in Communication, Electronic Engineering, and Computer Science at Jiangxi Normal University, was a visiting scholar at the University of California, Irvine, the USA, and interested in Distributed computing, Wireless Sensor Network, and Indoor Positioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Xiwei Xu is a Senior Research Scientist with Architecture& Analytics Platforms Team, Data61, CSIRO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' She is also a Conjoint Lecturer with UNSW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' She started working on blockchain since 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Her main research interest is software architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' She also does research in the areas of service computing, business process, and cloud computing and dependability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Qinghua Lu is a Senior Research Scientist with Data61, CSIRO, Eveleigh, NSW, Australia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' She has published more than 100 academic papers in international journals and conferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Her research interests include the software architecture, blockchain, software engineering for AI, and AI ethics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' , Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' 1, Article .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} +page_content=' Publication date: January 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQflQeg/content/2301.03987v1.pdf'} diff --git a/CNE1T4oBgHgl3EQfVwTa/content/tmp_files/2301.03107v1.pdf.txt b/CNE1T4oBgHgl3EQfVwTa/content/tmp_files/2301.03107v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..6e54d6c3e59aa499569e67c6a5349ab1debdf066 --- /dev/null +++ b/CNE1T4oBgHgl3EQfVwTa/content/tmp_files/2301.03107v1.pdf.txt @@ -0,0 +1,2976 @@ +The q-neighbor Ising model on multiplex networks with partial overlap of nodes +A. Krawiecki and T. Gradowski +Faculty of Physics, Warsaw University of Technology, +Koszykowa 75, PL-00-662 Warsaw, Poland +The q-neighbor Ising model for the opinion formation on multiplex networks with two layers in +the form of random graphs (duplex networks), the partial overlap of nodes, and LOCAL&AND spin +update rule was investigated by means of the pair approximation and approximate Master equations +as well as Monte Carlo simulations. Both analytic and numerical results show that for different fixed +sizes of the q-neighborhood and finite mean degrees of nodes within the layers the model exhibits +qualitatively similar critical behavior as the analogous model on multiplex networks with layers in +the form of complete graphs. However, as the mean degree of nodes is decreased the discontinuous +ferromagnetic transition, the tricritical point separating it from the continuous transition and the +possible coexistence of the paramagnetic and ferromagnetic phases at zero temperature occur for +smaller relative sizes of the overlap. Predictions of the simple homogeneous pair approximation +concerning the critical behavior of the model under study show good qualitative agreement with +numerical results; predictions based on the approximate Master equations are usually quantitatively +more accurate, but yet not exact. Two versions of the heterogeneous pair approximation are also +derived for the model under study, which, surprisingly, yield predictions only marginally different +or even identical to those of the simple homogeneous pair approximation. In general, predictions of +all approximations show better agreement with the results of Monte Carlo simulations in the case +of continuous than discontinuous ferromagnetic transition. +I. +INTRODUCTION +Investigation of the opinion formation process by means of nonequilibrium models has become a firmly established +research field in statistical physics in the last decades [1]. Many results in this area were obtained using models with +agents’ opinions represented by spins with discrete (in most cases two) states obeying stochastic dynamics described +by various rates at which agents change (e.g., flip) their opinions, e.g., the majority-vote model [2–7], the noisy voter +model [8–10], different versions of the noisy nonlinear and q-voter model [11–20] and the q-neighbor Ising model [21– +24]. In particular, much effort was devoted to determining conditions under which the above-mentioned models exhibit +phase transition from a disordered paramagnetic (PM) state in which each opinion appears with the same probability +to an ordered ferromagnetic (FM) state with one dominant opinion as the parameter controlling the level of stochastic +noise in the model is varied, measuring the agents’ uncertainty in decision making. In this context the presence of +the first-order FM transition, or even transition to a frozen FM phase is of prime importance, with abrupt occurrence +of a dominant opinion as well as possible hysteresis and bistability of the PM and FM phases [4, 5, 12–21, 24]. +Following the growing interest in the dynamical processes on complex networks [25] agents in the models for the +opinion formation are often located in the nodes and interact via edges of complex networks reflecting a complicated +structure of social interactions [3–7, 9, 13–16, 18–20, 23, 24]. In this case analytic predictions concerning the critical +behavior of the models based on the mean-field approximation (MFA) need not exhibit quantitative agreement with +results of Monte Carlo (MC) simulations, hence, more accurate approaches based on the pair approximation (PA) +[26–30] and approximate Master equations (AMEs) [28–30] were applied to describe theoretically the observed phase +transitions [10, 14–16, 18–20, 24]. +Recently much attention has been devoted to combining complex networks in order to create even more complicated +and heterogeneous structures known in general as ”networks of networks” [31]. An important class of such structures +is formed by multiplex networks (MNs) which consist of a fixed set of nodes connected by various sets of edges called +layers [31–33]. In the simplest case, the layers are independently generated random networks with a full overlap of +nodes, i.e., with each node belonging to all layers, which means it has at least one attached edge from each layer. +In turn, in MNs with partial overlap of nodes, there are nodes belonging only to some rather than all layers. In +particular, in the case of MNs with two layers (duplex networks) and partial overlap of nodes, the nodes are divided +into a class of nodes belonging to both layers and forming the overlap, and two other classes, each consisting of nodes +belonging only to one of the two layers [34–36] (the node overlap should not be confused with the link overlap [37–39] +which is negligible in the case of independently generated layers). FM phase transition in equilibrium models on MNs +was studied, e.g., in the Ising model [40, 41] and a related Ashkin-Teller model [42]. Analogously, FM transition in +nonequilibrium models for the opinion formation on MNs was studied, e.g., in the majority vote model [44, 45], the +q-voter model [46–48] and the q-neighbor Ising model [49]. As expected, the critical properties of the nonequilibrium +models, in particular the extension or confinement of the range of parameters for which the first-order transition occurs, +strongly depend on the way in which the multiplexity affects the spin-flip rate. In this respect, very interesting seems +arXiv:2301.03107v1 [cond-mat.stat-mech] 8 Jan 2023 + +2 +the q-neighbor Ising model with LOCAL&AND spin update rule [50], which so far has been studied by MC simulations +and in the MF approximation on duplex networks with full and partial overlap of nodes and with layers in the form of +fully connected graphs [49]. In this model, the flip probability per unit time for the spins in nodes belonging to only +one layer (i.e., outside the overlap) is given by a Metropolis-like rate, but with a local field produced only by a subset +of q randomly chosen neighboring spins (q-neighborhood), and for the spins in nodes belonging to both layers (i.e., +within the overlap) it is given by a product of two above-mentioned rates evaluated separately for each layer. With +the increase of the relative size of the overlap, and depending on the size of the q-neighborhood, suppression of the +first-order transition, appearance of a tricritical point separating first- and second-order FM transition, and possible +coexistence of the PM and FM phases even in zero temperature were observed in the model [49]. +In this paper, the q-neighbor Ising model on MNs with partial overlap of nodes, with layers in the form of complex +networks and with the LOCAL&AND spin update rule is studied by means of MC simulations and theoretically in +the framework of the PA and AMEs. It should be noted that the q-neighbor Ising model is used here as a convenient +example since the results can be readily compared with the above-mentioned ones for the limiting case of the model +on MNs with layers in the form of complete graphs [49], and the PA and AMEs used here can be easily generalized +to other models for the opinion formation with similar structure of interactions. In order to make large systems of +AMEs numerically tractable in this paper only the case of duplex networks with layers in the form of homogeneous +random networks is considered; nevertheless, such MNs exhibit certain overlap-induced inhomogeneity since the nodes +within and outside the overlap form distinct classes characterized by different degrees within the individual layers +(both non-zero or one zero and one non-zero). Thus also the flip rates for the spins located in nodes belonging to +distinct classes are different; a related q-voter model with quenched disorder, with agents divided into subpopulations +according to different rates of the opinion change, has been recently considered [15]. +The aim of this paper is first to provide a general formulation of the PA and AMEs, which take into account to +a different extent the above-mentioned inhomogeneity of nodes, for models on MNs with partial overlap of nodes. +For this purpose, first, the homogeneous PA for models on MNs with a full overlap of nodes [47] is extended to the +case with partial overlap. For nodes belonging to different classes this simplest form of the PA takes into account the +inhomogeneity of the average directions of spins (opinions) but neglects possible inhomogeneity of the distributions +of directions of neighboring spins within each layer. For the q-neighbor Ising model predictions of this approximation +concerning the FM phase transition show surprisingly good agreement with results of MC simulations for a wide +range of the size of the q-neighborhood, the mean degrees of nodes within layers and the size of the overlap. Then, +the most advanced approximation based on the AMEs for models on MNs with the full overlap of nodes [45] and +weighted networks [51] is extended to the case of models on MNs with partial overlap of nodes. Finally, two kinds +of heterogeneous PA, the fully heterogeneous PA [15] and the AMEs-based heterogeneous PA [28–30] are applied +to models on MNs with partial overlap of nodes. Both versions of the PA take into account, to a different extent, +the above-mentioned inhomogeneity of distributions of directions of neighboring spins within each layer and are in +general intermediate with respect to the accuracy of predictions between the homogeneous PA and the AMEs. For +the q-neighbor Ising model under study, it turns out that their predictions are only marginally different or even +identical with these of the homogeneous PA. On the other hand, predictions based on the AMEs show slightly better +quantitative agreement with the results of MC simulations, in particular for smaller mean degrees of nodes within +layers. In general, predictions of all approximations concerning the first-order FM transition (e.g., location and width +of the hysteresis loop) are quantitatively worse than those concerning the second-order transition (e.g., location of +the critical point). Besides, the aim of this paper is also to study in detail the phase diagram for the q-neighbor Ising +model on MNs with partial overlap of nodes and with layers with a finite mean degree of nodes. It is shown that the +critical behavior of this model resembles qualitatively that of the analogous model on MNs with layers in the form +of fully connected graphs [49]. However, as the mean degree of nodes is decreased, the first-order FM transition, the +tricritical point separating it from the second-order transition, and the possible coexistence of the PM and FM phases +occur for smaller relative sizes of the overlap, while the range of the occurrence of the second-order FM transition is +broadened correspondingly. +II. +THE MODEL +A. +Multiplex networks with partial overlap of nodes +MNs consist of a fixed set of nodes connected by several sets of edges; the set of nodes with each set of edges +forms a network which is called a layer of a MN [32, 33]. Henceforth, the nodes are indexed by i, i = 1, 2, . . . N, +and the subsequent layers are denoted as G(L), L = A, B, . . . Lmax. In the case of MNs with a full overlap of nodes +each node belongs to all layers, i.e., each node has at least one edge from each layer attached to it. In general, MNs +with partial overlap of nodes are defined as MNs in which nodes may belong to (i.e., may have attached edges from) + +3 +some rather than all layers, given that each node belongs to at least one layer. Henceforth, the number of nodes +belonging to the layer G(L) is denoted as N (L). In this paper, it is assumed that the sets of edges for the subsequent +layers G(L) are generated independently and form complex random networks with N (L) nodes. As a result, multiple +connections between nodes are not allowed within the same layer, but the same nodes belonging to several layers can +be accidentally connected by multiple edges belonging to different layers. A simple example of the MN with partial +overlap of nodes is that with only two layers G(A), G(B), called a duplex network, and with n nodes belonging to +both layers which form the overlap (0 ≤ n ≤ N); then, N = N (A) + N (B) − n. Furthermore, if both layers contain +the same number of nodes N (A) = N (B) = ˜N it is possible to introduce a single parameter r = n/ ˜N, also called the +overlap. Then, the nodes are divided into three subsets: ˜N − n = N(1 − r)/(2 − r) nodes belonging only to the layer +G(A), N(1 − r)/(2 − r) nodes belonging only to the layer G(B) and n = Nr/(2 − r) nodes belonging both to G(A) and +G(B). +The numbers of edges attached to the node i (degrees) within the individual layers G(L) are denoted as k(L) +i +; if the +node i does not belong to the layer G(L) then k(L) +i += 0. In the case of MNs with independently generated layers the +degrees of nodes belonging to the individual layers G(L), i.e., these with k(L) +i +> 0, are drawn from probability distri- +butions P +� +k(L)� +which characterize the layers as complex networks. For a given node i a vector of its degrees within +the individual layers ki = +� +k(A) +i +, k(B) +i +, . . . k(Lmax) +i +� +, with possible zero components in the case of MNs with partial +overlap of nodes, is called a multidegree of the node. The multidegree distribution P(k) = P +� +k(A) +i +, k(B) +i +, . . . k(Lmax) +i +� +characterizes the MN as a complex ”network of networks”; in the case of MNs with the full overlap of nodes and +independently generated layers, it is obviously P(k) = �Lmax +L=A P(k(L)). In the formulas below, averages are evaluated +over the multidegree distribution, e.g., ⟨k(L)⟩ = N −1 �N +i=1 k(L) +i += � +k P(k)k(L) is the mean degree of nodes within +the layer G(L) (note that the average is over all N nodes rather than N (L) nodes belonging to the layer G(L)). As a +simple example, in this paper the q-neighbor Ising model is considered on a duplex network with partial overlap of +nodes and with the two independently generated layers in the form of random regular graphs (RRGs) with K edges +attached to each node belonging to the layer, the same numbers of nodes N (A) = N (B) = ˜N and the overlap r, for +which the multidegree distribution is +P (k) = P +� +k(A), k(B)� += 1 − r +2 − rδk(A),Kδk(B),0 + +r +2 − rδk(A),Kδk(B),K + 1 − r +2 − rδk(A),0δk(B),K, +(1) +and ⟨k(A)⟩ = ⟨k(B)⟩ = ˜NK/N = K/(2 − r). +B. +The q-neighbor Ising model on multiplex networks with partial overlap of nodes +The q-neighbor Ising model [21–24, 49] is a nonequilibrium variant of the Ising model used to investigate the process +of opinion formation. In this paper the above-mentioned model is considered on MNs with partial overlap of nodes +and layers in the form of complex networks; the MF version of this model, on MNs with layers in the form of fully +connected graphs, was studied in Ref. [49]. The main interest is in the FM transition which can occur in the q-neighbor +Ising model with decreasing effective temperature T, which measures the level of internal noise (uncertainty in agents’ +decision making). +In order to introduce the model under study, it is convenient to start with the q-neighbor Ising model on (monoplex) +networks which can be regular, complex, or fully connected graphs [21–24, 49]. In this model agents with two possible +opinions on a given subject are represented by two-state spins σi = ±1, i = 1, 2, . . . N placed in the nodes and +interacting via edges of the network. It is assumed that these interactions prefer identical orientations of spins in the +connected nodes, which is reflected in the spin-flip rate. Thus, interactions between spins with opposite directions in +general increase the probability that one of the spins flips, i.e., the corresponding agent changes opinion, and edges +representing these interactions are called active links. The dynamics of the q-neighbor Ising model on networks is a +modification of that of the kinetic Ising model with the Metropolis spin-flip rate in which, at each time step, each +spin interacts only with its q randomly chosen neighbors. MC simulations of the model are performed using random +asynchronous updating of spins, with each MC simulation step (MCSS) corresponding to updating all N spins. Nodes +are picked randomly and for each picked node q its neighbors are chosen randomly and without repetitions, which +form the q-neighborhood of the picked node. Then, the spin in the picked node is flipped with probability given by a +Metropolis-like formula, +E (l; T, q) = min {1, exp[−2(q − 2l)/T]} , +(2) +where l is the number of nodes belonging to the q-neighborhood occupied by spins with a direction opposite to that +of the spin in the picked node, i.e., the number of active links attached to the picked node leading to nodes within + +4 +the chosen q-neighborhood (notation in Eq. (2) emphasizes that T, q, are parameters of the model). As a result, the +flip rate for a picked spin given that it is placed in a node with degree k which has in total i active links attached +(0 ≤ i ≤ k) is +f (i; T|k) = +1 +�k +q +� +q +� +l=0 +�i +l +��k − i +q − l +� +E (l; T, q) = +1 +�k +i +� +q +� +l=0 +�k − q +i − l +��q +l +� +E (l; T, q) . +(3) +The q-neighbor Ising model on complete graphs for q = 3 exhibits second-order FM transition, while for q ≥ 4 +first-order FM transition occurs with a clearly visible hysteresis loop. Width of the hysteresis loop in general increases +with q, though for q > 4 there are oscillations superimposed on this trend such that loops for the consecutive odd +values of q are narrower than for the neighboring even values of q [21]. The same is true for the model on networks +with finite mean degree ⟨k⟩ provided that q ≪ ⟨k⟩. However, as q is increased and becomes comparable with ⟨k⟩ the +hysteresis loop becomes narrower and eventually disappears, and the FM transition becomes second-order [24]. +In the q-neighbor Ising model on MNs with full or partial overlap of nodes, interactions take place within individual +layers with respective, independently chosen q-neighborhoods. Then, spins flip according to a probabilistic rule which +combines the effect of the above-mentioned interactions. In this paper the LOCAL&AND spin update rule is used [50] +according to which the spin in the picked node flips if interaction with every q-neighborhood from every layer suggests +flip; consequently, the probability of the spin-flip is given by a product of the Metropolis-like factors (2) corresponding +to all layers containing the picked node. The LOCAL&AND rule is assumed in this paper since it usually leads to +richer phase diagrams than other methods of including the multiplex character of the network of interactions in the +spin-flip rate [46–49]. Eventually, in numerical simulations of the q-neighbor Ising model on MNs with partial overlap +of nodes and the LOCAL&AND spin update rule, each MCSS is performed as follows. +(i.) A node i, 1 ≤ i ≤ N, with multidegree ki is picked randomly. +(ii.) From each layer G(L) containing the picked node a set of its q neighbors (q-neighborhood) is chosen randomly +and without repetitions; it is assumed that 0 < q ≤ k(L) +i +. Sets from different layers are chosen independently, +thus the same node can by chance belong to two or more q-neighborhoods if it is a neighbor of the picked node +within two or more layers. +(iii.) The Metropolis-like factor for the picked node is evaluated separately for each layer G(L), +E +� +l(L); T, q +� += min +� +1, exp[−2(q − 2l(L))/T] +� +(4) +where l(L) is the number of nodes in the q-neighborhood in the layer G(L) occupied by spins with direction +opposite to that of the spin in the picked node; note that if a node does not belong to G(L) then q = l(L) = 0 +and E(T, 0, 0) = 1. +(iv.) Due to the LOCAL&AND spin update rule, the spin σi in the picked node flips with probability +E (l; T, q) = +Lmax +� +L=A +E +� +l(L); T, q +� +, +(5) +where l = +� +l(A), l(B), . . . l(Lmax)� +; and obviously l(L) = 0 if the picked node does not belong to the layer G(L) +(i.e., l is a vector of numbers of active links from the individual layers attached to the picked node which lead +to nodes within the respective q-neighborhoods). +(v.) Steps (i.)-(iv.) are repeated until all N spins are updated without repetition. +Hence, the flip rate for a spin placed in a node with multidegree k = +� +k(A), k(B), . . . k(Lmax)� +and with the numbers of +attached active links within the individual layers i(L), 0 ≤ i(L) ≤ k(L), given by the corresponding components of the +vector i = +� +i(A), i(B), . . . i(Lmax)� +assumes a multiplicative form, +f (i; T |k) = +Lmax +� +L=A +f +� +i(L); T|k(L)� +(6) +(note that if a node does not belong to the layer G(L) there is k(L) = i(L) = 0 and f (0; T|0) ≡ 1). +The q-neighbor Ising model on a duplex network with layers in the form of complete graphs and partial overlap +of nodes, and with the LOCAL&AND spin update rule exhibits FM phase transition already for q ≥ 1 [49]. This + +5 +transition is in general second-order, with some exceptions. For q = 2 the transition is first-order for 1/2 < r < 1, +with a clearly visible hysteresis loop, and for rc < r ≤ 1/2, where rc = 2(3 +√ +2 − 4) = 0.4853 . . ., the coexistence of +the FM and PM phases is observed as the temperature is decreased below a critical value down to T = 0; for r < rc +there is no phase transition and the PM phase remains the only stable phase down to T = 0. For q ≥ 4 the transition +for small r is first-order and for larger r is second-order. The first- and second-order transitions are separated by a +tricritical point at r = rT CP (q) which for q = 4 occurs at a particularly high value of r, and for q > 4 is an increasing +function of q, but again with oscillations between the consecutive odd and even values of q superimposed on this +trend. Remarkably, for r = 1 the FM transition is always second-order for any q, i.e., full overlap of nodes suppresses +discontinuous transition. In this paper, it is investigated how the phase diagram of the model changes if the layers of +the MN are complex networks with a finite mean degree of nodes. +III. +THEORY +A. +Pair approximation +In the case of spin models on networks, the effect of the network topology (e.g, of the degree distribution or the mean +degree of nodes) on the observed phase transitions often can be more accurately described in the framework of the +PA than by the usual MFA [26–30]. In particular, this was demonstrated for the q-neighbor Ising model on complex +networks [24] and a sort of stochastic q-voter model on MNs with a full overlap of nodes [47]. In both above-mentioned +studies the networks, or the layers of the MNs, were homogeneous complex networks (e.g., RRGs), thus the simplest +homogeneous PA was enough to reproduce quantitatively results of MC simulations in a wide range of the parameters +of the models. As mentioned in Sec. I & II MNs with partial overlap of nodes retain some multiplexity-induced +inhomogeneity even if the layers are homogeneous complex networks. Nevertheless, in this section the homogeneous +PA derived in Ref. [47] for a wide class of models with various spin update rules on MNs with the full overlap of nodes +is presented in a more general form which makes it applicable to models on MNs with partial overlap of nodes, in +order to find, inter alia, to what extent it can be used to explain critical behavior of systems with multiplicity-induced +inhomogeneity. +The advantage of the PA consists in that it takes into account dynamical correlations between pairs of interacting +agents (spins). In the framework of the homogeneous PA, macroscopic quantities characterizing a model with two- +state spins on MNs are concentrations ck of spins directed up located in nodes with multidegree k (with possible zero +components in the case of MNs with partial overlap of nodes) as well as concentrations b(L) of active links within +separate layers G(L). The homogeneous character of the PA allows for the simplification that the latter concentrations +are averaged over all nodes belonging to a given layer and do not depend on the multidegrees of the connected nodes. +Consequently, it is assumed that conditional probabilities θ(L) +j +, j ∈ {↑, ↓}, that an active link within the layer G(L) is +attached to a node given that it is occupied by spin with direction j are also independent of the multidegree of the +node. These probabilities can be evaluated as ratios of the number of attachments of active links to nodes with spins +with direction j, independently of their multidegrees, within the layer G(L), which is N⟨k(L)⟩b(L)/2, and the number +of attachments of all links within GL to such nodes, which is � +k NP (k) k(L)ck,j, where ck,↑ = ck, ck,↓ = 1 − ck, thus +θ(L) +↑ += +b(L) +2 � +k P (k) k(L)ck,↑/⟨k(L)⟩ = +b(L) +2 � +k P (k) k(L)ck/⟨k(L)⟩, +(7) +θ(L) +↓ += +b(L) +2 � +k P (k) k(L)ck,↓/⟨k(L)⟩ = +b(L) +2 +� +1 − � +k P (k) k(L)ck/⟨k(L)⟩ +� +(8) +The core approximation made in the PA for models on MNs is that the numbers of active links i(L) attached +to a node with degrees k(L) within individual layers G(L) (0 ≤ i(L) ≤ k(L)) occupied by spin with direction j obey +independent binomial distributions with parameters θ(L) +j +given by Eq. (7,8). Then, the rates at which the concentration +ck increases or decreases are given by averages of the spin-flip rate, Eq. (6), over the appropriate joint distributions +of the number of active links within all layers which have a multiplicative form +P(j, i|k) = +Lmax +� +L=A +Bk(L),i(L) +� +θ(L) +j +� +, +(9) +where Bk,i(θ) = +�k +i +� +θi(1 − θ)k−i denotes the binomial factor and, formally, B0,0(θ) ≡ 1. Hence, the equation for the +time dependence of ck can be written as a rate equation, + +6 +∂ck +∂t = +� +j∈{↑,↓} +(−1)δj,↑ck,j +� +i +Lmax +� +L=A +Bk(L),i(L) +� +θ(L) +j +� +f (i; T |k) , +(10) +where � +i ≡ �k(A) +i(A)=0 . . . �k(Lmax) +i(Lmax)=0. +In order to obtain an equation for the time dependence of the concentrations of active links b(L) one should observe +that each flip of a spin (irrespective of its direction) in a picked node with multidegree k with the numbers of active +links attached given by the components of the vector i results in the change of the numbers of active links within +the individual layers G(L) by k(L) − 2i(L), since then i(L) previously active links become inactive and k(L) − i(L) +previously inactive links become active. The corresponding changes in the concentrations of active links b(L) are thus +� +k(L) − 2i(L)� +/(N⟨k(L)⟩/2). As in Eq. (10), such changes connected with the flip of a spin with direction j occur at a +rate given by the average of the spin-flip rate, Eq. (6), over the appropriate joint distributions of the number of active +links attached to the picked node, Eq. (9). Due to the homogeneous character of the PA, in order to obtain time +dependence of b(L) further averaging over all nodes occupied by spins with direction j should be performed, which +is equivalent to averaging over the probability distribution P(k)ck,j that a node with multidegree k is occupied by +a spin with direction j. Eventually, taking into account that nodes are picked and spins are updated within time +intervals 1/N, for a given layer G(L′) it is obtained that +∂b(L′) +∂t += +2 +⟨k(L′)⟩ +� +j∈{↑,↓} +� +k +P (k) ck,j +� +i +Lmax +� +L=A +Bk(L),i(L) +� +θ(L) +j +� +f (i; T |k) +� +k(L′) − 2i(L′)� +, +(11) +where L′ = A, B . . . Lmax. +In particular, let us consider the q-neighbor Ising model on a MN with two layers in the form of RRGs and partial +overlap of nodes, with the multidegree distribution given by Eq. (1). Then, the nodes are divided into three classes, +these belonging only to the layer G(A) with multidegree k = (K, 0), only to the layer G(B) with k = (0, K) and to +the overlapping part of G(A) and G(B), with k = (K, K). The macroscopic quantities to be used in the homogeneous +PA are thus concentrations of spins directed up in the nodes belonging to the subsequent classes c(K,0), c(0,K), c(K,K) +and concentrations of active links in the two layers b(A), b(B). Since both layers are identical, with N (A) = N (B) = ˜N, +stable solutions of the system of equations (10), (11) are limited to the subspace with c(0,K) = c(K,0), b(A) = b(B) ≡ b; +moreover, according to Eq. (7,8) there is θ(A) +j += θ(B) +j +≡ θj. Using Eq. (1), (3), (6), performing summations in Eq. +(10), (11) as in Ref. [24] and introducing functions R(θ; T, q) and S(θ; T, K, q) to shorten notation, +R(θ; T, q) = +q +� +l=0 +Bq,l (θ) E(l; T, q), +(12) +S(θ; T, K, q) = +q +� +l=0 +Bq,l (θ) [(K − q)θ + l]E(l; T, q), +(13) +the following system of three equations for the time dependence of the macroscopic quantities in the homogeneous +PA is obtained, +dc(K,0) +dt += +� +1 − c(K,0) +� +R (θ↓; T, q) − c(K,0)R (θ↑; T, q) +(14) +dc(K,K) +dt += +� +1 − c(K,K) +� +[R (θ↓; T, q)]2 − c(K,K) [R (θ↑; T, q)]2 +(15) +db +dt = 2 +K (1 − r) +�� +1 − c(K,0) +� +[KR (θ↓; T, q) − 2S (θ↓; T, K, q)] + c(K,0) [KR (θ↑; T, q) − 2S (θ↑; T, K, q)] +� ++ 2 +K r +�� +1 − c(K,K) +� +[KR (θ↓; T, q) − 2S (θ↓; T, K, q)] R (θ↓; T, q) ++ c(K,K) [KR (θ↑; T, q) − 2S (θ↑; T, K, q)] R (θ↑; T, q) +� +, +(16) +where +θ↑ = +b +2 +� +(1 − r)c(K,0) + rc(K,K) +�, +(17) +θ↓ = +b +2 +� +1 − (1 − r)c(K,0) − rc(K,K) +�. +(18) + +7 +Other macroscopic quantities of interest are the concentration of spins directed up in each layer, i.e., the fraction of ˜N +nodes occupied by such spins, which is ˜c = (1 − r)c(K,0) + rc(K,K), the concentration of spins directed up in the whole +MN, i.e., the fraction of N nodes occupied by such spins, which is c = 2(1−r) +2−r c(K,0) + +r +2−rc(K,K), and the resulting +magnetization of the MN m = 2c − 1. Note that in the limiting case of layers in the form of fully connected graphs +there is b = ˜N 2˜c(1 − ˜c)/[ ˜N( ˜N − 1)/2] ≈ 2˜c(1 − ˜c) and θ↓ = ˜c, θ↑ = 1 − ˜c; after inserting this into Eq. (14) and (15) +equations for the concentrations c(K,0), c(K,K) in the MF approximation are reproduced [49], as expected. +Natural extension of the homogeneous PA consists in taking into account heterogeneity of the concentrations of +the (possibly active) links connecting classes of nodes with different multidegrees, so that, instead of the average +concentration b(L) of active links within the layer G(L), e.g., concentrations of classes of active links connecting spins +in nodes with multidegrees k, k′ within the layer G(L) become separate macroscopic quantities characterizing the +model. +This leads to the most advanced and accurate version of the PA called fully heterogeneous PA [15, 27]; +corresponding equations for the macroscopic quantities for spin models on MNs with partial overlap of nodes, in +particular for the q-neighbor Ising model under study, are given in Appendix A. In the latter case solutions of these +equations show that in the stationary state concentrations of active links (strictly speaking, of their ends called bonds) +belonging to different classes indeed show noticeable heterogeneity; nevertheless, this does not lead to the values of +magnetization noticeably different from these predicted by the homogeneous PA. Thus, magnetization curves and +phase diagrams for the model under study obtained from the fully heterogeneous PA are practically indistinguishable +from those obtained from the homogeneous PA and do not show better agreement with the results of MC simulations. +B. +Approximate Master equations +A more accurate approximation for the study of spin models on MNs with partial overlap of nodes is based on +approximate Master equations (AMEs) for the densities of spins directed up ck,m and down sk,m which are located +in nodes with multidegree k and have m(L) neighboring spins directed up within the consecutive layers G(L), which +is denoted as m = +� +m(A), m(B) . . . m(Lmax)� +. In the thermodynamic limit and for mutually uncorrelated layers in the +form of random networks with finite mean degrees +� +k(L)� +possibility that a pair of nodes is connected simultaneously +by edges within different layers can be neglected. Thus, in the AMEs it is assumed that in a single simulation step +for a given node the allowed changes of the number of neighboring spins directed up are m → m ± e(L), where e(L) +is a unit vector with Lmax components and only L-th component equal to one, while simultaneous changes of many +components of m, e.g., m → m ± e(L) ± e(L′), L ̸= L′, etc., cannot occur. Under the above-mentioned assumptions, +the AMEs in a general form are [45, 51] +dsk,m +dt += −Fk,msk,m + Rk,mck,m ++ +Lmax +� +L=A +� +−β(L) +s +� +k(L) − m(L)� +sk,m + β(L) +s +� +k(L) − m(L) + 1 +� +sk,m−e(L) +� ++ +Lmax +� +L=A +� +−γ(L) +s +m(L)sk,m + γ(L) +s +� +m(L) + 1 +� +sk,m+e(L) +� +, +(19) +dck,m +dt += −Rk,mck,m + Fk,msk,m ++ +Lmax +� +L=A +� +−β(L) +i +� +k(L) − m(L)� +ck,m + β(L) +i +� +k(L) − m(L) + 1 +� +ck,m−e(L) +� ++ +Lmax +� +L=A +� +−γ(L) +i +m(L)ck,m + γ(L) +i +� +m(L) + 1 +� +ck,m+e(L) +� +. +(20) +In Eq. (19), (20) the first two terms account for the effect of a flip of a spin in a node with multidegree k and the +remaining terms account for the average effect of the flips of spins in the neighboring nodes, irrespective of their +multidegrees. In terms of Sec. II B the flip rate for a spin directed down occupying a node with multidegree k with +m neighboring spins directed up is Fk,m = f (m; T |k) and that for a spin directed up Rk,m = f (k − m; T |k). +The remaining average rates can be estimated by evaluating the ratios (at a given time step) of the average number +of edges connecting spins with a given direction such that one of these spins flips to the average total numbers +of these edges [28, 29]; in the case of models on MNs this should be done separately for each layer [45, 51]. +Thus β(L) +s += +� � +m +� +k(L) − m(L)� +Fk,msk,m +� +/ +� � +m +� +k(L) − m(L)� +sk,m +� +, γ(L) +s += +� � +m +� +k(L) − m(L)� +Rk,mck,m +� +/ + +8 +� � +m +� +k(L) − m(L)� +ck,m +� +, +β(L) +i += +� � +m m(L)Fk,msk,m +� +/ +� � +m m(L)sk,m +� +, +γ(L) +i += +� � +m m(L)Rk,mck,m +� +/ +� � +m m(L)ck,m +� +, where L = A, B, . . . Lmax, � +m ≡ �k(A) +m(A)=0 +�k(B) +m(B)=0 . . . �k(Lmax) +m(Lmax)=0 and ⟨. . .⟩ denotes average +over the multidegree distribution P (k), as usually. Natural initial conditions for the system of equations (19), (20) +are sk,m(0) = (1−c(0)) �Lmax +L=A Bk(L),m(L)(c(0)), ck,m(0) = c(0) �Lmax +L=A Bk(L),m(L)(c(0)), where 0 < c(0) < 1 is arbitrary. +In particular, in the case of the q-neighbor Ising model on a MN with two layers in the form of RRGs and partial over- +lap of nodes, with the multidegree distribution P(k) given by Eq. (1), there are three classes of nodes with k = (0, K), +k = (K, 0) and k = (K, K). The corresponding spin flip rates are F(K,0),(m(A);0) = f +� +m(A); T |K +� +, F(0,K),(0;m(B)) = +f +� +m(B); T |K +� +, F(K,K),(m(A),m(B)) = f +� +m(A); T |K +� +f +� +m(B); T |K +� +and R(K,0),(m(A),0) = f +� +K − m(A); T |K +� +, +R(0,K),(0,m(B)) = f +� +K − m(B); T |K +� +, R(K,K),(m(A),m(B)) = f +� +K − m(A); T |K +� +f +� +K − m(B); T |K +� +, with f (m; T |K ) +given by Eq. (3). Hence, the system (19), (20) consists of 2(K +1)2+4(K +1) equations and can be solved numerically +for moderate K. The quantities of interest, e.g., the concentration c of spins directed up in the MN and the magnetiza- +tion m = 2c−1 can be evaluated as in Sec. III A using c(K,0) = �K +m(A)=0 c(K,0),(m(A),0), c(0,K) = �K +m(B)=0 c(0,0),(0,m(B)), +c(K,K) = �K +m(A)=0 +�K +m(B)=0 c(K,K),(m(A),m(B)). +The AMEs are a starting point for a more elaborate approximation representing another formulation of the het- +erogeneous PA [28–30, 45, 51] which takes into account the possible heterogeneity due to different multidegrees k +of nodes of both the concentrations ck of spins directed up and of the conditional probabilities that a link attached +to a node is active or, equivalently, leads to a spin with a given (say, up) direction. A general formulation of such +AMEs-based heterogeneous PA for spin (two-state) models on (monoplex) networks by Gleeson [28, 29] was extended +to the case of weighted networks [51] and, partly, MNs [45]. It is believed that due to the approximations made the +AMEs-based heterogeneous PA is in general more accurate than the homogeneous PA and less accurate than the +fully heterogeneous PA mentioned in Sec. III A. In this paper the AMEs-based heterogeneous PA is applied to spin +models on MNs with partial overlap of nodes, in particular to the q-neighbor Ising model under study; equations +for the macroscopic quantities are given in Appendix B. Surprisingly, it turns out that in the stationary state the +above-mentioned conditional probabilities that a node has a link leading to a spin directed up do not depend on +whether the node belongs or not to the overlap. Hence, predictions of the AMEs-based heterogeneous PA concerning +the FM transition in the model under study are identical to those of the homogeneous PA from Sec. III A, so they +are not further discussed. +IV. +RESULTS +The main results concerning the FM transition in the q-neighbor Ising model on MNs with partial overlap of nodes +and with layers in the form of complete graphs have been summarized in Sec. II B. These results were obtained in the +MF approximation and confirmed by MC simulations [49]. In this section first predictions of the homogeneous PA +of Sec. III A concerning the FM transition in the q-neighbor Ising model on MNs with partial overlap of nodes and +with layers in the form of RRGs are presented and compared with results of MC simulations. In this case, noticeable +discrepancies occur between theoretical and numerical results, in particular concerning the first-order FM transition. +As pointed out in Sec. III, the more advanced fully and AMEs-based heterogeneous PA yield results practically +indistinguishable or even identical to the homogeneous PA, thus their predictions are only briefly mentioned in the +Appendix. Finally, it is verified in which cases and to what extent theoretical predictions are improved by using the +AMEs of Sec. III B. +In the framework of the homogeneous PA of Sec. III A stationary values of the magnetization m vs. T, corresponding +to different thermodynamic phases, are given by stable fixed points of the system of equations (14-16) with ˙c(K,0) = +˙c(K,K) = ˙b = 0; for certain ranges of parameters r, q, K many stable fixed points can coexist for given T, and their +basins of attraction are then separated by stable manifolds of unstable fixed points. The homogeneous PA predicts +various critical behavior of the model under study as the temperature T is varied, depending on r, q, K which are +fixed: first- and second-order FM phase transition, the coexistence of the PM and FM phases for T → 0 and absence +of the FM transition. At high temperatures, the only stable fixed point is that with m = 0 corresponding to the PM +phase. In the case of the second-order FM transition, this fixed point loses stability as the temperature is decreased +below the critical value Tc, and simultaneously a pair of symmetric stable fixed points with m > 0 and m < 0 occurs +via a supercritical pitchfork bifurcation, corresponding to the two symmetric FM phases. In the case of the first-order +transition two symmetric pairs of stable and unstable fixed points with m > 0 and m < 0 occur simultaneously +via two saddle-node bifurcations as the temperature is decreased below the upper critical value T (2) +c +, and the two +above-mentioned stable fixed points correspond to the two symmetric FM phases. As the temperature is further +decreased both the FM and PM fixed points remain stable (coexist) until the PM point loses stability via a subcritical + +9 +(c) +(d) +(b) +(a) +FIG. 1. +The curves show magnetization m vs. temperature T obtained from the homogeneous PA for different K (green +solid lines, both stable and unstable fixed points of the system of equations (14-16) are shown) and from the MFA of Ref. +[49] (black solid lines), for q = 2, K = 200, 100, 50, 20, 10, 4 (from left) and (a) r = 0.49, (b) r = 0.5, as well as for q = 4, +K = 500, 200, 100, 50, 20, 10 and (c) r = 0.05, (d) r = 0.15. +pitchfork bifurcation at the lower critical temperature T (1) +c +(T (1) +c +< T (2) +c +) by colliding simultaneously with the two +above-mentioned unstable fixed points; coexistence of the PM and FM phases for T (1) +c +< T < T (2) +c +leads to the +occurrence of the hysteresis loop in the magnetization curves m(T). Eventually, for T < T (1) +c +the only stable fixed +points remain these corresponding to the two symmetric FM phases. In the case of the coexistence of the FM and +PM phases for T → 0 a pair of symmetric stable FM fixed points occurs at T = T (2) +c +as in the case of the first-order +transition, but these FM points, as well as the PM fixed point, remain stable (coexist) as T → 0. Finally, it can also +happen that fixed points corresponding to the FM phase do not exist for any T > 0, thus the FM transition is absent +and the only stable phase for T → 0 is the PM one. +Exemplary curves m(T) predicted by the homogeneous PA for the model under study with different K and selected +values of r are shown in Fig. 1 for the most interesting cases q = 2 and q = 4; in the former case, the MFA (valid +for the model on MNs with layers in the form of fully connected graphs with K → ∞) predicts occurrence of all +above-mentioned kinds of the critical behavior for different ranges of r, while in the latter one it predicts occurrence +of the first-order transition for a particularly wide range of small r. The curves m(T) for q = 2 are drawn in Fig. 1(a) +for r = 0.49, and in Fig. 1(b) for r = 0.5, i.e., for the values of r within or at the border of the interval rc < r < 0.5 +where the MFA predicts coexistence of the FM and PM phases for T → 0. In contrast, for the model on MNs with +layers in the form of RRGs the homogeneous PA for r = 0.49 (Fig. 1(a)) predicts second- or first-order FM transition +for small and moderate K, respectively; the critical temperature(s) decrease and the width of the hysteresis loop +increases with K. Only for large K coexistence of the FM and PM phases for T → 0 is predicted by the PA, and the +curves m(T) approach those resulting from the MFA, as expected. For r = 0.5 (Fig. 1(b)) only second- or first-order +FM transitions for finite K are predicted by the PA, with the lower critical temperature for the first-order transition + +10 +(a) +(b) +FIG. 2. Critical behavior predicted by the homogeneous PA for the model with q = 2 (left and middle panels) and q = 4 (right +panel) and different r, K; filled circles — continuous FM transition, open circles — discontinuous FM transition, filled squares +— coexistence of the FM and PM phases for T → 0, crosses — absence of the transition. +(a) +(b) +(c) +(d) +(e) +(f) +FIG. 3. Results of MC simulations, predictions of the PA and AMEs for the model with q = 2, K = 20 and (a) r = 0.45, (b) +r = 0.46, (c) r = 0.47, (d) r = 0.50, (e) r = 0.60, (f) r = 0.70; blue dots — results of MC simulations with FM initial conditions +and increasing temperature, red dots — results of MC simulations with PM initial conditions and decreasing temperature, black +dots — predictions of the AMEs for both FM (c(0) = 1) and PM (c(0) = 0.5) initial conditions and increasing or decreasing +temperature, respectively, green solid lines — predictions of the PA as in Fig. 1. + +60Q0-0000-011 +(a) +(b) +(c) +(d) +(e) +(f) +FIG. 4. As in Fig. 3 but for q = 4. (a) K = 20, r = 0.05, (b) K = 20, r = 0.10, (c) K = 20, r = 0.15, (d) K = 10, r = 0.10, +(e) K = 50, r = 0.10, (f) k = 10, r = 0.05. +T (1) +c +> 0. The curves m(T) for q = 4 are drawn in Fig. 1(c) for r = 0.05, and in Fig. 1(d) for r = 0.15, i.e., for the +values of r where the MFA predicts first-order FM transition with a wide and narrow hysteresis loop, respectively. +For the model on MNs with layers in the form of RRGs the homogeneous PA for small r = 0.05 (Fig. 1(c)) similarly +predicts the first-order FM transition for moderate and large K, while for larger r = 0.15 (Fig. 1(d)) it predicts +the second-order FM transition already for moderate K and the first-order FM transition only for large K. Again, +the critical temperature(s) decrease, and the width of the hysteresis loop increases with K, and the curves m(T) +eventually approach these resulting from the MFA. +It may be inferred from Fig. 1 that the homogeneous PA predicts for the q-neighbor Ising model on MNs with +partial overlap of nodes and layers in the form of RRGs with finite K the same critical behavior as the MFA for +the model on analogous MNs with layers in the form of complete graphs, only for different ranges of the overlap r. +This conclusion is supported by Fig. 2 where the critical behavior predicted by the PA is summarized for the former +model with fixed q = 2 and q = 4 and different K, r. For all K and r = 1 (full overlap of nodes), both PA and MFA +predict continuous FM transition with decreasing T, i.e., the first-order transition observed in the model on monoplex +networks is suppressed. However, for both q = 2, 4 and finite K the PA predicts the second-order FM transition also +for a range of r below r = 1 which is broadened with decreasing K. As a consequence, for q = 2 (Fig. 2, left and +middle panels) the PA predicts that the range of the occurrence of the first-order FM transition is shifted toward +smaller values of r. Similarly, for a narrow range of still smaller values of the overlap the PA predicts the coexistence +of the FM and PM phases for T → 0, but for small K this kind of critical behavior is expected at r significantly +below the interval rc < r < 0.5 obtained from the MFA. Finally, it is predicted that the range of small r for which +the FM transition is absent for decreasing K is narrowed. Eventually, for very small K = 4 comparable with q only +continuous FM transition is expected for any r, and all other kinds of critical behavior are suppressed. For q = 4 (Fig. +2, right panel) the range of small r for which the PA predicts the first-order FM transition is substantially diminished +with decreasing K. +In order to verify predictions of the homogeneous PA, MC simulations of the q-neighbor Ising model with q = 2, 4 +on large MNs with various parameters r, K were performed and the magnetization curves m(T) were obtained for +random PM initial conditions σi = ±1, i = 1, 2, . . . N and decreasing temperature as well as for FM initial conditions +σi = +1, i = 1, 2 . . . N and increasing temperature. Comparison with MC simulations shows that the homogeneous +PA qualitatively captures modification of the critical behavior of the model under study due to finite values of the + +00000000-012 +mean degree of the layers K, but its quantitative predictions, though much improved in comparison with those from +the MFA valid for large K, are not exact (Fig. 3, 4). For fixed q, K the PA approximately predicts the ranges +of the overlap r where different kinds of critical behavior should occur. However, as a rule, these predictions are +overestimated and in MC simulations the particular kinds of critical behavior appear for smaller values of r than +estimated from the PA. For example, for q = 2 the ranges of appearance of the coexistence of the FM and PM phases +for T → 0 (Fig. 3(a,b)) and of the second-order FM transition (Fig. 3(e,f)) in MC simulations are, respectively, shifted +and extended toward smaller values of r than expected from the PA. Consequently, in the case of the first-order FM +transition for q = 2 (Fig. 3(c,d)) and q = 4 (Fig. 4(a,b,e,f)) the lower and upper critical temperatures T (1) +c +, T (2) +c +are +underestimated and the width of the hysteresis loop is overestimated by the PA in comparison with these obtained +from MC simulations; it is interesting to note that discrepancies between the theoretical and numerical values of T (2) +c +are usually smaller than those for T (1) +c +. Similarly, in the case of the second-order FM transition for q = 2 (Fig. 3(f)) +and q = 4 (Fig. 4(c,d)) the critical temperature Tc is underestimated by the PA in comparison with that obtained +from MC simulations. In general, the curves m(T) evaluated from the PA show better agreement with those obtained +from MC simulations in the case of the second-order than the first-order FM transition. +In order to investigate the critical behavior of the model under study by means of appropriate AMEs as defined +in Sec. III B, Eq. (19,20) were solved numerically with various initial conditions and the curves m(T) were obtained +using long-time asymptotic values of the concentrations of spins directed up c(K,0),(m(A),0), etc., to evaluate stationary +values of the magnetization. As expected, predictions of the AMEs usually show comparable or better agreement +with the results of MC simulations than those of the homogeneous PA. This is particularly visible in the case of the +second-order FM transition in the model under study with small K and q = 2 (Fig. 3(f)) and q = 4 (Fig. 4(d)), +where the theoretical and numerical curves m(T) coincide very well and the critical temperature Tc is predicted +correctly. However, the ranges of the overlap predicted by the AMEs for which different kinds of critical behavior +occur are still shifted toward slightly higher values of r than obtained from MC simulations (see Fig. 3(a), where the +AMEs predict the absence of the FM transition rather than coexistence of the FM and PM phases observed in MC +simulations, and Fig. 3(e), where the AMEs predict the first-order FM transition with a narrow hysteresis loop rather +than the second-order transition). In the case of the coexistence of the FM and PM phases for T → 0 for q = 2 (Fig. +3(b)) and the first-order FM transition for q = 2 (Fig. 3(c,d)) and q = 4 (Fig. 4(a,b,e,f)) predictions of the AMEs +concerning the upper critical temperature T (2) +c +are usually better than those of the homogeneous PA, but the lower +critical temperature T (1) +c +is again usually underestimated and the width of the hysteresis loop is overestimated. In +general, some improvement of theoretical predictions by the AMEs in comparison with the homogeneous PA can be +seen for small and moderate K; for large K the curves m(T) obtained from the AMEs and PA coincide (Fig. 4(e)). +V. +DISCUSSION AND CONCLUSIONS +In this paper the q-neighbor Ising model on MNs with partial overlap of nodes and layers in the form of random +networks was investigated; as an example, the model on MNs with two layers in the form of RRGs was studied in +detail. Both theoretical considerations based on the homogeneous PA and AMEs as well as MC simulations show +that for given q ≥ 1 and finite mean degree of nodes K, and for varying overlap r and temperature T the model +exhibits qualitatively similar critical behavior as the q-neighbor Ising model on MNs with partial overlap of nodes +and layers in the form of complete graphs. In particular, for any q and full overlap of nodes r = 1 the first-order +FM transition is suppressed and only the second-order transition appears with decreasing T. Besides, for decreasing +K continuous rather than discontinuous FM transition is observed for an increasing range of large (for q = 2) and +large and moderate (for q > 2) values of r below r = 1. As a consequence, for decreasing K the ranges of r for +which the model exhibits the first-order FM transition (for q ≥ 2) and the coexistence of the FM and PM phases for +T → 0 (for q = 2) are shifted toward smaller values. It should be mentioned that in the q-neighbor Ising model on +(monoplex) networks the first-order FM transition is also suppressed for small K comparable with q [24]; in contrast, +in the model on MNs this suppression is due to the overlap of nodes and occurs for any K. The q-neighbor Ising +model was used here as an example, and related models for the opinion formation on MNs with partial overlap of +nodes can be studied using similar numerical and analytic methods; however, the expected qualitative changes of the +observed critical behavior with r will be probably less spectacular, since, e.g., in the case of the q-voter model even +for r = 1 the first-order FM transition is not suppressed [47]. +For the model under study with large K predictions of the simple homogeneous PA and more advanced system of +AMEs converge to these of the MFA and agree quantitatively with the results of MC simulations. For finite K the +predicted curves m(T) and critical temperature(s) differ quantitatively from the numerically obtained ones; usually, +the particular kinds of critical behavior are predicted to occur for smaller values of the overlap than observed in MC +simulations. In general, predictions of both PA and AMEs show better agreement with the results of MC simulations + +13 +in the case of the continuous than discontinuous FM transition. Predictions based on the AMEs are comparable to +or better than those of the PA; in particular, the critical temperature for the second-order FM transition and the +upper critical temperature for the first-order transition are more accurately predicted. Nevertheless, both PA and +AMEs qualitatively correctly capture changes of the critical behavior of the model with varying parameters K, r +characterizing the underlying MN. +Two versions of the heterogeneous PA were derived for the model under study, the more accurate fully heterogeneous +PA and the less accurate AMEs-based heterogeneous PA, which take to a different extent into account heterogeneity +of the distribution of the active links due to inhomogeneity of the nodes. In both cases, systems of equations for +the macroscopic quantities characterizing the model are significantly larger and more complicated than that in the +homogeneous PA but do not lead to a noticeable improvement of theoretical predictions concerning the magnetization +curves and phase diagrams for the model. This suggests that the simple homogeneous PA is as reliable as more +advanced versions of the PA in the study of the critical behavior of systems with multiplexity-induced inhomogeneity. +Only using much larger systems of AMEs in certain cases quantitatively improves agreement between theoretical +predictions and results of MC simulations of the above-mentioned models. +APPENDIX +A. +Fully heterogeneous pair approximation +The following outline of the fully heterogeneous PA for models on MNs is an extension of that for the q-voter models +with quenched disorder on networks, with two populations of agents differing by the spin-flip rates [15]. The fully +heterogeneous PA uses the assumption that the probability that a spin directed up or down in a node with multidegree +k has a given number of attached edges leading to spins directed up or down in nodes with multidegree ˜k obeys a +binomial distribution; this assumption is valid for each layer and for any pair k, ˜k separately, and the related binomial +distributions are assumed to be independent. The macroscopic quantities characterizing a model with two-state spins +on a MN are concentrations ck of spins directed up in nodes with multidegree k and concentrations ek,˜k,(L) +j,˜j +(= e +˜k,k,(L) +˜j,j +) +of bonds (ends of edges) attached within the layer G(L) to nodes with multidegree k containing spins with direction +j ∈ {↓, ↑} such that at the other end of the edge there is a node with multidegree ˜k containing spin with direction +˜j (normalized to the total number of bonds N⟨k(L)⟩ within G(L)). According to the above-mentioned assumptions +the joint probability that a node with multidegree k containing spin with direction j has i = +� +i(A), i(B), . . . i(Lmax)� +active bonds (ends of active links) attached within the consecutive layers, pointing at nodes with arbitrary multidegree +containing spins with opposite direction −j, has a multiplicative form +P(j, i|k) = +Lmax +� +L=A +Bk(L),i(L) +� +αk,(L) +j +� +, +(21) +where αk,(L) +j += � +k′ ek,k′,(L) +j,−j +/ � +k′ +� +j′∈{↓,↑} ek,k′,(L) +j,j′ +are conditional probabilities that an active bond is attached to +a node with multidegree k and spin with direction j (similar to θ(L) +↑ +, θ(L) +↓ +given by Eq. (7, 8)). In order to evaluate +the change in the concentration ek,˜k,(L) +j,˜j +due to, e.g., flipping the spin with direction j in a node with multidegree +k, it is necessary to know the numbers of bonds y, z attached to this node within the layer G(L) pointing at nodes +with multidegree ˜k given that these bonds are active (˜j = −j) or inactive (˜j = j), respectively. These numbers +obey binomial distributions Bi(L),y +� +βk,˜k,(L) +j,−j +� +, Bk(L)−i(L),z +� +γk,˜k,(L) +j,j +� +, respectively, where the conditional probabilities +are βk,˜k,(L) +j,−j += ek,˜k,(L) +j,−j +/ � +k′ ek,k′,(L) +j,−j +, γk,˜k,(L) +j,j += ek,˜k,(L) +j,j +/ � +k′ ek,k′,(L) +j,j +. Then the rate equations for the macroscopic +concentrations of spins directed up are +∂ck +∂t = +� +j∈{↑,↓} +(−1)δj,↑ck,j +� +i +Lmax +� +L=A +Bk(L),i(L) +� +αk,(L) +j +� +f (i; T |k) , +(22) +while the rate equations for the concentrations of active and inactive bonds within the layers contain terms such as +d +dtek,˜k,(L′) +j,−j += +1 +⟨k(L′)⟩P(k)ck,j +� +i +Lmax +� +L=A +Bk(L),i(L) +� +αk,(L) +j +� i(L′) +� +y=0 +Bi(L′),y +� +βk,˜k,(L′) +j,−j +� +(−y)f (i; T |k) + . . . +(23) + +14 +d +dtek,˜k,(L′) +j,j += +1 +⟨k(L′)⟩P(k)ck,j +� +i +Lmax +� +L=A +Bk(L),i(L) +� +αk,(L) +j +� k(L′)−i(L′) +� +z=0 +Bk(L′)−i(L′),z +� +γk,˜k,(L′) +j,j +� +(−z)f (i; T |k) + . . . +(24) +It can be seen that Eq. (22) resembles Eq. (10) in the homogeneous PA. Concerning the equations for the concentrations +of bonds, e.g., Eq. (23) states that a flip of the spin with direction j in node with multidegree k, which has i(L′) +active bonds attached within the layer G(L′), out of which y bonds point at nodes with multidegrees ˜k, decreases the +concentration ek,˜k,(L′) +j,−j +by y/ +� +N⟨k(L′)⟩ +� +; such a flip occurs with probability P(k)ck,jf (i; T |k) within a time interval +1/N; and the final input to the rate equation (23) is obtained by averaging the above-mentioned change over the +probability distributions Bk(L′),i(L′) +� +αk,(L) +j +� +for i(L′) and Bi(L′),y +� +βk,˜k,(L′) +j,−j +� +for y; etc. +In the case of the q-neighbor Ising model on MNs with partial overlap of nodes and with two layers in the form +of RRGs, with the multidegree distribution P (k) given by Eq. (1), there are three classes of nodes with k = (K, 0), +k = (0, K) and k = (K, K), and two layers G(L), L = A, B. Taking into account the symmetry of the model under +study and general symmetry conditions for the concentrations ek,˜k,(L) +j,˜j +the solutions of the system of equations (22 - 24) +can be constrained to a 12-dimensional subspace c(K,0) = c(0,K), e(K,0),(K,0),(A) +j +, +j′ += e(K,0),(K,0),(A) +j′ +, +j += e(0,K),(0,K),(B) +j +, +j′ += +e(0,K),(0,K),(B) +j′ +, +j +≡ e(K,0),(K,0) +j +, +j′ , e(K,0),(K,K),(A) +j +, +j′ += e(K,K),(K,0),(A) +j′ +, +j += e(0,K),(K,K),(B) +j +, +j′ += e(K,K),(0,K),(B) +j′ +, +j +≡ e(K,0),(K,K) +j +, +j′ , +e(K,K),(K,K),(A) +j +, +j′ += e(K,K),(K,K),(A) +j′ +, +j += e(K,K),(K,K),(B) +j +, +j′ += e(K,K),(K,K),(B) +j′ +, +j +≡ e(K,K),(K,K) +j +, +j′ +, j, j′ ∈ {↓, ↑}. Thus, as in +Ref. [15], there are effectively only two classes of agents located in nodes with k = (K, 0) and k = (K, K), differing +by the spin-flip rates (6). Besides, the distributions of the number of links pointing at nodes belonging to each class +given that these links are active or inactive are fully determined by the conditional probabilities βk,k +j,−j, γk,k +j,j for the +links within each class. Taking this into account and performing summations in Eq. (22 - 24) as in Ref. [24] the +following system of equations for the macroscopic quantities is obtained in the fully heterogeneous PA for the model +under study, +dc(K,0) +dt += +� +1 − c(K,0) +� +R +� +α(K,0) +↓ ; T, q +� +− c(K,0)R +� +α(K,0) +↑ ; T, q +� +(25) +dc(K,K) +dt += +� +1 − c(K,K) +� � +R +� +α(K,K) +↓ +; T, q +��2 +− c(K,K) +� +R +� +α(K,K) +↑ +; T, q +��2 +(26) +d +dte(K,0),(K,0) +↑ +, +↑ += −2(1 − r) +K +c(K,0)γ(K,0),(K,0) +↑ +, +↑ +� +KR +� +α(K,0) +↑ ; T, q +� +− S +� +α(K,0) +↑ ; T, K, q +�� ++ 2(1 − r) +K +� +1 − c(K,0) +� +β(K,0),(K,0) +↓ +, +↑ S +� +α(K,0) +↓ ; T, K, q +� +(27) +d +dte(K,0),(K,0) +↓ +, +↓ += 2(1 − r) +K +c(K,0)β(K,0),(K,0) +↑ +, +↓ S +� +α(K,0) +↑ ; T, K, q +� +− 2(1 − r) +K +� +1 − c(K,0) +� +γ(K,0),(K,0) +↓ +, +↓ +� +KR +� +α(K,0) +↓ ; T, q +� +− S +� +α(K,0) +↓ ; T, K, q +�� +(28) +d +dte(K,K),(K,K) +↑ +, +↑ += −2r +K c(K,K)γ(K,K),(K,K) +↑ +, +↑ +� +KR +� +α(K,K) +↑ +; T, q +� +− S +� +α(K,K) +↑ +; T, K, q +�� +R +� +α(K,K) +↑ +; T, q +� ++ 2r +K +� +1 − c(K,K) +� +β(K,K),(K,K) +↓ +, +↑ +S +� +α(K,K) +↓ +; T, K, q +� +R +� +α(K,K) +↓ +; T, K, q +� +(29) +d +dte(K,K),(K,K) +↓ +, +↓ += 2r +K c(K,K)β(K,K),(K,K) +↑ +, +↓ +S +� +α(K,K) +↑ +; T, K, q +� +R +� +α(K,K) +↑ +; T, K, q +� +− 2r +K +� +1 − c(K,K) +� +γ(K,K),(K,K) +↓ +, +↓ +� +KR +� +α(K,K) +↓ +; T, q +� +− S +� +α(K,K) +↓ +; T, K, q +�� +R +� +α(K,K) +↓ +; T, q +� +(30) +d +dte(K,0),(K,0) +↑ +, +↓ += 1 − r +K +c(0,0) +� +−β(K,0),(K,0) +↑ +, +↓ S +� +α(K,0) +↑ ; T, K, q +� ++ γ(K,0),(K,0) +↑ +, +↑ +� +KR +� +α(K,0) +↑ ; T, q +� +− S +� +α(K,0) +↑ ; T, K, q +��� ++ 1 − r +K +� +1 − c(K,0) +� � +γ(K,0),(K,0) +↓ +, +↓ +� +KR +� +α(K,0) +↓ ; T, q +� +− S +� +α(K,0) +↓ ; T, K, q +�� +− β(K,0),(K,0) +↓ +, +↑ S +� +α(K,0) +↓ ; T, K, q +�� +(31) +d +dte(K,K),(K,K) +↑ +, +↓ += r +K c(K,K) +� +−β(K,K),(K,K) +↑ +, +↓ +S +� +α(K,K) +↑ +; T, K, q +� + +15 ++ γ(K,K),(K,K) +↑ +, +↑ +� +KR +� +α(K,K) +↑ +; T, q +� +− S +� +α(K,K) +↑ +; T, K, q +��� +R +� +α(K,K) +↑ +; T, q +� ++ r +K +� +1 − c(K,K) +� � +γ(K,K),(K,K) +↓ +, +↓ +� +KR +� +α(K,K) +↓ +; T, q +� +− S +� +α(K,K) +↓ +; T, K, q +�� +− β(K,K),(K,K) +↓ +, +↑ +S +� +α(K,K) +↓ +; T, K, q +�� +R +� +α(K,K) +↓ +; T, q +� +(32) +d +dte(K,0),(K,K) +↑ +, +↑ += −1 − r +K +c(K,0) +� +1 − γ(K,0),(K,0) +↑ +, +↑ +� � +KR +� +α(K,0) +↑ ; T, q +� +− S +� +α(K,0) +↑ ; T, K, q +�� ++ 1 − r +K +� +1 − c(K,0) +� � +1 − β(K,0),(K,0) +↓ +, +↑ +� +S +� +α(K,0) +↓ ; T, K, q +� +− r +K c(K,K) +� +1 − γ(K,K),(K,K) +↑ +, +↑ +� � +KR +� +α(K,K) +↑ +; T, q +� +− S +� +α(K,K) +↑ +; T, K, q +�� +R +� +α(K,K) +↑ +; T, q +� ++ r +K +� +1 − c(K,K) +� � +1 − β(K,K),(K,K) +↓ +, +↑ +� +S +� +α(K,K) +↓ +; T, K, q +� +R +� +α(K,K) +↓ +; T, q +� +(33) +d +dte(K,0),(K,K) +↓ +, +↓ += 1 − r +K +c(K,0) +� +1 − β(K,0),(K,0) +↑ +, +↓ +� +S +� +α(K,0) +↑ ; T, K, q +� +− 1 − r +K +� +1 − c(K,0) +� � +1 − γ(K,0),(K,0) +↓ +, +↓ +� � +KR +� +α(K,0) +↓ ; T, q +� +− S +� +α(K,0) +↓ ; T, K, q +�� ++ r +K c(K,K) +� +1 − β(K,K),(K,K) +↑ +, +↓ +� +S +� +α(K,K) +↑ +; T, K, q +� +R +� +α(K,K) +↑ +; T, q +� +− r +K +� +1 − c(K,K) +� � +1 − γ(K,K),(K,K) +↓ +, +↓ +� � +KR +� +α(K,K) +↓ +; T, q +� +− S +� +α(K,K) +↓ +; T, K, q +�� +R +� +α(K,K) +↓ +; T, q +� +(34) +d +dte(K,0),(K,K) +↑ +, +↓ += −1 − r +K +c(K,0) +� +1 − β(K,0),(K,0) +↑ +, +↓ +� +S +� +α(K,0) +↑ ; T, K, q +� ++ 1 − r +K +� +1 − c(K,0) +� � +1 − γ(K,0),(K,0) +↓ +, +↓ +� � +KR +� +α(K,0) +↓ ; T, q +� +− S +� +α(K,0) +↓ ; T, K, q +�� ++ r +K c(K,K) +� +1 − γ(K,K),(K,K) +↑ +, +↑ +� � +KR +� +α(K,K) +↑ +; T, q +� +− S +� +α(K,K) +↑ +; T, K, q +�� +R +� +α(K,K) +↑ +; T, q +� +− r +K +� +1 − c(K,K) +� � +1 − β(K,K),(K,K) +↓ +, +↑ +� +S +� +α(K,K) +↓ +; T, K, q +� +R +� +α(K,K) +↓ +; T, q +� +(35) +d +dte(K,0),(K,K) +↓ +, +↑ += 1 − r +K +c(K,0) +� +1 − γ(K,0),(K,0) +↑ +, +↑ +� � +KR +� +α(K,0) +↑ ; T, q +� +− S +� +α(K,0) +↑ ; T, K, q +�� +− 1 − r +K +� +1 − c(K,0) +� � +1 − β(K,0),(K,0) +↓ +, +↑ +� +S +� +α(K,0) +↓ ; T, K, q +� +− r +K c(K,K) +� +1 − β(K,K),(K,K) +↑ +, +↓ +� +S +� +α(K,K) +↑ +; T, K, q +� +R +� +α(K,K) +↑ +; T, q +� ++ r +K +� +1 − c(K,K) +� � +1 − γ(K,K),(K,K) +↓ +, +↓ +� � +KR +� +α(K,K) +↓ +; T, q +� +− S +� +α(K,K) +↓ +; T, K, q +�� +R +� +α(K,K) +↓ +; T, q +� +, +(36) +where the significant conditional probabilities are +α(K,0) +↓ += +e(K,0),(K,0) +↓ +, +↑ ++ e(K,0),(K,K) +↓ +, +↑ +e(K,0),(K,0) +↓ +, +↑ ++ e(K,0),(K,K) +↓ +, +↑ ++ e(K,0),(K,0) +↓ +, +↓ ++ e(K,0),(K,K) +↓ +, +↓ +α(K,0) +↑ += +e(K,0),(K,0) +↑ +, +↓ ++ e(K,0),(K,K) +↑ +, +↓ +e(K,0),(K,0) +↑ +, +↓ ++ e(K,0),(K,K) +↑ +, +↓ ++ e(K,0),(K,0) +↑ +, +↑ ++ e(K,0),(K,K) +↑ +, +↑ +α(K,K) +↓ += +e(K,K),(K,K) +↓ +, +↑ ++ e(K,K),(K,0) +↓ +, +↑ +e(K,K),(K,K) +↓ +, +↑ ++ e(K,K),(K,0) +↓ +, +↑ ++ e(K,K),(K,K) +↓ +, +↓ ++ e(K,K),(K,0) +↓ +, +↓ +α(K,K) +↑ += +e(K,K),(K,K) +↑ +, +↓ ++ e(K,K),(K,0) +↑ +, +↓ +e(K,K),(K,K) +↑ +, +↓ ++ e(K,K),(K,0) +↑ +, +↓ ++ e(K,K),(K,K) +↑ +, +↑ ++ e(K,K),(K,0) +↑ +, +↑ +, +(37) +β(K,0),(K,0) +↓ +, +↑ = +e(K,0),(K,0) +↓ +, +↑ +e(K,0),(K,0) +↓ +, +↑ ++ e(K,0),(K,K) +↓ +, +↑ +, β(K,0),(K,0) +↑ +, +↓ = +e(K,0),(K,0) +↑ +, +↓ +e(K,0),(K,0) +↑ +, +↓ ++ e(K,0),(K,K) +↑ +, +↓ + +16 +2.1 +2.2 +2.3 +2.4 +2.5 +2.6 +2.7 +T +−1.0 +−0.5 +0.0 +0.5 +1.0 +m +FIG. 5. The curves show magnetization m vs. temperature T obtained from the homogeneous PA (solid lines) and from the +heterogeneous PA (symbols) for q = 4, K = 20 and r = 0.1, 0.15, 0.2 (from left to right). +β(K,K),(K,K) +↓ +, +↑ += +e(K,K),(K,K) +↓ +, +↑ +e(K,K),(K,K) +↓ +, +↑ ++ e(K,K),(K,0) +↓ +, +↑ +, β(K,K),(K,K) +↑ +, +↓ += +e(K,K),(K,K) +↑ +, +↓ +e(K,K),(K,K) +↑ +, +↓ ++ e(K,K),(K,0) +↑ +, +↓ +, +(38) +γ(K,0),(K,0) +↓ +, +↓ = +e(K,0),(K,0) +↓ +, +↓ +e(K,0),(K,0) +↓ +, +↓ ++ e(K,0),(K,K) +↓ +, +↓ +, γ(K,0),(K,0) +↑ +, +↑ = +e(K,0),(K,0) +↑ +, +↑ +e(K,0),(K,0) +↑ +, +↑ ++ e(K,0),(K,K) +↑ +, +↑ +γ(K,K),(K,K) +↓ +, +↓ += +e(K,K),(K,K) +↓ +, +↓ +e(K,K),(K,K) +↓ +, +↓ ++ e(K,K),(K,0) +↓ +, +↓ +, γ(K,K),(K,K) +↑ +, +↑ += +e(K,K),(K,K) +↑ +, +↑ +e(K,K),(K,K) +↑ +, +↑ ++ e(K,K),(K,0) +↑ +, +↑ +. +(39) +Concentration ˜c of spins directed up within each layer and concentration c of spins directed up in the MN are defined +in the same way as in Sec. III A. Natural initial conditions for the system of equations (25 - 36) are c(K,0)(0) = +c(K,K)(0) = ρ0, e(K,0),(K,0) +↑ +, +↑ (0) = (1 − r)2ρ2 +0, e(K,0),(K,0) +↑ +, +↓ (0) = (1 − r)2ρ0(1 − ρ0), e(K,0),(K,0) +↓ +, +↓ (0) = (1 − r)2(1 − ρ0)2, +e(K,K),(K,K) +↑ +, +↑ +(0) = r2ρ2 +0, e(K,K),(K,K) +↑ +, +↓ +(0) = r2ρ0(1 − ρ0), e(K,K),(K,K) +↓ +, +↓ +(0) = r2(1 − ρ0)2, e(K,0),(K,K) +↑ +, +↑ +(0) = (1 − r)rρ2 +0, +e(K,0),(K,K) +↓ +, +↓ +(0) = (1 − r)r(1 − ρ0)2, e(K,0),(K,K) +↑ +, +↓ +(0) = (1 − r)2ρ0(1 − ρ0) = e(K,0),(K,0) +↓ +, +↑ (0) = (1 − r)rρ0(1 − ρ0), where +ρ0 can be chosen arbitrarily. +As mentioned in Sec. III A the magnetization curves obtained from the fully heterogeneous PA are practically +indistinguishable from those obtained from the homogeneous PA. This is illustrated by examples in Fig. 5. +B. +AMEs-based heterogeneous pair approximation +The AMEs-based heterogeneous PA again uses the assumption that the probability that a spin directed up or down +in a node with multidegree k has a given number of neighboring spins directed up obeys a binomial distribution; for +models on MNs this assumption is made for each layer separately, and the related binomial distributions are assumed +to be independent. Hence, in contrast with the homogeneous PA, in the AMEs-based heterogeneous PA it is taken +into account that for a node with multidegree k occupied by a spin with downward or upward direction the respective +probabilities ϑ(L) +k , η(L) +k +that a randomly chosen neighboring node within the layer G(L) is occupied by a spin directed +upward can depend on k. However, in contrast with the fully heterogeneous PA developed in Appendix A, all active +or inactive edges attached to a given node within a given layer are treated in the same way and obey common binomial +distributions [28, 29]. As mentioned in Sec. III B, these two assumptions should make the AMEs-based heterogeneous +PA more accurate than the homogeneous PA and less accurate than the fully heterogeneous PA. Eventually, in the +AMEs-based heterogeneous PA the time-dependent macroscopic quantities are the density ck of spins directed up in +nodes with multidegree k as well as the above-mentioned probabilities ϑ(L) +k , η(L) +k . +In terms of the densities ck,m and sk,m used in the AMEs, Eq. (19), (20) the above-mentioned macroscopic +quantities can be expressed as ck = � +m ck,m = 1 − � +m sk,m, ϑ(L) +k += � +m m(L)sk,m/ +� +k(L) (1 − ck) +� +, η(L) +k += + +17 +� +m m(L)ck,m/ +� +k(L)ck +� +. +Then, the core approximation for the AMEs-based heterogeneous PA can be made, ac- +cording to which sk,m ≈ (1 − ck) �Lmax +L=A Bk(L),m(L) +� +ϑ(L) +k +� +, ck,m ≈ ck +�Lmax +L=A Bk(L),m(L) +� +η(L) +k +� +. The latter approxi- +mation should be made in Eq. (19), (20) as well as in the definitions of the average rates β(L) +s +, . . . , γ(L) +i +, so that, +e.g., β(L′) +s +≈ ¯β(L′) +s += +� +(1 − ck) � +m +� +k(L′) − m(L′)� +Fk,m +�Lmax +L=A Bk(L),m(L) +� +ϑ(L) +k +� � +/ +� +(1 − ck) k(L′) � +1 − ϑ(L′) +k +� � +, etc. +Differentiating the definitions of ck, ϑ(L) +k , η(L) +k +with respect to time and using Eq. (19), (20) with the above-mentioned +approximations yields the following system of equations for the time dependence of the macroscopic quantities in the +heterogeneous PA, +dck +dt = −ck +� +m +Rk,m +Lmax +� +L=A +Bk(L),m(L) +� +η(L) +k +� ++ (1 − ck) +� +m +Fk,m +Lmax +� +L=A +Bk(L),m(L) +� +ϑ(L) +k +� +, +(40) +dϑ(L′) +k +dt += +� +m +� +ϑ(L′) +k +− m(L′) +k(L′) +� � +Fk,m +Lmax +� +L=A +Bk(L),m(L) +� +ϑ(L) +k +� +− +ck +1 − ck +Rk,m +Lmax +� +L=A +Bk(L),m(L) +� +η(L) +k +�� ++¯β(L′) +s +� +1 − ϑ(L′) +k +� +− ¯γ(L′) +s +ϑ(L′) +k +, +(41) +dη(L′) +k +dt += +� +m +� +η(L′) +k +− m(L′) +k(L′) +� � +Rk,m +Lmax +� +L=A +Bk(L),m(L) +� +η(L) +k +� +− 1 − ck +ck +Fk,m +Lmax +� +L=A +Bk(L),m(L) +� +ϑ(L) +k +�� ++¯β(L′) +i +� +1 − η(L′) +k +� +− ¯γ(L′) +i +η(L′) +k +, +(42) +where L′ = A, B . . . Lmax. The above equations are very similar to those obtained in the AMEs-based heterogeneous +PA for the spin models on (monoplex) networks [28, 29]; in particular, terms containing β(L) +s +, . . . , γ(L) +i +with L ̸= L′ +do not occur in Eq. (41), (42) for ϑ(L′) +k +, η(L′) +k +since the respective terms from Eq. (19), (20) sum up to zero in the +derivation. It should be mentioned that the AMEs can also be a starting point to obtain the homogeneous PA from +Sec. III A by assuming that the probability that a spin directed down has within the layer G(L) a neighboring spin +directed up does not depend on k and can be expressed as the average θ(L) +↓ += ⟨� +m m(L)sk,m⟩/⟨k(L) (1 − ck)⟩ [28, 29]. +In the case of the q-neighbor Ising model on MNs with partial overlap of nodes and with layers in the form of +RRGs, with the multidegree distribution P (k) given by Eq. (1), there are three classes of nodes with k = (K, 0), +k = (0, K) and k = (K, K), and two layers G(L), L = A, B, thus the system of equations (40-42) is 11-dimensional. +Due to the symmetry of the model solutions of these equations should be constrained to a subspace c(0,K) = c(K,0), +ϑ(A) +(K,0) = ϑ(B) +(0,K) ≡ ϑ(0,K), η(A) +(K,0) = η(B) +(0,K) ≡ η(0,K), ϑ(A) +(K,K) = ϑ(B) +(K,K) ≡ ϑ(K,K), η(A) +(K,K) = η(B) +(K,K) ≡ η(K,K) which +reduces the number of equations to six. Performing summations in Eq. (40-42) as in Ref. [24] the following system of +equations for the macroscopic quantities is obtained in the AMEs-based heterogeneous PA for the model under study, +dc(K,0) +dt += −c(K,0)R +� +1 − η(K,0); T, q +� ++ +� +1 − c(K,0) +� +R +� +ϑ(K,0); T, q +� +, +(43) +dϑ(K,0) +dt += ϑ(K,0) +� +R +� +ϑ(K,0); T, q +� +− +c(K,0) +1 − c(K,0) +R +� +1 − η(K,0); T, q +�� +− 1 +K +� +S +� +ϑ(K,0); T, K, q +� +− +c(K,0) +1 − c(K,0) +� +KR +� +1 − η(K,0); T, q +� +− S +� +1 − η(K,0); T, K, q +��� ++ ¯βs +� +1 − ϑ(K,0) +� +− ¯γsϑ(K,0), +(44) +dη(K,0) +dt += η(K,0) +� +R +� +1 − η(K,0); T, q +� +− 1 − c(K,0) +c(K,0) +R +� +ϑ(K,0); T, q +�� +− 1 +K +�� +KR +� +1 − η(K,0); T, q +� +− S +� +1 − η(K,0); T, K, q +�� +− 1 − c(K,0) +c(K,0) +S +� +ϑ(K,0); T, K, q +�� ++ ¯βi +� +1 − η(K,0) +� +− ¯γiη(K,0), +(45) +dc(K,K) +dt += −c(K,K) +� +R +� +1 − η(K,K); T, q +��2 + +� +1 − c(K,K) +� � +R +� +ϑ(K,K); T, q +��2 , +(46) +dϑ(K,K) +dt += ϑ(K,K) +�� +R +� +ϑ(K,K); T, q +��2 − +c(K,K) +1 − c(K,K) +� +R +� +1 − η(K,K); T, q +��2 +� +− 1 +K +� +S +� +ϑ(K,K); T, K, q +� +R +� +ϑ(K,K); T, q +� + +18 +− +c(K,K) +1 − c(K,K) +� +KR +� +1 − η(K,0); T, q +� +− S +� +1 − η(K,0); T, K, q +�� +R +� +1 − η(K,K); T, q +�� ++ ¯βs +� +1 − ϑ(K,K) +� +− ¯γsϑ(K,K), +(47) +dη(K,K) +dt += η(K,K) +�� +R +� +1 − η(K,K); T, q +��2 − 1 − c(K,K) +c(K,K) +� +R +� +ϑ(K,K); T, q +��2 +� +− 1 +K +�� +KR +� +1 − η(K,0); T, q +� +− S +� +1 − η(K,0); T, K, q +�� +R +� +1 − η(K,K); T, q +� +−1 − c(K,K) +c(K,K) +S +� +ϑ(K,K); T, K, q +� +R +� +ϑ(K,K); T, q +�� ++ ¯βi +� +1 − η(K,K) +� +− ¯γiη(K,K), +(48) +where the average rates are +¯βs = +�1 − r +2 − r +� +1 − c(K,0) +� +K +� +1 − ϑ(K,0) +� ++ +r +2 − r +� +1 − c(K,K) +� +K +� +1 − ϑ(K,K) +��−1 +× +�1 − r +2 − r +� +1 − c(K,0) +� � +KR +� +ϑ(K,0); T, q +� +− S +� +ϑ(K,0); T, K, q +�� ++ +r +2 − r +� +1 − c(K,K) +� � +KR +� +ϑ(K,K); T, q +� +− S +� +ϑ(K,K); T, K, , q +�� +R +� +ϑ(K,K); T, q +�� +, +(49) +¯γs = +�1 − r +2 − rc(K,0)K +� +1 − η(K,0) +� ++ +r +2 − rc(K,K)K +� +1 − η(K,K) +��−1 +× +�1 − r +2 − rc(K,0)S +� +1 − η(K,0); T, K, q +� ++ +r +2 − rc(K,K)S +� +1 − η(K,K); T, K, q +� +R +� +1 − η(K,K); T, q +�� +, +(50) +¯βi = +�1 − r +2 − r +� +1 − c(K,0) +� +Kϑ(K,0) + +r +2 − r +� +1 − c(K,K) +� +Kϑ(K,K) +�−1 +× +�1 − r +2 − r +� +1 − c(K,0) +� +S +� +ϑ(K,0); T, K, q +� ++ +r +2 − r +� +1 − c(K,K) +� +S +� +ϑ(K,K); T, K, q +� +R +� +ϑ(K,K); T, q +�� +, +(51) +¯γi = +�1 − r +2 − rc(K,0)Kη(K,0) + +r +2 − rc(K,K)Kη(K,K) +�−1 +× +�1 − r +2 − rc(K,0) +� +KR +� +1 − η(K,0); T, q +� +− S +� +1 − η(K,0); T, K, q +�� ++ +r +2 − rc(K,K) +� +KR +� +1 − η(K,K); T, q +� +− S +� +1 − η(K,K); T, K, q +�� +R +� +1 − η(K,K); T, q +�� +. +(52) +Concentration ˜c of spins directed up within each layer and concentration c of spins directed up in the MN are defined in +the same way as in Sec. 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Rep. 8 (2018) 3094. + diff --git a/CNE1T4oBgHgl3EQfVwTa/content/tmp_files/load_file.txt b/CNE1T4oBgHgl3EQfVwTa/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..51d4207a1471278a8f29a3607398166121a87adc --- /dev/null +++ b/CNE1T4oBgHgl3EQfVwTa/content/tmp_files/load_file.txt @@ -0,0 +1,1504 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf,len=1503 +page_content='The q-neighbor Ising model on multiplex networks with partial overlap of nodes A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' Krawiecki and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' Gradowski Faculty of Physics, Warsaw University of Technology, Koszykowa 75, PL-00-662 Warsaw, Poland The q-neighbor Ising model for the opinion formation on multiplex networks with two layers in the form of random graphs (duplex networks), the partial overlap of nodes, and LOCAL&AND spin update rule was investigated by means of the pair approximation and approximate Master equations as well as Monte Carlo simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' Both analytic and numerical results show that for different fixed sizes of the q-neighborhood and finite mean degrees of nodes within the layers the model exhibits qualitatively similar critical behavior as the analogous model on multiplex networks with layers in the form of complete graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' However, as the mean degree of nodes is decreased the discontinuous ferromagnetic transition, the tricritical point separating it from the continuous transition and the possible coexistence of the paramagnetic and ferromagnetic phases at zero temperature occur for smaller relative sizes of the overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' Predictions of the simple homogeneous pair approximation concerning the critical behavior of the model under study show good qualitative agreement with numerical results;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' predictions based on the approximate Master equations are usually quantitatively more accurate, but yet not exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' Two versions of the heterogeneous pair approximation are also derived for the model under study, which, surprisingly, yield predictions only marginally different or even identical to those of the simple homogeneous pair approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' In general, predictions of all approximations show better agreement with the results of Monte Carlo simulations in the case of continuous than discontinuous ferromagnetic transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' INTRODUCTION Investigation of the opinion formation process by means of nonequilibrium models has become a firmly established research field in statistical physics in the last decades [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' Many results in this area were obtained using models with agents’ opinions represented by spins with discrete (in most cases two) states obeying stochastic dynamics described by various rates at which agents change (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=', flip) their opinions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=', the majority-vote model [2–7], the noisy voter model [8–10], different versions of the noisy nonlinear and q-voter model [11–20] and the q-neighbor Ising model [21– 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' In particular, much effort was devoted to determining conditions under which the above-mentioned models exhibit phase transition from a disordered paramagnetic (PM) state in which each opinion appears with the same probability to an ordered ferromagnetic (FM) state with one dominant opinion as the parameter controlling the level of stochastic noise in the model is varied, measuring the agents’ uncertainty in decision making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' In this context the presence of the first-order FM transition, or even transition to a frozen FM phase is of prime importance, with abrupt occurrence of a dominant opinion as well as possible hysteresis and bistability of the PM and FM phases [4, 5, 12–21, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' Following the growing interest in the dynamical processes on complex networks [25] agents in the models for the opinion formation are often located in the nodes and interact via edges of complex networks reflecting a complicated structure of social interactions [3–7, 9, 13–16, 18–20, 23, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' In this case analytic predictions concerning the critical behavior of the models based on the mean-field approximation (MFA) need not exhibit quantitative agreement with results of Monte Carlo (MC) simulations, hence, more accurate approaches based on the pair approximation (PA) [26–30] and approximate Master equations (AMEs) [28–30] were applied to describe theoretically the observed phase transitions [10, 14–16, 18–20, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' Recently much attention has been devoted to combining complex networks in order to create even more complicated and heterogeneous structures known in general as ”networks of networks” [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' An important class of such structures is formed by multiplex networks (MNs) which consist of a fixed set of nodes connected by various sets of edges called layers [31–33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' In the simplest case, the layers are independently generated random networks with a full overlap of nodes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=', with each node belonging to all layers, which means it has at least one attached edge from each layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' In turn, in MNs with partial overlap of nodes, there are nodes belonging only to some rather than all layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' In particular, in the case of MNs with two layers (duplex networks) and partial overlap of nodes, the nodes are divided into a class of nodes belonging to both layers and forming the overlap, and two other classes, each consisting of nodes belonging only to one of the two layers [34–36] (the node overlap should not be confused with the link overlap [37–39] which is negligible in the case of independently generated layers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' FM phase transition in equilibrium models on MNs was studied, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=', in the Ising model [40, 41] and a related Ashkin-Teller model [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' Analogously, FM transition in nonequilibrium models for the opinion formation on MNs was studied, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=', in the majority vote model [44, 45], the q-voter model [46–48] and the q-neighbor Ising model [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' As expected, the critical properties of the nonequilibrium models, in particular the extension or confinement of the range of parameters for which the first-order transition occurs, strongly depend on the way in which the multiplexity affects the spin-flip rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' In this respect, very interesting seems arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='03107v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='stat-mech] 8 Jan 2023 2 the q-neighbor Ising model with LOCAL&AND spin update rule [50], which so far has been studied by MC simulations and in the MF approximation on duplex networks with full and partial overlap of nodes and with layers in the form of fully connected graphs [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' In this model, the flip probability per unit time for the spins in nodes belonging to only one layer (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=', outside the overlap) is given by a Metropolis-like rate, but with a local field produced only by a subset of q randomly chosen neighboring spins (q-neighborhood), and for the spins in nodes belonging to both layers (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=', within the overlap) it is given by a product of two above-mentioned rates evaluated separately for each layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' With the increase of the relative size of the overlap, and depending on the size of the q-neighborhood, suppression of the first-order transition, appearance of a tricritical point separating first- and second-order FM transition, and possible coexistence of the PM and FM phases even in zero temperature were observed in the model [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' In this paper, the q-neighbor Ising model on MNs with partial overlap of nodes, with layers in the form of complex networks and with the LOCAL&AND spin update rule is studied by means of MC simulations and theoretically in the framework of the PA and AMEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' It should be noted that the q-neighbor Ising model is used here as a convenient example since the results can be readily compared with the above-mentioned ones for the limiting case of the model on MNs with layers in the form of complete graphs [49], and the PA and AMEs used here can be easily generalized to other models for the opinion formation with similar structure of interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' In order to make large systems of AMEs numerically tractable in this paper only the case of duplex networks with layers in the form of homogeneous random networks is considered;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' nevertheless, such MNs exhibit certain overlap-induced inhomogeneity since the nodes within and outside the overlap form distinct classes characterized by different degrees within the individual layers (both non-zero or one zero and one non-zero).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' Thus also the flip rates for the spins located in nodes belonging to distinct classes are different;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' a related q-voter model with quenched disorder, with agents divided into subpopulations according to different rates of the opinion change, has been recently considered [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' The aim of this paper is first to provide a general formulation of the PA and AMEs, which take into account to a different extent the above-mentioned inhomogeneity of nodes, for models on MNs with partial overlap of nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' For this purpose, first, the homogeneous PA for models on MNs with a full overlap of nodes [47] is extended to the case with partial overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' For nodes belonging to different classes this simplest form of the PA takes into account the inhomogeneity of the average directions of spins (opinions) but neglects possible inhomogeneity of the distributions of directions of neighboring spins within each layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' For the q-neighbor Ising model predictions of this approximation concerning the FM phase transition show surprisingly good agreement with results of MC simulations for a wide range of the size of the q-neighborhood, the mean degrees of nodes within layers and the size of the overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' Then, the most advanced approximation based on the AMEs for models on MNs with the full overlap of nodes [45] and weighted networks [51] is extended to the case of models on MNs with partial overlap of nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' Finally, two kinds of heterogeneous PA, the fully heterogeneous PA [15] and the AMEs-based heterogeneous PA [28–30] are applied to models on MNs with partial overlap of nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' Both versions of the PA take into account, to a different extent, the above-mentioned inhomogeneity of distributions of directions of neighboring spins within each layer and are in general intermediate with respect to the accuracy of predictions between the homogeneous PA and the AMEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' For the q-neighbor Ising model under study, it turns out that their predictions are only marginally different or even identical with these of the homogeneous PA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' On the other hand, predictions based on the AMEs show slightly better quantitative agreement with the results of MC simulations, in particular for smaller mean degrees of nodes within layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' In general, predictions of all approximations concerning the first-order FM transition (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=', location and width of the hysteresis loop) are quantitatively worse than those concerning the second-order transition (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=', location of the critical point).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' Besides, the aim of this paper is also to study in detail the phase diagram for the q-neighbor Ising model on MNs with partial overlap of nodes and with layers with a finite mean degree of nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' It is shown that the critical behavior of this model resembles qualitatively that of the analogous model on MNs with layers in the form of fully connected graphs [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' However, as the mean degree of nodes is decreased, the first-order FM transition, the tricritical point separating it from the second-order transition, and the possible coexistence of the PM and FM phases occur for smaller relative sizes of the overlap, while the range of the occurrence of the second-order FM transition is broadened correspondingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' THE MODEL A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' Multiplex networks with partial overlap of nodes MNs consist of a fixed set of nodes connected by several sets of edges;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' the set of nodes with each set of edges forms a network which is called a layer of a MN [32, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' Henceforth, the nodes are indexed by i, i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' N, and the subsequent layers are denoted as G(L), L = A, B, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' Lmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' In the case of MNs with a full overlap of nodes each node belongs to all layers, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=', each node has at least one edge from each layer attached to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' In general, MNs with partial overlap of nodes are defined as MNs in which nodes may belong to (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=', may have attached edges from) 3 some rather than all layers, given that each node belongs to at least one layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' Henceforth, the number of nodes belonging to the layer G(L) is denoted as N (L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' In this paper, it is assumed that the sets of edges for the subsequent layers G(L) are generated independently and form complex random networks with N (L) nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' As a result, multiple connections between nodes are not allowed within the same layer, but the same nodes belonging to several layers can be accidentally connected by multiple edges belonging to different layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' A simple example of the MN with partial overlap of nodes is that with only two layers G(A), G(B), called a duplex network, and with n nodes belonging to both layers which form the overlap (0 ≤ n ≤ N);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' then, N = N (A) + N (B) − n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' Furthermore, if both layers contain the same number of nodes N (A) = N (B) = ˜N it is possible to introduce a single parameter r = n/ ˜N, also called the overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' Then, the nodes are divided into three subsets: ˜N − n = N(1 − r)/(2 − r) nodes belonging only to the layer G(A), N(1 − r)/(2 − r) nodes belonging only to the layer G(B) and n = Nr/(2 − r) nodes belonging both to G(A) and G(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' The numbers of edges attached to the node i (degrees) within the individual layers G(L) are denoted as k(L) i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' if the node i does not belong to the layer G(L) then k(L) i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' In the case of MNs with independently generated layers the degrees of nodes belonging to the individual layers G(L), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=', these with k(L) i > 0, are drawn from probability distri- butions P � k(L)� which characterize the layers as complex networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' For a given node i a vector of its degrees within the individual layers ki = � k(A) i , k(B) i , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' k(Lmax) i � , with possible zero components in the case of MNs with partial overlap of nodes, is called a multidegree of the node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' The multidegree distribution P(k) = P � k(A) i , k(B) i , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' k(Lmax) i � characterizes the MN as a complex ”network of networks”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' in the case of MNs with the full overlap of nodes and independently generated layers, it is obviously P(k) = �Lmax L=A P(k(L)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' In the formulas below, averages are evaluated over the multidegree distribution, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=', ⟨k(L)⟩ = N −1 �N i=1 k(L) i = � k P(k)k(L) is the mean degree of nodes within the layer G(L) (note that the average is over all N nodes rather than N (L) nodes belonging to the layer G(L)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' As a simple example,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' in this paper the q-neighbor Ising model is considered on a duplex network with partial overlap of nodes and with the two independently generated layers in the form of random regular graphs (RRGs) with K edges attached to each node belonging to the layer,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' the same numbers of nodes N (A) = N (B) = ˜N and the overlap r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' for which the multidegree distribution is P (k) = P � k(A),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' k(B)� = 1 − r 2 − rδk(A),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='Kδk(B),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='0 + r 2 − rδk(A),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='Kδk(B),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='K + 1 − r 2 − rδk(A),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='0δk(B),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' (1) and ⟨k(A)⟩ = ⟨k(B)⟩ = ˜NK/N = K/(2 − r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' The q-neighbor Ising model on multiplex networks with partial overlap of nodes The q-neighbor Ising model [21–24, 49] is a nonequilibrium variant of the Ising model used to investigate the process of opinion formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' In this paper the above-mentioned model is considered on MNs with partial overlap of nodes and layers in the form of complex networks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' the MF version of this model, on MNs with layers in the form of fully connected graphs, was studied in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' The main interest is in the FM transition which can occur in the q-neighbor Ising model with decreasing effective temperature T, which measures the level of internal noise (uncertainty in agents’ decision making).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' In order to introduce the model under study, it is convenient to start with the q-neighbor Ising model on (monoplex) networks which can be regular, complex, or fully connected graphs [21–24, 49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' In this model agents with two possible opinions on a given subject are represented by two-state spins σi = ±1, i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' N placed in the nodes and interacting via edges of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' It is assumed that these interactions prefer identical orientations of spins in the connected nodes, which is reflected in the spin-flip rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' Thus, interactions between spins with opposite directions in general increase the probability that one of the spins flips, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=', the corresponding agent changes opinion, and edges representing these interactions are called active links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' The dynamics of the q-neighbor Ising model on networks is a modification of that of the kinetic Ising model with the Metropolis spin-flip rate in which, at each time step, each spin interacts only with its q randomly chosen neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' MC simulations of the model are performed using random asynchronous updating of spins, with each MC simulation step (MCSS) corresponding to updating all N spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' Nodes are picked randomly and for each picked node q its neighbors are chosen randomly and without repetitions, which form the q-neighborhood of the picked node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' Then, the spin in the picked node is flipped with probability given by a Metropolis-like formula, E (l;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, q) = min {1, exp[−2(q − 2l)/T]} , (2) where l is the number of nodes belonging to the q-neighborhood occupied by spins with a direction opposite to that of the spin in the picked node, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=', the number of active links attached to the picked node leading to nodes within 4 the chosen q-neighborhood (notation in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' (2) emphasizes that T, q, are parameters of the model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' As a result, the flip rate for a picked spin given that it is placed in a node with degree k which has in total i active links attached (0 ≤ i ≤ k) is f (i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T|k) = 1 �k q � q � l=0 �i l ��k − i q − l � E (l;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, q) = 1 �k i � q � l=0 �k − q i − l ��q l � E (l;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, q) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' (3) The q-neighbor Ising model on complete graphs for q = 3 exhibits second-order FM transition, while for q ≥ 4 first-order FM transition occurs with a clearly visible hysteresis loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' Width of the hysteresis loop in general increases with q, though for q > 4 there are oscillations superimposed on this trend such that loops for the consecutive odd values of q are narrower than for the neighboring even values of q [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' The same is true for the model on networks with finite mean degree ⟨k⟩ provided that q ≪ ⟨k⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' However, as q is increased and becomes comparable with ⟨k⟩ the hysteresis loop becomes narrower and eventually disappears, and the FM transition becomes second-order [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' In the q-neighbor Ising model on MNs with full or partial overlap of nodes, interactions take place within individual layers with respective, independently chosen q-neighborhoods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' Then, spins flip according to a probabilistic rule which combines the effect of the above-mentioned interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' In this paper the LOCAL&AND spin update rule is used [50] according to which the spin in the picked node flips if interaction with every q-neighborhood from every layer suggests flip;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' consequently, the probability of the spin-flip is given by a product of the Metropolis-like factors (2) corresponding to all layers containing the picked node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' The LOCAL&AND rule is assumed in this paper since it usually leads to richer phase diagrams than other methods of including the multiplex character of the network of interactions in the spin-flip rate [46–49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' Eventually, in numerical simulations of the q-neighbor Ising model on MNs with partial overlap of nodes and the LOCAL&AND spin update rule, each MCSS is performed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=') A node i, 1 ≤ i ≤ N, with multidegree ki is picked randomly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' (ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=') From each layer G(L) containing the picked node a set of its q neighbors (q-neighborhood) is chosen randomly and without repetitions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' it is assumed that 0 < q ≤ k(L) i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' Sets from different layers are chosen independently, thus the same node can by chance belong to two or more q-neighborhoods if it is a neighbor of the picked node within two or more layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' (iii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=') The Metropolis-like factor for the picked node is evaluated separately for each layer G(L), E � l(L);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, q � = min � 1, exp[−2(q − 2l(L))/T] � (4) where l(L) is the number of nodes in the q-neighborhood in the layer G(L) occupied by spins with direction opposite to that of the spin in the picked node;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' note that if a node does not belong to G(L) then q = l(L) = 0 and E(T, 0, 0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' (iv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=') Due to the LOCAL&AND spin update rule, the spin σi in the picked node flips with probability E (l;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, q) = Lmax � L=A E � l(L);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, q � , (5) where l = � l(A), l(B), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' l(Lmax)� ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' and obviously l(L) = 0 if the picked node does not belong to the layer G(L) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=', l is a vector of numbers of active links from the individual layers attached to the picked node which lead to nodes within the respective q-neighborhoods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' (v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=') Steps (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=')-(iv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=') are repeated until all N spins are updated without repetition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' Hence, the flip rate for a spin placed in a node with multidegree k = � k(A), k(B), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' k(Lmax)� and with the numbers of attached active links within the individual layers i(L), 0 ≤ i(L) ≤ k(L), given by the corresponding components of the vector i = � i(A), i(B), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' i(Lmax)� assumes a multiplicative form, f (i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T |k) = Lmax � L=A f � i(L);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T|k(L)� (6) (note that if a node does not belong to the layer G(L) there is k(L) = i(L) = 0 and f (0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T|0) ≡ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' The q-neighbor Ising model on a duplex network with layers in the form of complete graphs and partial overlap of nodes, and with the LOCAL&AND spin update rule exhibits FM phase transition already for q ≥ 1 [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' This 5 transition is in general second-order, with some exceptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' For q = 2 the transition is first-order for 1/2 < r < 1, with a clearly visible hysteresis loop, and for rc < r ≤ 1/2, where rc = 2(3 √ 2 − 4) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='4853 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=', the coexistence of the FM and PM phases is observed as the temperature is decreased below a critical value down to T = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' for r < rc there is no phase transition and the PM phase remains the only stable phase down to T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' For q ≥ 4 the transition for small r is first-order and for larger r is second-order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' The first- and second-order transitions are separated by a tricritical point at r = rT CP (q) which for q = 4 occurs at a particularly high value of r, and for q > 4 is an increasing function of q, but again with oscillations between the consecutive odd and even values of q superimposed on this trend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' Remarkably, for r = 1 the FM transition is always second-order for any q, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=', full overlap of nodes suppresses discontinuous transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' In this paper, it is investigated how the phase diagram of the model changes if the layers of the MN are complex networks with a finite mean degree of nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' THEORY A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' Pair approximation In the case of spin models on networks, the effect of the network topology (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='g, of the degree distribution or the mean degree of nodes) on the observed phase transitions often can be more accurately described in the framework of the PA than by the usual MFA [26–30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' In particular, this was demonstrated for the q-neighbor Ising model on complex networks [24] and a sort of stochastic q-voter model on MNs with a full overlap of nodes [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' In both above-mentioned studies the networks, or the layers of the MNs, were homogeneous complex networks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=', RRGs), thus the simplest homogeneous PA was enough to reproduce quantitatively results of MC simulations in a wide range of the parameters of the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' As mentioned in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' I & II MNs with partial overlap of nodes retain some multiplexity-induced inhomogeneity even if the layers are homogeneous complex networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' Nevertheless, in this section the homogeneous PA derived in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' [47] for a wide class of models with various spin update rules on MNs with the full overlap of nodes is presented in a more general form which makes it applicable to models on MNs with partial overlap of nodes, in order to find, inter alia, to what extent it can be used to explain critical behavior of systems with multiplicity-induced inhomogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' The advantage of the PA consists in that it takes into account dynamical correlations between pairs of interacting agents (spins).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' In the framework of the homogeneous PA, macroscopic quantities characterizing a model with two- state spins on MNs are concentrations ck of spins directed up located in nodes with multidegree k (with possible zero components in the case of MNs with partial overlap of nodes) as well as concentrations b(L) of active links within separate layers G(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' The homogeneous character of the PA allows for the simplification that the latter concentrations are averaged over all nodes belonging to a given layer and do not depend on the multidegrees of the connected nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' Consequently, it is assumed that conditional probabilities θ(L) j , j ∈ {↑, ↓}, that an active link within the layer G(L) is attached to a node given that it is occupied by spin with direction j are also independent of the multidegree of the node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' These probabilities can be evaluated as ratios of the number of attachments of active links to nodes with spins with direction j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' independently of their multidegrees,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' within the layer G(L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' which is N⟨k(L)⟩b(L)/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' and the number of attachments of all links within GL to such nodes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' which is � k NP (k) k(L)ck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' where ck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='↑ = ck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' ck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='↓ = 1 − ck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' thus θ(L) ↑ = b(L) 2 � k P (k) k(L)ck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='↑/⟨k(L)⟩ = b(L) 2 � k P (k) k(L)ck/⟨k(L)⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' (7) θ(L) ↓ = b(L) 2 � k P (k) k(L)ck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='↓/⟨k(L)⟩ = b(L) 2 � 1 − � k P (k) k(L)ck/⟨k(L)⟩ � (8) The core approximation made in the PA for models on MNs is that the numbers of active links i(L) attached to a node with degrees k(L) within individual layers G(L) (0 ≤ i(L) ≤ k(L)) occupied by spin with direction j obey independent binomial distributions with parameters θ(L) j given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' (7,8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' Then, the rates at which the concentration ck increases or decreases are given by averages of the spin-flip rate, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' (6), over the appropriate joint distributions of the number of active links within all layers which have a multiplicative form P(j, i|k) = Lmax � L=A Bk(L),i(L) � θ(L) j � , (9) where Bk,i(θ) = �k i � θi(1 − θ)k−i denotes the binomial factor and, formally, B0,0(θ) ≡ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' Hence, the equation for the time dependence of ck can be written as a rate equation, 6 ∂ck ∂t = � j∈{↑,↓} (−1)δj,↑ck,j � i Lmax � L=A Bk(L),i(L) � θ(L) j � f (i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T |k) , (10) where � i ≡ �k(A) i(A)=0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' �k(Lmax) i(Lmax)=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' In order to obtain an equation for the time dependence of the concentrations of active links b(L) one should observe that each flip of a spin (irrespective of its direction) in a picked node with multidegree k with the numbers of active links attached given by the components of the vector i results in the change of the numbers of active links within the individual layers G(L) by k(L) − 2i(L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' since then i(L) previously active links become inactive and k(L) − i(L) previously inactive links become active.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' The corresponding changes in the concentrations of active links b(L) are thus � k(L) − 2i(L)� /(N⟨k(L)⟩/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' As in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' (10), such changes connected with the flip of a spin with direction j occur at a rate given by the average of the spin-flip rate, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' (6), over the appropriate joint distributions of the number of active links attached to the picked node, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' Due to the homogeneous character of the PA, in order to obtain time dependence of b(L) further averaging over all nodes occupied by spins with direction j should be performed, which is equivalent to averaging over the probability distribution P(k)ck,j that a node with multidegree k is occupied by a spin with direction j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' Eventually, taking into account that nodes are picked and spins are updated within time intervals 1/N, for a given layer G(L′) it is obtained that ∂b(L′) ∂t = 2 ⟨k(L′)⟩ � j∈{↑,↓} � k P (k) ck,j � i Lmax � L=A Bk(L),i(L) � θ(L) j � f (i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T |k) � k(L′) − 2i(L′)� , (11) where L′ = A, B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' Lmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' In particular, let us consider the q-neighbor Ising model on a MN with two layers in the form of RRGs and partial overlap of nodes, with the multidegree distribution given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' Then, the nodes are divided into three classes, these belonging only to the layer G(A) with multidegree k = (K, 0), only to the layer G(B) with k = (0, K) and to the overlapping part of G(A) and G(B), with k = (K, K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' The macroscopic quantities to be used in the homogeneous PA are thus concentrations of spins directed up in the nodes belonging to the subsequent classes c(K,0), c(0,K), c(K,K) and concentrations of active links in the two layers b(A), b(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' Since both layers are identical, with N (A) = N (B) = ˜N, stable solutions of the system of equations (10), (11) are limited to the subspace with c(0,K) = c(K,0), b(A) = b(B) ≡ b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' moreover, according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' (7,8) there is θ(A) j = θ(B) j ≡ θj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' (1), (3), (6), performing summations in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' (10), (11) as in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' [24] and introducing functions R(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, q) and S(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, K, q) to shorten notation, R(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, q) = q � l=0 Bq,l (θ) E(l;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, q), (12) S(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, K, q) = q � l=0 Bq,l (θ) [(K − q)θ + l]E(l;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, q), (13) the following system of three equations for the time dependence of the macroscopic quantities in the homogeneous PA is obtained, dc(K,0) dt = � 1 − c(K,0) � R (θ↓;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, q) − c(K,0)R (θ↑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, q) (14) dc(K,K) dt = � 1 − c(K,K) � [R (θ↓;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, q)]2 − c(K,K) [R (θ↑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, q)]2 (15) db dt = 2 K (1 − r) �� 1 − c(K,0) � [KR (θ↓;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, q) − 2S (θ↓;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, K, q)] + c(K,0) [KR (θ↑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, q) − 2S (θ↑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, K, q)] � + 2 K r �� 1 − c(K,K) � [KR (θ↓;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, q) − 2S (θ↓;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, K, q)] R (θ↓;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, q) + c(K,K) [KR (θ↑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, q) − 2S (θ↑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, K, q)] R (θ↑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, q) � , (16) where θ↑ = b 2 � (1 − r)c(K,0) + rc(K,K) �, (17) θ↓ = b 2 � 1 − (1 − r)c(K,0) − rc(K,K) �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' (18) 7 Other macroscopic quantities of interest are the concentration of spins directed up in each layer, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=', the fraction of ˜N nodes occupied by such spins, which is ˜c = (1 − r)c(K,0) + rc(K,K), the concentration of spins directed up in the whole MN, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=', the fraction of N nodes occupied by such spins, which is c = 2(1−r) 2−r c(K,0) + r 2−rc(K,K), and the resulting magnetization of the MN m = 2c − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' Note that in the limiting case of layers in the form of fully connected graphs there is b = ˜N 2˜c(1 − ˜c)/[ ˜N( ˜N − 1)/2] ≈ 2˜c(1 − ˜c) and θ↓ = ˜c, θ↑ = 1 − ˜c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' after inserting this into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' (14) and (15) equations for the concentrations c(K,0), c(K,K) in the MF approximation are reproduced [49], as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' Natural extension of the homogeneous PA consists in taking into account heterogeneity of the concentrations of the (possibly active) links connecting classes of nodes with different multidegrees, so that, instead of the average concentration b(L) of active links within the layer G(L), e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=', concentrations of classes of active links connecting spins in nodes with multidegrees k, k′ within the layer G(L) become separate macroscopic quantities characterizing the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' This leads to the most advanced and accurate version of the PA called fully heterogeneous PA [15, 27];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' corresponding equations for the macroscopic quantities for spin models on MNs with partial overlap of nodes, in particular for the q-neighbor Ising model under study, are given in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' In the latter case solutions of these equations show that in the stationary state concentrations of active links (strictly speaking, of their ends called bonds) belonging to different classes indeed show noticeable heterogeneity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' nevertheless, this does not lead to the values of magnetization noticeably different from these predicted by the homogeneous PA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' Thus, magnetization curves and phase diagrams for the model under study obtained from the fully heterogeneous PA are practically indistinguishable from those obtained from the homogeneous PA and do not show better agreement with the results of MC simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' Approximate Master equations A more accurate approximation for the study of spin models on MNs with partial overlap of nodes is based on approximate Master equations (AMEs) for the densities of spins directed up ck,m and down sk,m which are located in nodes with multidegree k and have m(L) neighboring spins directed up within the consecutive layers G(L), which is denoted as m = � m(A), m(B) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' m(Lmax)� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' In the thermodynamic limit and for mutually uncorrelated layers in the form of random networks with finite mean degrees � k(L)� possibility that a pair of nodes is connected simultaneously by edges within different layers can be neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' Thus, in the AMEs it is assumed that in a single simulation step for a given node the allowed changes of the number of neighboring spins directed up are m → m ± e(L), where e(L) is a unit vector with Lmax components and only L-th component equal to one, while simultaneous changes of many components of m, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=', m → m ± e(L) ± e(L′), L ̸= L′, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=', cannot occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' Under the above-mentioned assumptions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' the AMEs in a general form are [45,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' 51] dsk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='m dt = −Fk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='msk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='m + Rk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='mck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='m + Lmax � L=A � −β(L) s � k(L) − m(L)� sk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='m + β(L) s � k(L) − m(L) + 1 � sk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='m−e(L) � + Lmax � L=A � −γ(L) s m(L)sk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='m + γ(L) s � m(L) + 1 � sk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='m+e(L) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' (19) dck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='m dt = −Rk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='mck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='m + Fk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='msk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='m + Lmax � L=A � −β(L) i � k(L) − m(L)� ck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='m + β(L) i � k(L) − m(L) + 1 � ck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='m−e(L) � + Lmax � L=A � −γ(L) i m(L)ck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='m + γ(L) i � m(L) + 1 � ck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='m+e(L) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' (20) In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' (19), (20) the first two terms account for the effect of a flip of a spin in a node with multidegree k and the remaining terms account for the average effect of the flips of spins in the neighboring nodes, irrespective of their multidegrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' In terms of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' II B the flip rate for a spin directed down occupying a node with multidegree k with m neighboring spins directed up is Fk,m = f (m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T |k) and that for a spin directed up Rk,m = f (k − m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T |k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' The remaining average rates can be estimated by evaluating the ratios (at a given time step) of the average number of edges connecting spins with a given direction such that one of these spins flips to the average total numbers of these edges [28, 29];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' in the case of models on MNs this should be done separately for each layer [45, 51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' Thus β(L) s = � � m � k(L) − m(L)� Fk,msk,m � / � � m � k(L) − m(L)� sk,m � , γ(L) s = � � m � k(L) − m(L)� Rk,mck,m � / 8 � � m � k(L) − m(L)� ck,m � , β(L) i = � � m m(L)Fk,msk,m � / � � m m(L)sk,m � , γ(L) i = � � m m(L)Rk,mck,m � / � � m m(L)ck,m � , where L = A, B, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' Lmax, � m ≡ �k(A) m(A)=0 �k(B) m(B)=0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' �k(Lmax) m(Lmax)=0 and ⟨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='⟩ denotes average over the multidegree distribution P (k), as usually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' Natural initial conditions for the system of equations (19), (20) are sk,m(0) = (1−c(0)) �Lmax L=A Bk(L),m(L)(c(0)), ck,m(0) = c(0) �Lmax L=A Bk(L),m(L)(c(0)), where 0 < c(0) < 1 is arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' In particular, in the case of the q-neighbor Ising model on a MN with two layers in the form of RRGs and partial over- lap of nodes, with the multidegree distribution P(k) given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' (1), there are three classes of nodes with k = (0, K), k = (K, 0) and k = (K, K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' The corresponding spin flip rates are F(K,0),(m(A);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='0) = f � m(A);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T |K � , F(0,K),(0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='m(B)) = f � m(B);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T |K � , F(K,K),(m(A),m(B)) = f � m(A);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T |K � f � m(B);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T |K � and R(K,0),(m(A),0) = f � K − m(A);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T |K � , R(0,K),(0,m(B)) = f � K − m(B);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T |K � , R(K,K),(m(A),m(B)) = f � K − m(A);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T |K � f � K − m(B);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T |K � , with f (m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T |K ) given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' Hence, the system (19), (20) consists of 2(K +1)2+4(K +1) equations and can be solved numerically for moderate K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' The quantities of interest, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=', the concentration c of spins directed up in the MN and the magnetiza- tion m = 2c−1 can be evaluated as in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' III A using c(K,0) = �K m(A)=0 c(K,0),(m(A),0), c(0,K) = �K m(B)=0 c(0,0),(0,m(B)), c(K,K) = �K m(A)=0 �K m(B)=0 c(K,K),(m(A),m(B)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' The AMEs are a starting point for a more elaborate approximation representing another formulation of the het- erogeneous PA [28–30, 45, 51] which takes into account the possible heterogeneity due to different multidegrees k of nodes of both the concentrations ck of spins directed up and of the conditional probabilities that a link attached to a node is active or, equivalently, leads to a spin with a given (say, up) direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' A general formulation of such AMEs-based heterogeneous PA for spin (two-state) models on (monoplex) networks by Gleeson [28, 29] was extended to the case of weighted networks [51] and, partly, MNs [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' It is believed that due to the approximations made the AMEs-based heterogeneous PA is in general more accurate than the homogeneous PA and less accurate than the fully heterogeneous PA mentioned in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' III A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' In this paper the AMEs-based heterogeneous PA is applied to spin models on MNs with partial overlap of nodes, in particular to the q-neighbor Ising model under study;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' equations for the macroscopic quantities are given in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' Surprisingly, it turns out that in the stationary state the above-mentioned conditional probabilities that a node has a link leading to a spin directed up do not depend on whether the node belongs or not to the overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' Hence, predictions of the AMEs-based heterogeneous PA concerning the FM transition in the model under study are identical to those of the homogeneous PA from Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' III A, so they are not further discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' RESULTS The main results concerning the FM transition in the q-neighbor Ising model on MNs with partial overlap of nodes and with layers in the form of complete graphs have been summarized in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' II B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' These results were obtained in the MF approximation and confirmed by MC simulations [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' In this section first predictions of the homogeneous PA of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' III A concerning the FM transition in the q-neighbor Ising model on MNs with partial overlap of nodes and with layers in the form of RRGs are presented and compared with results of MC simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' In this case, noticeable discrepancies occur between theoretical and numerical results, in particular concerning the first-order FM transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' As pointed out in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' III, the more advanced fully and AMEs-based heterogeneous PA yield results practically indistinguishable or even identical to the homogeneous PA, thus their predictions are only briefly mentioned in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' Finally, it is verified in which cases and to what extent theoretical predictions are improved by using the AMEs of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' III B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' In the framework of the homogeneous PA of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' III A stationary values of the magnetization m vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, corresponding to different thermodynamic phases, are given by stable fixed points of the system of equations (14-16) with ˙c(K,0) = ˙c(K,K) = ˙b = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' for certain ranges of parameters r, q, K many stable fixed points can coexist for given T, and their basins of attraction are then separated by stable manifolds of unstable fixed points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' The homogeneous PA predicts various critical behavior of the model under study as the temperature T is varied, depending on r, q, K which are fixed: first- and second-order FM phase transition, the coexistence of the PM and FM phases for T → 0 and absence of the FM transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' At high temperatures, the only stable fixed point is that with m = 0 corresponding to the PM phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' In the case of the second-order FM transition, this fixed point loses stability as the temperature is decreased below the critical value Tc, and simultaneously a pair of symmetric stable fixed points with m > 0 and m < 0 occurs via a supercritical pitchfork bifurcation, corresponding to the two symmetric FM phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' In the case of the first-order transition two symmetric pairs of stable and unstable fixed points with m > 0 and m < 0 occur simultaneously via two saddle-node bifurcations as the temperature is decreased below the upper critical value T (2) c , and the two above-mentioned stable fixed points correspond to the two symmetric FM phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' As the temperature is further decreased both the FM and PM fixed points remain stable (coexist) until the PM point loses stability via a subcritical 9 (c) (d) (b) (a) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' The curves show magnetization m vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' temperature T obtained from the homogeneous PA for different K (green solid lines, both stable and unstable fixed points of the system of equations (14-16) are shown) and from the MFA of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' [49] (black solid lines), for q = 2, K = 200, 100, 50, 20, 10, 4 (from left) and (a) r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='49, (b) r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='5, as well as for q = 4, K = 500, 200, 100, 50, 20, 10 and (c) r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='05, (d) r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' pitchfork bifurcation at the lower critical temperature T (1) c (T (1) c < T (2) c ) by colliding simultaneously with the two above-mentioned unstable fixed points;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' coexistence of the PM and FM phases for T (1) c < T < T (2) c leads to the occurrence of the hysteresis loop in the magnetization curves m(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' Eventually, for T < T (1) c the only stable fixed points remain these corresponding to the two symmetric FM phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' In the case of the coexistence of the FM and PM phases for T → 0 a pair of symmetric stable FM fixed points occurs at T = T (2) c as in the case of the first-order transition, but these FM points, as well as the PM fixed point, remain stable (coexist) as T → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' Finally, it can also happen that fixed points corresponding to the FM phase do not exist for any T > 0, thus the FM transition is absent and the only stable phase for T → 0 is the PM one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' Exemplary curves m(T) predicted by the homogeneous PA for the model under study with different K and selected values of r are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' 1 for the most interesting cases q = 2 and q = 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' in the former case, the MFA (valid for the model on MNs with layers in the form of fully connected graphs with K → ∞) predicts occurrence of all above-mentioned kinds of the critical behavior for different ranges of r, while in the latter one it predicts occurrence of the first-order transition for a particularly wide range of small r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' The curves m(T) for q = 2 are drawn in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' 1(a) for r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='49, and in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' 1(b) for r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='5, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=', for the values of r within or at the border of the interval rc < r < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='5 where the MFA predicts coexistence of the FM and PM phases for T → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' In contrast, for the model on MNs with layers in the form of RRGs the homogeneous PA for r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='49 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' 1(a)) predicts second- or first-order FM transition for small and moderate K, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' the critical temperature(s) decrease and the width of the hysteresis loop increases with K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' Only for large K coexistence of the FM and PM phases for T → 0 is predicted by the PA, and the curves m(T) approach those resulting from the MFA, as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' For r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='5 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' 1(b)) only second- or first-order FM transitions for finite K are predicted by the PA, with the lower critical temperature for the first-order transition 10 (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' Critical behavior predicted by the homogeneous PA for the model with q = 2 (left and middle panels) and q = 4 (right panel) and different r, K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' filled circles — continuous FM transition, open circles — discontinuous FM transition, filled squares — coexistence of the FM and PM phases for T → 0, crosses — absence of the transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' (a) (b) (c) (d) (e) (f) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' Results of MC simulations, predictions of the PA and AMEs for the model with q = 2, K = 20 and (a) r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='45, (b) r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='46, (c) r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='47, (d) r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='50, (e) r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='60, (f) r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='70;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' blue dots — results of MC simulations with FM initial conditions and increasing temperature, red dots — results of MC simulations with PM initial conditions and decreasing temperature, black dots — predictions of the AMEs for both FM (c(0) = 1) and PM (c(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='5) initial conditions and increasing or decreasing temperature, respectively, green solid lines — predictions of the PA as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' 60Q0-0000-011 (a) (b) (c) (d) (e) (f) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' As in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' 3 but for q = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' (a) K = 20, r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='05, (b) K = 20, r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='10, (c) K = 20, r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='15, (d) K = 10, r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='10, (e) K = 50, r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='10, (f) k = 10, r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T (1) c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' The curves m(T) for q = 4 are drawn in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' 1(c) for r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='05, and in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' 1(d) for r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='15, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=', for the values of r where the MFA predicts first-order FM transition with a wide and narrow hysteresis loop, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' For the model on MNs with layers in the form of RRGs the homogeneous PA for small r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='05 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' 1(c)) similarly predicts the first-order FM transition for moderate and large K, while for larger r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='15 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' 1(d)) it predicts the second-order FM transition already for moderate K and the first-order FM transition only for large K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' Again, the critical temperature(s) decrease, and the width of the hysteresis loop increases with K, and the curves m(T) eventually approach these resulting from the MFA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' It may be inferred from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' 1 that the homogeneous PA predicts for the q-neighbor Ising model on MNs with partial overlap of nodes and layers in the form of RRGs with finite K the same critical behavior as the MFA for the model on analogous MNs with layers in the form of complete graphs, only for different ranges of the overlap r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' This conclusion is supported by Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' 2 where the critical behavior predicted by the PA is summarized for the former model with fixed q = 2 and q = 4 and different K, r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' For all K and r = 1 (full overlap of nodes), both PA and MFA predict continuous FM transition with decreasing T, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=', the first-order transition observed in the model on monoplex networks is suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' However, for both q = 2, 4 and finite K the PA predicts the second-order FM transition also for a range of r below r = 1 which is broadened with decreasing K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' As a consequence, for q = 2 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' 2, left and middle panels) the PA predicts that the range of the occurrence of the first-order FM transition is shifted toward smaller values of r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' Similarly, for a narrow range of still smaller values of the overlap the PA predicts the coexistence of the FM and PM phases for T → 0, but for small K this kind of critical behavior is expected at r significantly below the interval rc < r < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='5 obtained from the MFA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' Finally, it is predicted that the range of small r for which the FM transition is absent for decreasing K is narrowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' Eventually, for very small K = 4 comparable with q only continuous FM transition is expected for any r, and all other kinds of critical behavior are suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' For q = 4 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' 2, right panel) the range of small r for which the PA predicts the first-order FM transition is substantially diminished with decreasing K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' In order to verify predictions of the homogeneous PA, MC simulations of the q-neighbor Ising model with q = 2, 4 on large MNs with various parameters r, K were performed and the magnetization curves m(T) were obtained for random PM initial conditions σi = ±1, i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' N and decreasing temperature as well as for FM initial conditions σi = +1, i = 1, 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' N and increasing temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' Comparison with MC simulations shows that the homogeneous PA qualitatively captures modification of the critical behavior of the model under study due to finite values of the 00000000-012 mean degree of the layers K, but its quantitative predictions, though much improved in comparison with those from the MFA valid for large K, are not exact (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' 3, 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' For fixed q, K the PA approximately predicts the ranges of the overlap r where different kinds of critical behavior should occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' However, as a rule, these predictions are overestimated and in MC simulations the particular kinds of critical behavior appear for smaller values of r than estimated from the PA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' For example, for q = 2 the ranges of appearance of the coexistence of the FM and PM phases for T → 0 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' 3(a,b)) and of the second-order FM transition (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' 3(e,f)) in MC simulations are, respectively, shifted and extended toward smaller values of r than expected from the PA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' Consequently, in the case of the first-order FM transition for q = 2 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' 3(c,d)) and q = 4 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' 4(a,b,e,f)) the lower and upper critical temperatures T (1) c , T (2) c are underestimated and the width of the hysteresis loop is overestimated by the PA in comparison with these obtained from MC simulations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' it is interesting to note that discrepancies between the theoretical and numerical values of T (2) c are usually smaller than those for T (1) c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' Similarly, in the case of the second-order FM transition for q = 2 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' 3(f)) and q = 4 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' 4(c,d)) the critical temperature Tc is underestimated by the PA in comparison with that obtained from MC simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' In general, the curves m(T) evaluated from the PA show better agreement with those obtained from MC simulations in the case of the second-order than the first-order FM transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' In order to investigate the critical behavior of the model under study by means of appropriate AMEs as defined in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' III B, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' (19,20) were solved numerically with various initial conditions and the curves m(T) were obtained using long-time asymptotic values of the concentrations of spins directed up c(K,0),(m(A),0), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=', to evaluate stationary values of the magnetization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' As expected, predictions of the AMEs usually show comparable or better agreement with the results of MC simulations than those of the homogeneous PA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' This is particularly visible in the case of the second-order FM transition in the model under study with small K and q = 2 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' 3(f)) and q = 4 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' 4(d)), where the theoretical and numerical curves m(T) coincide very well and the critical temperature Tc is predicted correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' However, the ranges of the overlap predicted by the AMEs for which different kinds of critical behavior occur are still shifted toward slightly higher values of r than obtained from MC simulations (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' 3(a), where the AMEs predict the absence of the FM transition rather than coexistence of the FM and PM phases observed in MC simulations, and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' 3(e), where the AMEs predict the first-order FM transition with a narrow hysteresis loop rather than the second-order transition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' In the case of the coexistence of the FM and PM phases for T → 0 for q = 2 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' 3(b)) and the first-order FM transition for q = 2 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' 3(c,d)) and q = 4 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' 4(a,b,e,f)) predictions of the AMEs concerning the upper critical temperature T (2) c are usually better than those of the homogeneous PA, but the lower critical temperature T (1) c is again usually underestimated and the width of the hysteresis loop is overestimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' In general, some improvement of theoretical predictions by the AMEs in comparison with the homogeneous PA can be seen for small and moderate K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' for large K the curves m(T) obtained from the AMEs and PA coincide (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' 4(e)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' DISCUSSION AND CONCLUSIONS In this paper the q-neighbor Ising model on MNs with partial overlap of nodes and layers in the form of random networks was investigated;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' as an example, the model on MNs with two layers in the form of RRGs was studied in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' Both theoretical considerations based on the homogeneous PA and AMEs as well as MC simulations show that for given q ≥ 1 and finite mean degree of nodes K, and for varying overlap r and temperature T the model exhibits qualitatively similar critical behavior as the q-neighbor Ising model on MNs with partial overlap of nodes and layers in the form of complete graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' In particular, for any q and full overlap of nodes r = 1 the first-order FM transition is suppressed and only the second-order transition appears with decreasing T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' Besides, for decreasing K continuous rather than discontinuous FM transition is observed for an increasing range of large (for q = 2) and large and moderate (for q > 2) values of r below r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' As a consequence, for decreasing K the ranges of r for which the model exhibits the first-order FM transition (for q ≥ 2) and the coexistence of the FM and PM phases for T → 0 (for q = 2) are shifted toward smaller values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' It should be mentioned that in the q-neighbor Ising model on (monoplex) networks the first-order FM transition is also suppressed for small K comparable with q [24];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' in contrast, in the model on MNs this suppression is due to the overlap of nodes and occurs for any K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' The q-neighbor Ising model was used here as an example, and related models for the opinion formation on MNs with partial overlap of nodes can be studied using similar numerical and analytic methods;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' however, the expected qualitative changes of the observed critical behavior with r will be probably less spectacular, since, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=', in the case of the q-voter model even for r = 1 the first-order FM transition is not suppressed [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' For the model under study with large K predictions of the simple homogeneous PA and more advanced system of AMEs converge to these of the MFA and agree quantitatively with the results of MC simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' For finite K the predicted curves m(T) and critical temperature(s) differ quantitatively from the numerically obtained ones;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' usually, the particular kinds of critical behavior are predicted to occur for smaller values of the overlap than observed in MC simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' In general, predictions of both PA and AMEs show better agreement with the results of MC simulations 13 in the case of the continuous than discontinuous FM transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' Predictions based on the AMEs are comparable to or better than those of the PA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' in particular, the critical temperature for the second-order FM transition and the upper critical temperature for the first-order transition are more accurately predicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' Nevertheless, both PA and AMEs qualitatively correctly capture changes of the critical behavior of the model with varying parameters K, r characterizing the underlying MN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' Two versions of the heterogeneous PA were derived for the model under study, the more accurate fully heterogeneous PA and the less accurate AMEs-based heterogeneous PA, which take to a different extent into account heterogeneity of the distribution of the active links due to inhomogeneity of the nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' In both cases, systems of equations for the macroscopic quantities characterizing the model are significantly larger and more complicated than that in the homogeneous PA but do not lead to a noticeable improvement of theoretical predictions concerning the magnetization curves and phase diagrams for the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' This suggests that the simple homogeneous PA is as reliable as more advanced versions of the PA in the study of the critical behavior of systems with multiplexity-induced inhomogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' Only using much larger systems of AMEs in certain cases quantitatively improves agreement between theoretical predictions and results of MC simulations of the above-mentioned models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' APPENDIX A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' Fully heterogeneous pair approximation The following outline of the fully heterogeneous PA for models on MNs is an extension of that for the q-voter models with quenched disorder on networks, with two populations of agents differing by the spin-flip rates [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' The fully heterogeneous PA uses the assumption that the probability that a spin directed up or down in a node with multidegree k has a given number of attached edges leading to spins directed up or down in nodes with multidegree ˜k obeys a binomial distribution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' this assumption is valid for each layer and for any pair k, ˜k separately, and the related binomial distributions are assumed to be independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' The macroscopic quantities characterizing a model with two-state spins on a MN are concentrations ck of spins directed up in nodes with multidegree k and concentrations ek,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='˜k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(L) j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='˜j (= e ˜k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(L) ˜j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='j ) of bonds (ends of edges) attached within the layer G(L) to nodes with multidegree k containing spins with direction j ∈ {↓,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' ↑} such that at the other end of the edge there is a node with multidegree ˜k containing spin with direction ˜j (normalized to the total number of bonds N⟨k(L)⟩ within G(L)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' According to the above-mentioned assumptions the joint probability that a node with multidegree k containing spin with direction j has i = � i(A), i(B), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' i(Lmax)� active bonds (ends of active links) attached within the consecutive layers,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' pointing at nodes with arbitrary multidegree containing spins with opposite direction −j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' has a multiplicative form P(j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' i|k) = Lmax � L=A Bk(L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='i(L) � αk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(L) j � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' (21) where αk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(L) j = � k′ ek,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='k′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(L) j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='−j / � k′ � j′∈{↓,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='↑} ek,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='k′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(L) j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='j′ are conditional probabilities that an active bond is attached to a node with multidegree k and spin with direction j (similar to θ(L) ↑ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' θ(L) ↓ given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' (7, 8)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' In order to evaluate the change in the concentration ek,˜k,(L) j,˜j due to, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=', flipping the spin with direction j in a node with multidegree k, it is necessary to know the numbers of bonds y, z attached to this node within the layer G(L) pointing at nodes with multidegree ˜k given that these bonds are active (˜j = −j) or inactive (˜j = j), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' These numbers obey binomial distributions Bi(L),y � βk,˜k,(L) j,−j � , Bk(L)−i(L),z � γk,˜k,(L) j,j � , respectively, where the conditional probabilities are βk,˜k,(L) j,−j = ek,˜k,(L) j,−j / � k′ ek,k′,(L) j,−j , γk,˜k,(L) j,j = ek,˜k,(L) j,j / � k′ ek,k′,(L) j,j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' Then the rate equations for the macroscopic concentrations of spins directed up are ∂ck ∂t = � j∈{↑,↓} (−1)δj,↑ck,j � i Lmax � L=A Bk(L),i(L) � αk,(L) j � f (i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T |k) , (22) while the rate equations for the concentrations of active and inactive bonds within the layers contain terms such as d dtek,˜k,(L′) j,−j = 1 ⟨k(L′)⟩P(k)ck,j � i Lmax � L=A Bk(L),i(L) � αk,(L) j � i(L′) � y=0 Bi(L′),y � βk,˜k,(L′) j,−j � (−y)f (i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T |k) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' (23) 14 d dtek,˜k,(L′) j,j = 1 ⟨k(L′)⟩P(k)ck,j � i Lmax � L=A Bk(L),i(L) � αk,(L) j � k(L′)−i(L′) � z=0 Bk(L′)−i(L′),z � γk,˜k,(L′) j,j � (−z)f (i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T |k) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' (24) It can be seen that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' (22) resembles Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' (10) in the homogeneous PA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' Concerning the equations for the concentrations of bonds, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=', Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' (23) states that a flip of the spin with direction j in node with multidegree k, which has i(L′) active bonds attached within the layer G(L′), out of which y bonds point at nodes with multidegrees ˜k, decreases the concentration ek,˜k,(L′) j,−j by y/ � N⟨k(L′)⟩ � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' such a flip occurs with probability P(k)ck,jf (i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T |k) within a time interval 1/N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' and the final input to the rate equation (23) is obtained by averaging the above-mentioned change over the probability distributions Bk(L′),i(L′) � αk,(L) j � for i(L′) and Bi(L′),y � βk,˜k,(L′) j,−j � for y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' In the case of the q-neighbor Ising model on MNs with partial overlap of nodes and with two layers in the form of RRGs, with the multidegree distribution P (k) given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' (1), there are three classes of nodes with k = (K, 0), k = (0, K) and k = (K, K), and two layers G(L), L = A, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' Taking into account the symmetry of the model under study and general symmetry conditions for the concentrations ek,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='˜k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(L) j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='˜j the solutions of the system of equations (22 - 24) can be constrained to a 12-dimensional subspace c(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='0) = c(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='K),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' e(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(A) j ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' j′ = e(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(A) j′ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' j = e(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='K),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='K),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(B) j ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' j′ = e(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='K),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='K),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(B) j′ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' j ≡ e(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='0) j ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' j′ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' e(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='K),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(A) j ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' j′ = e(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='K),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(A) j′ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' j = e(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='K),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='K),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(B) j ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' j′ = e(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='K),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='K),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(B) j′ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' j ≡ e(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='K) j ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' j′ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' e(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='K),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='K),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(A) j ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' j′ = e(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='K),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='K),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(A) j′ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' j = e(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='K),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='K),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(B) j ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' j′ = e(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='K),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='K),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(B) j′ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' j ≡ e(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='K),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='K) j ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' j′ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' j′ ∈ {↓,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' ↑}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' Thus, as in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' [15], there are effectively only two classes of agents located in nodes with k = (K, 0) and k = (K, K), differing by the spin-flip rates (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' Besides, the distributions of the number of links pointing at nodes belonging to each class given that these links are active or inactive are fully determined by the conditional probabilities βk,k j,−j, γk,k j,j for the links within each class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' Taking this into account and performing summations in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' (22 - 24) as in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' [24] the following system of equations for the macroscopic quantities is obtained in the fully heterogeneous PA for the model under study, dc(K,0) dt = � 1 − c(K,0) � R � α(K,0) ↓ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, q � − c(K,0)R � α(K,0) ↑ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, q � (25) dc(K,K) dt = � 1 − c(K,K) � � R � α(K,K) ↓ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, q ��2 − c(K,K) � R � α(K,K) ↑ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, q ��2 (26) d dte(K,0),(K,0) ↑ , ↑ = −2(1 − r) K c(K,0)γ(K,0),(K,0) ↑ , ↑ � KR � α(K,0) ↑ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, q � − S � α(K,0) ↑ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, K, q �� + 2(1 − r) K � 1 − c(K,0) � β(K,0),(K,0) ↓ , ↑ S � α(K,0) ↓ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, K, q � (27) d dte(K,0),(K,0) ↓ , ↓ = 2(1 − r) K c(K,0)β(K,0),(K,0) ↑ , ↓ S � α(K,0) ↑ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, K, q � − 2(1 − r) K � 1 − c(K,0) � γ(K,0),(K,0) ↓ , ↓ � KR � α(K,0) ↓ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, q � − S � α(K,0) ↓ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, K, q �� (28) d dte(K,K),(K,K) ↑ , ↑ = −2r K c(K,K)γ(K,K),(K,K) ↑ , ↑ � KR � α(K,K) ↑ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, q � − S � α(K,K) ↑ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, K, q �� R � α(K,K) ↑ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, q � + 2r K � 1 − c(K,K) � β(K,K),(K,K) ↓ , ↑ S � α(K,K) ↓ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, K, q � R � α(K,K) ↓ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, K, q � (29) d dte(K,K),(K,K) ↓ , ↓ = 2r K c(K,K)β(K,K),(K,K) ↑ , ↓ S � α(K,K) ↑ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, K, q � R � α(K,K) ↑ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, K, q � − 2r K � 1 − c(K,K) � γ(K,K),(K,K) ↓ , ↓ � KR � α(K,K) ↓ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, q � − S � α(K,K) ↓ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, K, q �� R � α(K,K) ↓ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, q � (30) d dte(K,0),(K,0) ↑ , ↓ = 1 − r K c(0,0) � −β(K,0),(K,0) ↑ , ↓ S � α(K,0) ↑ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, K, q � + γ(K,0),(K,0) ↑ , ↑ � KR � α(K,0) ↑ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, q � − S � α(K,0) ↑ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, K, q ��� + 1 − r K � 1 − c(K,0) � � γ(K,0),(K,0) ↓ , ↓ � KR � α(K,0) ↓ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, q � − S � α(K,0) ↓ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, K, q �� − β(K,0),(K,0) ↓ , ↑ S � α(K,0) ↓ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, K, q �� (31) d dte(K,K),(K,K) ↑ , ↓ = r K c(K,K) � −β(K,K),(K,K) ↑ , ↓ S � α(K,K) ↑ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, K, q � 15 + γ(K,K),(K,K) ↑ , ↑ � KR � α(K,K) ↑ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, q � − S � α(K,K) ↑ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, K, q ��� R � α(K,K) ↑ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, q � + r K � 1 − c(K,K) � � γ(K,K),(K,K) ↓ , ↓ � KR � α(K,K) ↓ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, q � − S � α(K,K) ↓ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, K, q �� − β(K,K),(K,K) ↓ , ↑ S � α(K,K) ↓ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, K, q �� R � α(K,K) ↓ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, q � (32) d dte(K,0),(K,K) ↑ , ↑ = −1 − r K c(K,0) � 1 − γ(K,0),(K,0) ↑ , ↑ � � KR � α(K,0) ↑ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, q � − S � α(K,0) ↑ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, K, q �� + 1 − r K � 1 − c(K,0) � � 1 − β(K,0),(K,0) ↓ , ↑ � S � α(K,0) ↓ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, K, q � − r K c(K,K) � 1 − γ(K,K),(K,K) ↑ , ↑ � � KR � α(K,K) ↑ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, q � − S � α(K,K) ↑ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, K, q �� R � α(K,K) ↑ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, q � + r K � 1 − c(K,K) � � 1 − β(K,K),(K,K) ↓ , ↑ � S � α(K,K) ↓ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, K, q � R � α(K,K) ↓ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, q � (33) d dte(K,0),(K,K) ↓ , ↓ = 1 − r K c(K,0) � 1 − β(K,0),(K,0) ↑ , ↓ � S � α(K,0) ↑ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, K, q � − 1 − r K � 1 − c(K,0) � � 1 − γ(K,0),(K,0) ↓ , ↓ � � KR � α(K,0) ↓ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, q � − S � α(K,0) ↓ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, K, q �� + r K c(K,K) � 1 − β(K,K),(K,K) ↑ , ↓ � S � α(K,K) ↑ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, K, q � R � α(K,K) ↑ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, q � − r K � 1 − c(K,K) � � 1 − γ(K,K),(K,K) ↓ , ↓ � � KR � α(K,K) ↓ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, q � − S � α(K,K) ↓ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, K, q �� R � α(K,K) ↓ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, q � (34) d dte(K,0),(K,K) ↑ , ↓ = −1 − r K c(K,0) � 1 − β(K,0),(K,0) ↑ , ↓ � S � α(K,0) ↑ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, K, q � + 1 − r K � 1 − c(K,0) � � 1 − γ(K,0),(K,0) ↓ , ↓ � � KR � α(K,0) ↓ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, q � − S � α(K,0) ↓ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, K, q �� + r K c(K,K) � 1 − γ(K,K),(K,K) ↑ , ↑ � � KR � α(K,K) ↑ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, q � − S � α(K,K) ↑ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, K, q �� R � α(K,K) ↑ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, q � − r K � 1 − c(K,K) � � 1 − β(K,K),(K,K) ↓ , ↑ � S � α(K,K) ↓ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, K, q � R � α(K,K) ↓ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, q � (35) d dte(K,0),(K,K) ↓ , ↑ = 1 − r K c(K,0) � 1 − γ(K,0),(K,0) ↑ , ↑ � � KR � α(K,0) ↑ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, q � − S � α(K,0) ↑ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, K, q �� − 1 − r K � 1 − c(K,0) � � 1 − β(K,0),(K,0) ↓ , ↑ � S � α(K,0) ↓ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, K, q � − r K c(K,K) � 1 − β(K,K),(K,K) ↑ , ↓ � S � α(K,K) ↑ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, K, q � R � α(K,K) ↑ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, q � + r K � 1 − c(K,K) � � 1 − γ(K,K),(K,K) ↓ , ↓ � � KR � α(K,K) ↓ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, q � − S � α(K,K) ↓ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, K, q �� R � α(K,K) ↓ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' q � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' (36) where the significant conditional probabilities are α(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='0) ↓ = e(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='0) ↓ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' ↑ + e(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='K) ↓ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' ↑ e(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='0) ↓ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' ↑ + e(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='K) ↓ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' ↑ + e(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='0) ↓ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' ↓ + e(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='K) ↓ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' ↓ α(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='0) ↑ = e(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='0) ↑ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' ↓ + e(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='K) ↑ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' ↓ e(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='0) ↑ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' ↓ + e(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='K) ↑ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' ↓ + e(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='0) ↑ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' ↑ + e(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='K) ↑ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' ↑ α(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='K) ↓ = e(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='K),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='K) ↓ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' ↑ + e(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='K),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='0) ↓ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' ↑ e(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='K),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='K) ↓ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' ↑ + e(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='K),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='0) ↓ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' ↑ + e(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='K),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='K) ↓ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' ↓ + e(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='K),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='0) ↓ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' ↓ α(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='K) ↑ = e(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='K),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='K) ↑ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' ↓ + e(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='K),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='0) ↑ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' ↓ e(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='K),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='K) ↑ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' ↓ + e(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='K),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='0) ↑ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' ↓ + e(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='K),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='K) ↑ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' ↑ + e(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='K),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='0) ↑ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' ↑ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' (37) β(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='0) ↓ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' ↑ = e(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='0) ↓ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' ↑ e(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='0) ↓ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' ↑ + e(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='K) ↓ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' ↑ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' β(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='0) ↑ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' ↓ = e(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='0) ↑ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' ↓ e(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='0) ↑ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' ↓ + e(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='K) ↑ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' ↓ 16 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='7 T −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='0 m FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' The curves show magnetization m vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' temperature T obtained from the homogeneous PA (solid lines) and from the heterogeneous PA (symbols) for q = 4, K = 20 and r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='15, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='2 (from left to right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' β(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='K),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='K) ↓ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' ↑ = e(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='K),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='K) ↓ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' ↑ e(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='K),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='K) ↓ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' ↑ + e(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='K),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='0) ↓ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' ↑ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' β(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='K),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='K) ↑ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' ↓ = e(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='K),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='K) ↑ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' ↓ e(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='K),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='K) ↑ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' ↓ + e(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='K),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='0) ↑ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' ↓ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' (38) γ(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='0) ↓ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' ↓ = e(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='0) ↓ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' ↓ e(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='0) ↓ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' ↓ + e(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='K) ↓ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' ↓ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' γ(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='0) ↑ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' ↑ = e(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='0) ↑ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' ↑ e(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='0) ↑ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' ↑ + e(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='K) ↑ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' ↑ γ(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='K),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='K) ↓ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' ↓ = e(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='K),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='K) ↓ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' ↓ e(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='K),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='K) ↓ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' ↓ + e(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='K),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='0) ↓ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' ↓ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' γ(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='K),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='K) ↑ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' ↑ = e(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='K),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='K) ↑ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' ↑ e(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='K),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='K) ↑ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' ↑ + e(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='K),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='0) ↑ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' ↑ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' (39) Concentration ˜c of spins directed up within each layer and concentration c of spins directed up in the MN are defined in the same way as in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' III A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' Natural initial conditions for the system of equations (25 - 36) are c(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='0)(0) = c(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='K)(0) = ρ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' e(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='0) ↑ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' ↑ (0) = (1 − r)2ρ2 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' e(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='0) ↑ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' ↓ (0) = (1 − r)2ρ0(1 − ρ0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' e(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='0) ↓ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' ↓ (0) = (1 − r)2(1 − ρ0)2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' e(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='K),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='K) ↑ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' ↑ (0) = r2ρ2 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' e(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='K),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='K) ↑ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' ↓ (0) = r2ρ0(1 − ρ0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' e(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='K),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='K) ↓ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' ↓ (0) = r2(1 − ρ0)2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' e(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='K) ↑ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' ↑ (0) = (1 − r)rρ2 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' e(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='K) ↓ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' ↓ (0) = (1 − r)r(1 − ρ0)2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' e(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='K) ↑ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' ↓ (0) = (1 − r)2ρ0(1 − ρ0) = e(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='0) ↓ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' ↑ (0) = (1 − r)rρ0(1 − ρ0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' where ρ0 can be chosen arbitrarily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' As mentioned in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' III A the magnetization curves obtained from the fully heterogeneous PA are practically indistinguishable from those obtained from the homogeneous PA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' This is illustrated by examples in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' AMEs-based heterogeneous pair approximation The AMEs-based heterogeneous PA again uses the assumption that the probability that a spin directed up or down in a node with multidegree k has a given number of neighboring spins directed up obeys a binomial distribution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' for models on MNs this assumption is made for each layer separately, and the related binomial distributions are assumed to be independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' Hence, in contrast with the homogeneous PA, in the AMEs-based heterogeneous PA it is taken into account that for a node with multidegree k occupied by a spin with downward or upward direction the respective probabilities ϑ(L) k , η(L) k that a randomly chosen neighboring node within the layer G(L) is occupied by a spin directed upward can depend on k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' However, in contrast with the fully heterogeneous PA developed in Appendix A, all active or inactive edges attached to a given node within a given layer are treated in the same way and obey common binomial distributions [28, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' As mentioned in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' III B, these two assumptions should make the AMEs-based heterogeneous PA more accurate than the homogeneous PA and less accurate than the fully heterogeneous PA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' Eventually, in the AMEs-based heterogeneous PA the time-dependent macroscopic quantities are the density ck of spins directed up in nodes with multidegree k as well as the above-mentioned probabilities ϑ(L) k , η(L) k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' In terms of the densities ck,m and sk,m used in the AMEs, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' (19), (20) the above-mentioned macroscopic quantities can be expressed as ck = � m ck,m = 1 − � m sk,m, ϑ(L) k = � m m(L)sk,m/ � k(L) (1 − ck) � , η(L) k = 17 � m m(L)ck,m/ � k(L)ck � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' Then, the core approximation for the AMEs-based heterogeneous PA can be made, ac- cording to which sk,m ≈ (1 − ck) �Lmax L=A Bk(L),m(L) � ϑ(L) k � , ck,m ≈ ck �Lmax L=A Bk(L),m(L) � η(L) k � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' The latter approxi- mation should be made in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' (19), (20) as well as in the definitions of the average rates β(L) s , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' , γ(L) i , so that, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=', β(L′) s ≈ ¯β(L′) s = � (1 − ck) � m � k(L′) − m(L′)� Fk,m �Lmax L=A Bk(L),m(L) � ϑ(L) k � � / � (1 − ck) k(L′) � 1 − ϑ(L′) k � � , etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' Differentiating the definitions of ck, ϑ(L) k , η(L) k with respect to time and using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' (19),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' (20) with the above-mentioned approximations yields the following system of equations for the time dependence of the macroscopic quantities in the heterogeneous PA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' dck dt = −ck � m Rk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='m Lmax � L=A Bk(L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='m(L) � η(L) k � + (1 − ck) � m Fk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='m Lmax � L=A Bk(L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='m(L) � ϑ(L) k � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' (40) dϑ(L′) k dt = � m � ϑ(L′) k − m(L′) k(L′) � � Fk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='m Lmax � L=A Bk(L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='m(L) � ϑ(L) k � − ck 1 − ck Rk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='m Lmax � L=A Bk(L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='m(L) � η(L) k �� +¯β(L′) s � 1 − ϑ(L′) k � − ¯γ(L′) s ϑ(L′) k ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' (41) dη(L′) k dt = � m � η(L′) k − m(L′) k(L′) � � Rk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='m Lmax � L=A Bk(L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='m(L) � η(L) k � − 1 − ck ck Fk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='m Lmax � L=A Bk(L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content='m(L) � ϑ(L) k �� +¯β(L′) i � 1 − η(L′) k � − ¯γ(L′) i η(L′) k ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' (42) where L′ = A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' Lmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' The above equations are very similar to those obtained in the AMEs-based heterogeneous PA for the spin models on (monoplex) networks [28, 29];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' in particular, terms containing β(L) s , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' , γ(L) i with L ̸= L′ do not occur in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' (41), (42) for ϑ(L′) k , η(L′) k since the respective terms from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' (19), (20) sum up to zero in the derivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' It should be mentioned that the AMEs can also be a starting point to obtain the homogeneous PA from Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' III A by assuming that the probability that a spin directed down has within the layer G(L) a neighboring spin directed up does not depend on k and can be expressed as the average θ(L) ↓ = ⟨� m m(L)sk,m⟩/⟨k(L) (1 − ck)⟩ [28, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' In the case of the q-neighbor Ising model on MNs with partial overlap of nodes and with layers in the form of RRGs, with the multidegree distribution P (k) given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' (1), there are three classes of nodes with k = (K, 0), k = (0, K) and k = (K, K), and two layers G(L), L = A, B, thus the system of equations (40-42) is 11-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' Due to the symmetry of the model solutions of these equations should be constrained to a subspace c(0,K) = c(K,0), ϑ(A) (K,0) = ϑ(B) (0,K) ≡ ϑ(0,K), η(A) (K,0) = η(B) (0,K) ≡ η(0,K), ϑ(A) (K,K) = ϑ(B) (K,K) ≡ ϑ(K,K), η(A) (K,K) = η(B) (K,K) ≡ η(K,K) which reduces the number of equations to six.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' Performing summations in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' (40-42) as in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' [24] the following system of equations for the macroscopic quantities is obtained in the AMEs-based heterogeneous PA for the model under study, dc(K,0) dt = −c(K,0)R � 1 − η(K,0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, q � + � 1 − c(K,0) � R � ϑ(K,0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, q � , (43) dϑ(K,0) dt = ϑ(K,0) � R � ϑ(K,0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, q � − c(K,0) 1 − c(K,0) R � 1 − η(K,0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, q �� − 1 K � S � ϑ(K,0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, K, q � − c(K,0) 1 − c(K,0) � KR � 1 − η(K,0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, q � − S � 1 − η(K,0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, K, q ��� + ¯βs � 1 − ϑ(K,0) � − ¯γsϑ(K,0), (44) dη(K,0) dt = η(K,0) � R � 1 − η(K,0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, q � − 1 − c(K,0) c(K,0) R � ϑ(K,0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, q �� − 1 K �� KR � 1 − η(K,0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, q � − S � 1 − η(K,0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, K, q �� − 1 − c(K,0) c(K,0) S � ϑ(K,0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, K, q �� + ¯βi � 1 − η(K,0) � − ¯γiη(K,0), (45) dc(K,K) dt = −c(K,K) � R � 1 − η(K,K);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, q ��2 + � 1 − c(K,K) � � R � ϑ(K,K);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, q ��2 , (46) dϑ(K,K) dt = ϑ(K,K) �� R � ϑ(K,K);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, q ��2 − c(K,K) 1 − c(K,K) � R � 1 − η(K,K);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, q ��2 � − 1 K � S � ϑ(K,K);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, K, q � R � ϑ(K,K);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, q � 18 − c(K,K) 1 − c(K,K) � KR � 1 − η(K,0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, q � − S � 1 − η(K,0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, K, q �� R � 1 − η(K,K);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, q �� + ¯βs � 1 − ϑ(K,K) � − ¯γsϑ(K,K), (47) dη(K,K) dt = η(K,K) �� R � 1 − η(K,K);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, q ��2 − 1 − c(K,K) c(K,K) � R � ϑ(K,K);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, q ��2 � − 1 K �� KR � 1 − η(K,0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, q � − S � 1 − η(K,0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, K, q �� R � 1 − η(K,K);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, q � −1 − c(K,K) c(K,K) S � ϑ(K,K);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, K, q � R � ϑ(K,K);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, q �� + ¯βi � 1 − η(K,K) � − ¯γiη(K,K), (48) where the average rates are ¯βs = �1 − r 2 − r � 1 − c(K,0) � K � 1 − ϑ(K,0) � + r 2 − r � 1 − c(K,K) � K � 1 − ϑ(K,K) ��−1 × �1 − r 2 − r � 1 − c(K,0) � � KR � ϑ(K,0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, q � − S � ϑ(K,0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, K, q �� + r 2 − r � 1 − c(K,K) � � KR � ϑ(K,K);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, q � − S � ϑ(K,K);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, K, , q �� R � ϑ(K,K);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, q �� , (49) ¯γs = �1 − r 2 − rc(K,0)K � 1 − η(K,0) � + r 2 − rc(K,K)K � 1 − η(K,K) ��−1 × �1 − r 2 − rc(K,0)S � 1 − η(K,0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, K, q � + r 2 − rc(K,K)S � 1 − η(K,K);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, K, q � R � 1 − η(K,K);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, q �� , (50) ¯βi = �1 − r 2 − r � 1 − c(K,0) � Kϑ(K,0) + r 2 − r � 1 − c(K,K) � Kϑ(K,K) �−1 × �1 − r 2 − r � 1 − c(K,0) � S � ϑ(K,0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, K, q � + r 2 − r � 1 − c(K,K) � S � ϑ(K,K);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, K, q � R � ϑ(K,K);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, q �� , (51) ¯γi = �1 − r 2 − rc(K,0)Kη(K,0) + r 2 − rc(K,K)Kη(K,K) �−1 × �1 − r 2 − rc(K,0) � KR � 1 − η(K,0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, q � − S � 1 − η(K,0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, K, q �� + r 2 − rc(K,K) � KR � 1 − η(K,K);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, q � − S � 1 − η(K,K);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, K, q �� R � 1 − η(K,K);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' T, q �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' (52) Concentration ˜c of spins directed up within each layer and concentration c of spins directed up in the MN are defined in the same way as in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' III A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE1T4oBgHgl3EQfVwTa/content/2301.03107v1.pdf'} +page_content=' Natural initial conditions for the system of equations (43-48) are ϑ(K,0)(0) = η(K,0)(0) = ϑ(K,K)(0) = η(K,K)(0) = ˜c(0), while c(K,0)(0), c(K,K)(0) can be 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G. T. Taylor2, Johan De Keyser3, Yoshifumi Futaana4, Ruth A. +Bamford5, Graziella Branduardi-Raymont6, Jean-Yves Chaufray7, Dragos Constantinescu8, +Elisabetta De Angelis9, Pierre Devoto1, Jonathan Eastwood10, Marius Echim3, Philippe +Garnier1, Benjamin Grison11, David Hercik12, Helmut Lammer13, André Laurens14, François +Leblanc7, Anna Milillo9, Rumi Nakamura13, Lubomír Přech15, Elias Roussos16, Štěpán +Štverák17, 11, Julien Forest18, Arnaud Trouche18, Sébastien L. G. Hess19, Jean-Charles Mateo- +Vélez19, James Carpenter2, Josef Winter2 +1Institut de Recherche en Astrophysique et Planétologie, Université de Toulouse / CNRS / UPS / +CNES, Toulouse, France +2ESTEC / ESA, Noordwijk, The Netherlands +3Royal Belgian Institute for Space Aeronomy, Brussels, Belgium +4Swedish Institute of Space Physics, Kiruna, Sweden +5RAL Space, STFC, Rutherford Appleton Laboratory, Chilton, Didcot, Oxfordshire, UK +6Mullard Space Science Laboratory / UCL, Holmbury St Mary, UK +7LATMOS (Laboratoire Atmosphères, Milieux, Observations Spatiales) / IPSL, Paris, France +8TU-Braunschweig, Braunschweig, Germany +9Institute for Space Astrophysics and Planetology / INAF, Rome, Italy +10Imperial College, London, UK +11Institute of Atmospheric Physics / CAS, Prague, Czechia +12scibit s.r.o., Liberec, Czechia +13Space Research Institute / Austrian Academy of Sciences, Graz, Austria +14CNES, Toulouse, France +15Charles University, Prague, Czechia +16Max Planck Institute for Solar System Research, Göttingen, Germany +17Astronomical Institute / CAS, Prague, Czechia +18Artenum, Ramonville Saint-Agne, France +19ONERA, Toulouse, France +* Correspondence: +Corresponding Author +Iannis.Dandouras@irap.omp.eu +Keywords: Moon, Gateway, deep space, space plasmas, heliophysics, space weather +Submitted to "Frontiers in Astronomy and Space Sciences" +09 Dec. 2022 + +frontiers + +Lunar Gateway Space Plasma Physics + +2 +This is a provisional file, not the final typeset article +Abstract +The Lunar Orbital Platform - Gateway (LOP - Gateway, or simply Gateway) is a crewed platform +that will be assembled and operated in the vicinity of the Moon by NASA and international partner +organizations, including ESA, starting from the mid-2020s. It will offer new opportunities for +fundamental and applied scientific research. The Moon is a unique location to study the deep space +plasma environment. Moreover, the lunar surface and the surface-bounded exosphere are interacting +with this environment, constituting a complex multi-scale interacting system. This paper examines +the opportunities provided by externally mounted payloads on the Gateway in the field of space +plasma physics, heliophysics and space weather, but also examines the impact of the space +environment on an inhabited platform in the vicinity of the Moon. It then presents the conceptual +design of a model payload, required to perform these space plasma measurements and observations. +It results that the Gateway is very well-suited for space plasma physics research. It allows a series of +scientific objectives with a multi-disciplinary dimension to be addressed. +1 +Introduction +The Moon is a unique location to study the deep space plasma environment. During most part of its +orbit around the Earth the Moon is directly exposed to the solar wind. Due to the absence of a +substantial intrinsic magnetic field and of a collisional atmosphere, solar wind and solar energetic +particles (SEPs) arrive almost without any deviation or absorption and impact directly on its surface, +interacting with the lunar regolith and the tenuous lunar exosphere (e.g. Geiss et al., 2004; Futaana et +al., 2018). The same phenomenon occurs also with the galactic cosmic rays (GCRs), which present +fluxes and energy spectra typical of interplanetary space (e.g. Sohn et al., 2014). Downstream from +the Moon, a structured plasma umbra and penumbra region is formed, characterized by the gradual +decrease of the ion and electron densities (Bosqued et al., 1996; Nishino et al., 2010). The Moon’s +vicinity is an ideal environment to study galactic cosmic rays, solar wind and solar energetic +particles. This environment is typical of deep space (Plainaki et al., 2016), apart from the fact that the +Moon itself forms an obstruction to the GCRs. +During 5 – 6 days every orbit, however, the Moon crosses the tail of the terrestrial magnetosphere +(Figure 1). It is then exposed not to the solar wind but to the terrestrial magnetotail plasma +environment, offering the possibility to study in-situ magnetotail dynamics and its dependence on +solar and geomagnetic activity (e.g. Kallio and Facskó, 2015; Kallio et al., 2019). Phenomena such as +plasmoids released from the near-Earth magnetotail and propagating anti-Sunward, bursty bulk flows +(BBFs), energetic particle bursts, plasma waves, magnetic reconnection and plasma sheet dynamics +can thus be studied in-situ (e.g. Parks et al., 2001; Nakamura, 2006; Taylor et al., 2006; Nagai et al., +2009; Du et al., 2011; Cao et al., 2013; Grigorenko et al., 2019; Sitnov et al., 2019; Kronberg et al., +2021). +The Moon is then also very well situated to study atmospheric escape from the Earth into space +(Lammer et al., 2008; Harnett et al., 2013; Wei et al., 2020; Dandouras, 2021; André et al., 2021; +Wang et al., 2021), in the form of heavy ions upwelling from the terrestrial ionosphere and +transported and lost into the deep magnetotail. The wealth of data supplied from the THEMIS- +ARTEMIS and from the Kaguya (SELENE) spacecraft confirmed the observation of such ions, of +terrestrial origin, in the lunar environment (Poppe et al., 2016; Terada et al., 2017). The THEMIS- +ARTEMIS data, however, did not include the crucial information on the plasma composition +(Angelopoulos, 2011), and the Kaguya plasma measurements where limited to a less than two-year +mission and to low-energy plasma (Saito et al., 2010). Far magnetotail studies performed by the + + + + Lunar Gateway Space Plasma Physics + +3 +Geotail spacecraft supplied key information on the dynamics of ion beams streaming downtail +(Christon et al., 1994, 2020; Seki et al., 1998), but lacked the ion composition measurements at low +energies (below ~10 keV). +When the Moon gets again outside of the magnetotail, terrestrial magnetosphere dynamics can be +monitored through remote sensing, using a variety of magnetospheric imaging techniques. These +include Energetic Neutral Atom (ENA) imaging, which conveys information on the interaction +between energetic ions and the terrestrial exosphere (e.g. C:son Brandt et al., 2002; Vallat et al., +2004), solar wind charge exchange X-rays imaging of the interaction between the solar wind / +magnetosheath plasma and the terrestrial exosphere (Branduardi-Raymont et al., 2012; Sibeck et al., +2018), plasmasphere EUV imaging (Sandel et al., 2003), or exosphere Lyman-α imaging (e.g. +Zoennchen et al., 2017). +But, most important, the lunar environment offers a unique opportunity to study the Moon surface- +bounded exosphere (Figure 2), its production mechanisms, its dynamics, its interaction with the solar +wind and with the terrestrial magnetotail plasma, and its escape into space (Potter et al., 2000; Wurz +et al., 2007, 2022; Futaana et al., 2008; Leblanc and Chaufray, 2011, Lammer et al., 2022). The +LADEE (Lunar Atmosphere and Dust Environment Explorer) and LRO (Lunar Reconnaissance +Orbiter) observations have provided a glimpse of the complexity of the lunar exosphere and of the +associated physical mechanisms (Stern et al., 2013; Elphic et al., 2014; Benna et al., 2015; Hodges, +2016; Hurley et al., 2016). +The lunar surface offers also exciting possibilities for studying energetic ion implantation in the lunar +regolith (Ozima et al., 2005; Ireland et al., 2006), albedo energetic particles produced through the +interaction of SEPs and GCRs with the regolith (Schwadron et al., 2016, 2018; Zhang et al., 2020), +solar wind ion implantation or neutralization and reflection from the lunar regolith (Futaana et al., +2006, 2012; Vorburger et al., 2015; Tucker et al., 2019), formation of hydrogen bearing molecules +(McCord et al., 2011; Stern et al., 2013; McLain et al. 2021) possibly including water (Schörghofer +et al., 2021), solar wind interaction with crustal magnetic anomalies (Poppe et al., 2015; Bamford et +al., 2016), lunar pickup ion generation (Poppe et al., 2012a; Wang et al., 2011), or lunar surface +electrostatic charging and dust levitation (Stubbs et al., 2007; Hess et al., 2015; Popel et al., 2018), +just to mention few examples. +The analysis of implanted particles on the lunar surface, that originated from the Earth’s atmosphere, +will also reveal some knowledge of Earth’s early atmosphere (Marty et al., 2003; Ozima et al., 2005; +Lammer et al., 2018, 2022). It is expected that early Earth’s atmosphere experienced strong escape of +hydrogen, oxygen and carbon, that originated from the dissociation of water and methane molecules, +and of nitrogen due to the increased EUV flux from the young Sun (Lammer et al., 2018; Zahnle et +al., 2019; Gebauer et al., 2020; Kislyakova et al., 2020; Johnstone et al., 2021). As suggested by +Marty et al. (2003), nitrogen originating from the early Earth was implanted on the lunar surface. +This is based on the strong variations of N, He, Ne and Ar noble gas isotope implantations into the +regolith, of up to 30 % (Ozima et al., 2005). According to Marty et al. (2003) and Ozima et al. (2005) +these enhancements cannot be explained as due to solar wind implantation alone. +The Moon is also an ideal test case for studying planetary surface weathering resulting from the +exposure to energetic particles, i.e. surface - energetic particle interactions (e.g. Hapke, 2001; Pieters +and Noble, 2016; Nénon and Poppe, 2020). Given that the Moon is irradiated by GCRs quasi- +uniformly, any differences in the resulting interaction, including the emitted albedo particles, point to +the variable properties, physical or chemical, of the surface (Schwadron et al., 2016, 2018). From the + + + +Lunar Gateway Space Plasma Physics + +4 +This is a provisional file, not the final typeset article +perspective of the Gateway, surface - GCR interactions can be mainly probed through the albedo +particles. +Following the legacy of the Apollo missions and of the more recent missions to the Moon (THEMIS- +ARTEMIS, Kaguya, LADEE, LRO, Chandrayaan, Chang’e, etc.), a series of lunar missions is in +preparation, or already operating, building on their outstanding heritage (Dandouras et al., 2020a). +The Lunar Orbital Platform - Gateway (LOP - Gateway, or simply Gateway) is a crewed platform +that will be assembled and operated in the vicinity of the Moon by NASA and international partner +organizations, including ESA. Launch of the first modules will start in the mid-2020s (Phase 1), and +it will continue with the launch and assembly of additional modules during the late 2020s (Phase 2). +The Gateway will provide support for all lunar activities, including the Artemis program to return +humans to the Moon (Artemis III Science Definition Team Report, 2020). It will also offer new +opportunities for fundamental and applied scientific research (Carpenter et al., 2018; Dandouras et +al., 2020a). +In preparation of its scientific payload, ESA set up international science teams to prepare and to +support the definition of payload studies, including a topical team in the field of space plasma +physics. In the first part of this article (sections 2 and 3) we report on the outcome of this topical +team, which was entitled “Space Plasma Physics Science Opportunities for the Lunar Orbital +Platform - Gateway”. This part focuses on the science objectives identified by the topical team +(section 2), and on the corresponding instrumentation required to address them (section 3). In the +second part (section 4) we present a conceptual design study for a “Space Plasma Physics Payload +Package onboard the Gateway” (SP4GATEWAY) we undertook for ESA, addressing these +objectives and compatible with the technical requirements. +2 +Specific Objectives and Goals +The “Space Plasma Physics Science Opportunities for the Lunar Orbital Platform-Gateway” topical +team, set up by ESA in 2019, brought together the key expertise required for defining the space +plasma parameters to measure from lunar orbit, and the appropriate instrumentation required to +perform these observations. The science objectives that were identified include: +2.1 +Monitoring the solar wind and the lunar energetic particle environment +Due to the absence of a substantial intrinsic magnetic field and of a collisional atmosphere, the Moon +is directly exposed to: +− Solar Wind: ~keV particles +− Solar Energetic Particles (SEPs): ~MeV particles +− Galactic Cosmic Rays (GCRs): ~GeV particles +The monitoring of the solar wind (e.g. von Steiger, 2008), at the lunar environment, aims to evaluate +its role as a driver for the dynamics of the terrestrial and the lunar exospheres, of the dynamics of the +terrestrial magnetosphere, and of the lunar surface sputtering and charging. +The monitoring and characterization of the SEPs and GCRs, at lunar orbit, aims to evaluate the +radiation environment of the Moon and also the role of SEPs and GCRs as lunar surface sputtering +sources. Since the Moon does not have a substantial magnetic field it is possible, with an appropriate +particle detector, to measure the low energy part of the GCR spectrum (< 1 GeV) with high precision. + + + + Lunar Gateway Space Plasma Physics + +5 +This offers an advantage with respect to low-Earth orbits, where most of the advanced GCR +observatories like PAMELA and AMS-02 are located, where this low energy part is filtered out by +the Earth's magnetosphere. +Typical SEP proton intensities, measured during a solar event, are shown in Figure 3 (adapted from +Quinn et al., 2017). Some of the SEP protons (~MeV energy range) can interact, in the high solar +corona, with partially stripped coronal ions, charge exchange with them and produce ~MeV ENAs +(Energetic Neutral Atoms) (Mewaldt et al., 2009). +GCR Hydrogen and Oxygen nuclei fluxes are shown in Figure 4, presenting a clear solar cycle +modulation (adapted from Mrigakshi et al., 2012). The interaction of these SEPs and GCRs with the +lunar regolith produces albedo energetic particles, resolvable with current instruments up to a few +~100 MeV, and with fluxes that are sensitive to the regolith hydration (Looper et al., 2013; +Schwadron et al., 2016; Zaman et al., 2022), cf. Figure 5. The separation of the pristine energetic +particle fluxes from the albedo energetic particles (e.g. by zenith centered / nadir centered looking +directions respectively) appears thus as a requirement, in order to provide information on the deep +space SEP and GCR environment and on the interaction of the lunar regolith with this environment. +2.2 +Monitoring the terrestrial magnetosphere and exosphere +When the Moon is within the terrestrial magnetotail, in-situ measurements of the plasma sheet and +plasma sheet boundary layer dynamics are enabled. These consist of magnetic field and energetic ion +and electron monitoring, including the measurement of energetic ions of terrestrial origin streaming +downtail. +The evolution of the flux of O+ downtail streaming beams, as a function of the tailward distance from +the Earth, is shown in Figure 6 (from Seki et al., 1998). During high geomagnetic activity conditions +these beams include heavy atomic and molecular ions (Christon et al., 1994, 2020). Closer to the +Moon, O+ downtail streaming beams have been observed by the Kaguya Lunar Orbiter (Terada et al., +2017). The spectral characteristics of these streaming O+ ions show a clear distinction between the O+ +ions of lunar origin (few 10 eV to ~100 eV) and the terrestrial magnetospheric O+ ions (few keV), cf. +Figure 7 (adapted from Terada et al., 2017). Particle tracing simulations performed by Harnett et al. +(2013) using a 3D multi-fluid model, and by Poppe et al. (2016) using the MHD Open Global +Geospace Circulation Model, show how heavy ions, originating from the Earth’s inner +magnetosphere, can be ejected downtail during high geomagnetic activity events, reaching energies +of several keV to several 10 keV at lunar distances, cf. Figure 8. +When the Moon is outside of the magnetotail, terrestrial magnetosphere dynamics and response to +solar wind conditions can be monitored through remote sensing. This includes: +− Ring current and near-Earth plasma sheet monitoring, by imaging of the ENAs produced by +charge exchange between the plasma sheet or ring current energetic ions (few ~keV to few ~10 +keV) with the geocorona neutral hydrogen atoms, e.g. Brandt et al. (2004), Vallat et al. (2004), +Goldstein et al. (2022). +− Magnetopause and cusps monitoring by detecting and imaging the SWCX (solar wind charge +exchange) soft X-rays produced by charge exchange between highly-charged heavy ions, +originating from the solar wind, and the exospheric neutral atoms, e.g. Branduardi-Raymont et al. +(2012, 2021), Sibeck et al. (2018). +− Plasmasphere imaging, by resonant scattering of the solar EUV (30.4 nm) by the plasmaspheric +He+ ions, e.g. Sandel et al. (2003), Darrouzet et al. (2008), He et al. (2016). + + + +Lunar Gateway Space Plasma Physics + +6 +This is a provisional file, not the final typeset article +− Geocorona imaging at Lyman-α (121.6 nm), e.g. Rairden et al. (1986), Zoennchen et al. (2017, +2022). +2.3 +Monitoring the Moon’s surface-bounded exosphere +The Moon surface-bounded exosphere constitutes a complex multi-scale system (Figure 2), +characterised by its interactions with the solar radiation, the solar wind and terrestrial magnetotail +plasma, the meteoritic flux, dust, and the regolith (Futaana et al., 2018). The low number densities of +this very tenuous atmosphere, particularly of the minority species, and the complexity and +multiplicity of the source and loss mechanisms have resulted in a poor understanding of it (Wurz et +al., 2007, 2022; Poppe et al., 2022). Figure 9 provides the altitude density profiles of the major +species, separately for the atoms and molecules released thermally from the regolith and for the +atoms released through sputtering. In addition to atomic and molecular hydrogen, O, OH, CH4, noble +gases (He, Ne, Ar, Kr, Xe), metallic atoms (Na, K, Mg, Al) and other elements populate the lunar +exosphere (Leblanc and Chaufray, 2011; Benna et al., 2015; Grava et al., 2015, 2016, 2021; Halekas +et al., 2015; Hodges, 2016; Hurley and Benna, 2018; Leblanc et al., 2022; Wurz et al., 2022). +These neutral exospheric atoms and molecules can be subsequently ionized by the solar UV radiation +and generate pickup ions. These ions are promptly accelerated from their birthplace by the ambient +electric field E and drift across the magnetic field B. The unique orbital characteristics of the pickup +ions (cycloidal motion consisting of a combination of E × B drift and a gyration around B) make it +possible to infer important details about their sources (Hartle and Killen, 2006). Such lunar pickup +ions have been detected in the terrestrial magnetotail lobes (Poppe et al., 2012a) and in the solar wind +(Wang et al., 2011). +As the measurements performed onboard the LADEE and LRO spacecraft have shown, the lunar +exosphere can be monitored, either by in-situ measurements, using a neutral mass spectrometer, or by +remote sensing using a UV spectrometer (Chin et al., 2007; Elphic et al., 2014). +Other techniques for studying the lunar exosphere are: +− Remote sensing of the lunar exosphere by detecting and imaging the ENAs produced by charge +exchange interactions between the solar wind protons and the exospheric neutral atoms (Futaana +et al., 2008). The energies of these ENAs are comparable to the energies of the parent solar wind +protons, i.e. of the order of ~keV. +− Remote sensing of the lunar exosphere by detecting and imaging the SWCX soft X-rays produced +by charge exchange between highly-charged heavy solar wind ions and the exospheric neutral +atoms (Robertson et al., 2009). +− In-situ measurement of freshly ionized pickup ions, originating from the lunar exosphere neutral +species. (Hartle and Killen, 2006; Yokota et al., 2009; Wang et al., 2011; Poppe et al., 2022). At +high altitudes above the lunar surface, as those of the Gateway orbit (cf. section 4.1), this method +can provide higher sensitivity in the detection of low number density species than the direct +sampling of the parent neutrals (Halekas et al., 2015; Poppe et al., 2022). +2.4 +Monitoring the interaction of the solar wind with the Moon’s surface +Solar wind protons, arriving at the Moon’s surface, can be absorbed, or scattered, or can remove +another atom from the lunar regolith by sputtering or desorption (Wieser et al., 2009; McComas et +al., 2009a; Futaana et al., 2012). It results that a large fraction of the solar wind protons, up to 20%, is +reflected back to space as neutral hydrogen atoms (ENAs). It is noteworthy that backscattering of + + + + Lunar Gateway Space Plasma Physics + +7 +neutralized solar wind protons occurs not only when the Moon is in the pristine solar wind, but also +when the Moon enters into the terrestrial magnetosheath and is then exposed to the shocked and +thermalized solar wind (Allegrini et al., 2013). +Figure 10 shows typical energy spectra of the reflected hydrogen ENAs, compared to the parent +solar wind protons energy spectra. As shown, the flux of the reflected ENAs closely follows the +variations of the flux of the parent proton population. The energies of these ENAs are however a +fraction of the parent solar wind protons. +Since the solar wind proton trajectories are modulated by the surface electrostatic potential and by the +eventual local magnetic field anomalies (cf. Figure 11), the detection and imaging of these reflected +ENAs provides a tool to investigate the lunar surface electric and magnetic fields (Futaana et al., +2013; Vorburger et al., 2013, 2015, 2016; Bamford et al., 2016). Local crustal magnetic anomalies +(or “swirls”) constitute “mini-magnetospheres”, shielding locally the lunar regolith from the solar +wind protons and from the resulting space weathering (Wieser et al., 2010; Wang et al., 2012; Deca +et al., 2015; Glotch et al., 2015; Hemingway et al., 2015; Poppe et al., 2015; Pieters and Noble, 2016; +Hemingway and Tikoo, 2018). +The solar wind protons that do not scatter back, but are absorbed in the lunar regolith (top 20 – 30 nm +of the lunar grains), diffuse within the regolith. They can then interact with the oxygen atoms in the +regolith oxides and form OH (McCord et al., 2011; Farrell et al., 2017; Tucker et al., 2019; McLain +et al., 2021). These solar wind-produced hydroxyl radicals contribute to the formation and release of +molecular water, and thus to a solar wind-induced water cycle on the Moon (Crider and Vondrak, +2003; Liu et al., 2012; Futaana et al., 2018; Jones et al., 2018; Honniball et al., 2021). +The exposure of the lunar surface to the solar radiation and to the flux of charged particles results +also in an electrostatic surface charging. An electric potential thus develops between the lunar surface +and the ambient plasma, which manifests itself in a near-surface plasma sheath with a scale height of +the order of the Debye length (Halekas et al., 2011; Stubbs et al., 2013; Burinskaya, 2015; Harada et +al., 2017). This near-surface electric field becomes very complex and highly variable in the vicinity +of the terminator, with the surface polarity changing from mostly positive (few 10 V) on the dayside, +due to photoelectron emission, to highly negative (of the order of the ambient electron temperature, +i.e. up to several -100 V) on the nightside, and in the trailing lunar wake region (Farrell et al., 2007). +Local surface topography is also a factor contributing to a complex near-surface electrostatic and +plasma environment, particularly in the vicinity of permanently-shadowed craters (Poppe et al., +2012b; Nénon and Poppe, 2021). As the THEMIS-ARTEMIS observations have shown, the lunar +surface charging can be remotely sensed from a Moon orbiting spacecraft, even several 1000 km +away from the lunar surface, through the shifted energy spectra of the detected plasma particles when +the spacecraft crosses magnetic field lines connected to the lunar surface (Halekas et al., 2011). +Dust is another component of the lunar plasma environment (Stubbs et al., 2007; Grün et al., 2011; +Horányi et al., 2015; Popel et al., 2018, 2022). Dust grains on (or near) the lunar surface can either be +ejected from the regolith, due to the impact of interplanetary micrometeoroids, or be electrostatically +levitated due to grain charging, as discussed in the previous paragraph. This creates a dusty plasma +system consisting of neutrals of the lunar exosphere, solar-wind ions and electrons, ions and electrons +of the Earth's magnetotail (when the Moon gets inside the terrestrial magnetotail), photoelectrons +formed due to the interaction of the solar radiation with the lunar surface, and charged dust grains +flying over the lunar surface. + + + +Lunar Gateway Space Plasma Physics + +8 +This is a provisional file, not the final typeset article +3 +Measurement Requirements +Following the identification of the scientific objectives in the field of space plasma physics, that can +be addressed using instrumentation onboard the Lunar Orbital Platform - Gateway (cf. section 2), the +ESA topical team identified the physical parameters needed to be measured in order to address these +objectives, and the corresponding instrumentation required to perform these observations. The topical +team addressed thus the following two questions (Dandouras et al., 2020b): +− What plasma physics science questions can be addressed in the vicinity of the Lunar Orbital +Platform - Gateway? +− What are the instrument / payload requirements to achieve such science? +It identified measurements that can be performed either directly from the Gateway platform (3 200 × +70 000 km altitude lunar orbit), or from instrumented cubesats that could be released from the +platform and placed into lower lunar orbits, or directly from the Moon surface. Here we will focus on +the measurements that can be performed by instrumentation mounted onboard the Gateway, and +which can be either in-situ measurements or remote sensing observations, and then we briefly +mention the other two possibilities. Space plasma physics measurements that could be performed +directly from the Moon surface will be the object of a dedicated forthcoming paper. +Table 1 provides an overview of the physical parameters / observables identified, in the field of +space plasma physics, that can that be monitored by instrumentation onboard the Gateway. +Tables 2 and 3 are for the observations that could be performed, on a longer term, from lower lunar +orbits and from the Moon’s surface, respectively. +Table 4 focuses then on the science questions that can be addressed from instrumentation onboard +the Gateway, and shows how each science objective, identified by the topical team, translates into a +measurement requirement, and then to the corresponding instrument / payload requirement. +Additional objectives that could be eventually addressed by remote sensing instrumentation onboard +the Gateway, and could point to targets of opportunity, include: aurora imaging, heliosphere imaging +(through the ENA imager), and lunar surface imaging (e.g. meteor impact flashes). +4 +Conceptual design for a Space Plasma Physics Payload Package onboard the Gateway +Following the work of the topical team, and the identification of the measurement requirements, ESA +issued an Invitation to Tender for a “Deep Space Gateway Plasma Physics Payload Conceptual +Design” (ESA AO/1-9789/19/NL/FC). In response to it we proposed to ESA, were selected and then +undertook a conceptual design study for a “Space Plasma Physics Payload Package onboard the +Gateway” (SP4GATEWAY), addressing these objectives while being compatible with the technical +requirements. +The Gateway modules that are best-suited for hosting the in-situ measurement plasma instruments +were first identified, following a simulation we performed of the interaction between the Gateway +and its plasma environment (section 4.2). The proposed model payload, and its accommodation on +the Gateway modules, are presented in sections 4.3 and 4.4 respectively. The fields-of-view (FOVs) +of the remote sensing instruments, as projected on the sky and on the celestial objects, were then +analyzed by simulating their evolution along the Gateway orbit (section 4.5). + + + + Lunar Gateway Space Plasma Physics + +9 +4.1 +Gateway configuration, orbit and attitude +The Gateway will evolve during its lifetime, different modules being added during the successive +phases of the project. For the purpose of this study we considered a typical “Gateway Phase 2” +configuration, with the Orion spacecraft attached, shown in Figure 12. +The Gateway orbit will be a Near Rectilinear Halo Orbit (NRHO) around the Moon, with periapsis × +apoapsis altitudes 3 200 × 70 000 km (Whitley and Martinez, 2016). The orbital period is ~6.5 +(Earth) days, and the orbital inclination ~90°. The periapsis will be above the north pole of the Moon. +This orbit provides constant Earth visibility (9:2 resonance with the lunar synodic period). +The Gateway attitude will be with the +X axis (longitudinal axis, cf. Figure 12) pointed towards the +Sun. The +Z axis will be normal to the Moon orbit plane, pointing southwards. The pointing +accuracy requirement is that Orion remains in a tail-to-Sun attitude ±20°, i.e. the +X axis has a ±20° +pointing accuracy. +4.2 +Simulation of the Gateway plasma environment +The simulation of the interaction between the Gateway and the plasma environment was performed +by ONERA and the Artenum company (Hess et al., 2020). A 3D mesh model with approximately +64 000 elements was developed to represent the Gateway, and the properties of the surface materials +of the different Gateway modules were taken into account. The SPIS (Spacecraft Plasma Interaction +System) software tool was then used to simulate the Gateway interaction with its ambient plasma +environment. This open-source software, available at https://www.spis.org, computes the potential at +the surface of a spacecraft according to its exchange of charges with the space plasma, i.e. the +collection of charge from the plasma and the re-emission of photoelectrons and of secondary +electrons due to impacting energetic particles. It also simulates the perturbation induced by this +electrostatic charging on the natural plasma. This software was further developed to simulate the +charging of the regolith and the motion of lunar dust particles (Hess et al., 2015) and to simulate the +perturbation of the measurements by plasma instruments due to the charging (Sarrailh et al., 2015). +Two cases were simulated, that correspond to the two situations that will be typically encountered: +1. Gateway in the solar wind (most frequent case, cf. section 1). +Typical solar wind conditions considered were: +solar wind density: 7 cm-3 +solar wind velocity: 450 km/s +ion and electron temperatures: 10 eV +2. Gateway in the terrestrial magnetotail (5 – 6 days per lunar orbit). +The plasma environment considered, corresponding to active geomagnetic activity conditions +(conditions producing downtail plasma streaming, cf. section 2.2), was: +plasma density: 2.01 cm-3 +H+ density: 2 cm-3 +O+ density: 0.01 cm-3 +plasma streaming velocity: 250 km/s (away from the Earth) +ion temperature: 200 eV +electron temperature: 15 eV + + + +Lunar Gateway Space Plasma Physics + +10 +This is a provisional file, not the final typeset article +In each case both a nominal Gateway attitude (Gateway major axis aligned to the solar direction, cf. +section 4.1) and an extreme attitude excursion, with the Gateway major axis tilted by 20° with respect +to the solar direction, were considered. The simulation runs generated, for each case, maps of the +electrostatic potential (volume values in the Gateway environment and surface values on the Gateway +modules), and maps of the density values of H+, O+, photoelectrons and secondary electrons. The +details are given in the report by Hess et al. (2020). +The main results of this study are: +4.2.1 Gateway in the solar wind +The volume electrostatic potential distribution, when the Gateway is in the solar wind and the major +axis of the station is aligned to the solar direction (nominal attitude), following 600 s of interaction +time, is shown in Figure 13. +The Gateway structure gets to a 3.5 V equilibrium potential, while the major part of the solar panels +goes to a 10 V potential on the Sun facing side and -46 V on the rear side (Figure 13A). The PPE +(Power and Propulsion Element), bearing the two main solar panels, is thus inappropriate for space +plasmas instrumentation for low-energy plasmas. The wake effect, due to the solar wind flow, is +particularly visible behind the main solar panels, whereas in the front modules of the station the +potential perturbation appears to be moderate. To highlight the potential values away from the solar +panels (i.e. where the plasma instruments should be mounted), the surface and volume potentials are +plotted also in a scale saturated between +5 V and -3 V (Figure 13B). As shown there, the thickness +of the sheath formed by the plasma flow around the Gateway, on the station parts exposed to the +solar wind and away from the solar panels, is typically ~1.8 m and the electrostatic potential +perturbation is moderate (a few volts). This implies that the effect on the ion and electron +measurements will be very moderate, and only the lowest energy particles (< ~100 eV) will be +affected. Solar wind ions, which have energies of typically ~1 keV, will be almost not affected. It +implies also that a boom of ~2 – 3 m length is adequate for placing sensors as a magnetometer and a +wave antenna outside of the sheath. +The ambient proton density around the Gateway is shown in Figures 13C and 13D. The left panel +(Figure 13C) corresponds to the nominal Gateway attitude (Gateway major axis aligned to the solar +direction), whereas the right panel (Figure 13D) corresponds to an extreme attitude excursion of 20° +with respect to the solar direction. Note, in both cases, the plasma wake downstream of the station. +The tilted axis simulations show a small asymmetry between the illuminated and the shadowed sides +of the Gateway and, as expected, a tilted plasma wake. +4.2.2 Gateway in the terrestrial magnetotail +Here the Gateway is exposed to the terrestrial plasma sheet / magnetosheath plasma streaming +downtail. The volume electrostatic potential distribution, under these conditions and when the major +axis of the station is aligned to the solar direction (nominal attitude), following 400 s of interaction +time, is shown in Figure 14A. The surface equilibrium potential here is 6.5 V, while the major part +of the solar panels goes to a 13 V on the Sun facing side and -31 V on the rear side. Due to the lower +density of the ambient plasma, the sheath forming around the station is more extended, but the +potential barrier is weaker (-1.1 V) and more isotropic, compared to the solar wind case. However, +the overall results are not very different and the conclusions made in the solar wind case apply also +here. Figure 14B shows the emitted photoelectron density. + + + + Lunar Gateway Space Plasma Physics + +11 +The ambient H+ and O+ ion densities (terrestrial ions streaming downtail during active geomagnetic +conditions, cf. section 2.2) are shown in Figures 14C and 14D respectively. The O+ density +distribution shows here a high similarity with the proton density distribution in the solar wind, +presenting a very clear wake effect due to the higher ion mass. +4.2.3 Gateway - plasma environment interaction: synthesis +The interaction of the Gateway with its plasma environment has been simulated for the two cases that +will be encountered: Gateway in the solar wind and Gateway in the terrestrial magnetotail. In both +cases the surface potential of the Gateway away from the solar panels is moderate (3.5 V in the solar +wind and 6.5 V in the magnetotail). A sheath is formed by the plasma flow around the Gateway, +which for the solar wind case has a thickness of ~1.8 m when the Gateway is aligned to the solar +direction. However, when the Gateway major axis (X-axis) is tilted by 20° with respect to the solar +direction, which corresponds to an extreme excursion from the nominal attitude, this plasma sheath +becomes asymmetrical and much thicker in the “shadowed” side. +These results are very encouraging, because they allow to identify the Gateway modules on which +the perturbation of the natural plasma environment by the Gateway will be minimal, and are thus +well-suited for placing the plasma instruments. Figure 15 shows the positions identified for +instrument mounting in a color code, from green (most favorable) to red (least favorable positions). +The US Habitat and the International Habitat present small surface charging, are surrounded by a thin +plasma sheath and do not suffer from any plasma wake effect. They are thus suitable for placing the +plasma instruments sensitive to electrostatic charging, as the magnetospheric ion and electron +spectrometers (green / light green markers in Figure 15). +However, these positions on the cylindrical surfaces of the US Habitat and of the International +Habitat are tangent to the solar wind flow. When the Gateway is tilted with respect to the solar +direction, the solar wind flow is detached and thus not measurable from these positions (cf. Figure +13D). Solar wind measurements require, not only limited (less than ~10 V) surface charging and +absence of local plasma wake effects, but also a direct “face exposure” to the solar wind. The +X side +of the Logistics Module (lower light green marker in Figure 15) is thus the most suitable position for +the solar wind instruments. +Concerning the wave and field instruments, their positioning on ~2 – 3m booms, on the “green / light +green markers”, allows having them outside of the plasma sheath. +The remaining positions can be used for energetic particle and magnetospheric imaging instruments, +which are not sensitive to plasma charging effects. +The least favorable positions for placing plasma instruments (positions to avoid) are the PPE (Power +and Propulsion Element) and the close to it HALO (Habitation and Logistics Outpost), cf. red +markers in Figure 15, due to the large solar panels and associated circuitry, their “downstream” +positioning (with respect to the solar wind flow), the high surface charging, and the proximity to the +ion propulsion engine. +4.3 +Model payload +In order to address the scientific objectives identified by the topical team, we first defined a model +payload, consisting of a suite of instruments corresponding to the requirements shown in Table 4 (cf. +section 3). These measurement instruments are largely based either on existing flight-proven + + + +Lunar Gateway Space Plasma Physics + +12 +This is a provisional file, not the final typeset article +instruments, adapted here for the lunar plasma environment, or on tested and validated laboratory +prototypes (TRL (Technology Readiness Level) ≥ 5). Table 5 lists these instruments and provides an +overview of their main characteristics. The detailed description of the characteristics of the +instruments is given in a series of three ESA reports, corresponding respectively to a Requirements +Inventory (De Keyser et al., 2020), Conceptual Design Report (Devoto and Dandouras, 2020), and +Programmatic Assessment (Futaana et al., 2020). Here we present their principal characteristics. +4.3.1 cMAGF: 3-axis Fluxgate Magnetometer +This instrument will provide the ambient vector magnetic field (in solar wind, terrestrial magnetotail, +Moon vicinity, lunar wake, etc.). The proposed magnetometers package consists of three different +types of units. The main one has three pieces of boom-mounted 3-axis fluxgate magnetometers, on +two ~3 m retractable booms: one sensor at the tip of each boom, and a third sensor at the common +root of the two booms. This allows having two main sensors outside the plasma sheath formed by the +plasma flow around the Gateway (cf. section 4.2), while the boom-root sensor provides the +possibility for removing eventual Gateway-induced perturbations by using the gradiometer +technique. The presence of two boom-tip mounted sensors, on two booms, provides for further +corrections for eventual perturbations. These three units will provide the main measurements while +supporting the cleaning and processing of the measured data. +In order to monitor the perturbations from the station in more detail, several single magnetometer +sensors will be mounted on various places directly on the station (Constantinescu et al., 2020). +Additionally, we propose two current monitors, monitoring the currents flowing from the solar +panels, which are expected to contribute the largest magnetic field perturbations. +The proposed accommodation on the Gateway of this magnetometer package, and the corresponding +CAD figures, are shown as also for the other instruments in section 4.4. +Fluxgate magnetometers benefit from a strong heritage, as such instruments have flown on several +space missions, including Cluster (Balogh et al., 2001), Cassini (Dougherty et al., 2004), THEMIS- +ARTEMIS (Auster et al., 2008), BepiColombo (Glassmeier et al., 2010), etc. +4.3.2 cSWIS: Solar Wind Ion Spectrometer +The cSWIS instrument is a solar wind ion spectrometer that will determine the velocity distribution +functions (VDFs) of the solar wind ions and will provide the solar wind density, velocity and +temperature. +A top-hat electrostatic analyzer instrument is considered, covering the 0.1 – 40 keV/e energy range +and having a field-of-view (FOV) of 96° × 48° aligned with the solar wind arrival direction: 96° +angular range in azimuth (+24° to -72° in the ecliptic plane, as the Gateway points to the Sun but it +may sometimes drift away from this direction, after which it catches up) and 48° in elevation +(between -24° to +24°). The spatial resolution is 3° in both azimuth and elevation, and the energy +resolution is ΔE/E = 8%. In order to achieve high VDF acquisition cadence, we propose to use solar +wind beam tracking, along the lines of the Cold Solar Wind (CSW) instrument (Cara et al., 2017; De +Keyser et al., 2018), that was designed for the THOR (Turbulent Heating ObserveR) mission which +was proposed to ESA as a medium-class M4 mission. + + + + Lunar Gateway Space Plasma Physics + +13 +4.3.3 cSWFC: Solar Wind Faraday Cup +The cSWFC instrument will be used to determine the solar wind density, velocity and relative alpha- +particle content, based on simultaneous measurements of the collector currents provided by six +identical Faraday cups. +The energy of incoming ions is determined by the high voltages applied onto the control grids. The +Faraday cups are organized into three units, each of them containing two cups. One unit serves for +the determination of the total ion flux vector, the second unit uses high voltages applied on the +control grids and provides two points of energy distribution that are used for the determination of the +proton velocity and temperature in the Maxwellian approximation. The last unit serves for the +measurement of the 1D velocity distribution (integral distribution) of protons and alpha particles. +Each of the six Faraday cups will have a 45° × 45° FOV which, as for the cSWIS instrument, will be +aligned with the solar wind arrival direction. The energy resolution is 1% (< 50 eV). +The proposed cSWFC instrument is based on the BMSW (Bright Monitor of the Solar Wind) +Faraday cup instrument, that flew onboard the Spektr-R mission (Šafránková et al., 2013). +4.3.4 cMISP: Magnetospheric Ion Spectrometer +cMISP is a mass-discriminating ion spectrometer, that determines the velocity distribution functions +of the ambient plasma ions: terrestrial magnetosphere ions, lunar exosphere pickup ions and solar +wind ions. +The proposed instrument is a time-of-flight ion mass spectrometer capable of obtaining ion +distributions (about 10 eV/e to 40 keV/e) with a high-resolution mass-per-charge composition +determination (m/Δm > 15). Ions are selected as a function of their E/q (energy per charge) ratio, by +sweeping the high voltage applied between the two hemispheres of a rotationally symmetric toroidal +electrostatic analyzer (360° ×5° instantaneous FOV). Then they go through a post-acceleration of +about 5 kV and they subsequently enter into the time-of-flight (TOF) section, where the velocity of +the incoming ions is measured, which allows then the calculation of their m/q (mass per charge) ratio. +A specially designed thin microchannel plate (MCP), through which the ions pass, is used as a +conversion surface for the production the “start” TOF signal secondary electrons. The “stop” TOF +signal is provided by the ion detection on another MCP. The instrument provides for a ΔE/E ~7 % +energy resolution and a 22.5° angular resolution. +cMISP is based on the MIMS (MCP Ion Mass Spectrometer) instrument, that was designed for the +ESCAPE (European SpaceCraft for the study of Atmospheric Particle Escape) mission, which was +proposed to ESA as a medium-class M5 mission (Dandouras et al., 2018). MIMS in its turn was +based on a successfully tested prototype developed at IRAP (Devoto et al. 2008). MIMS is an +evolution of the CIS-CODIF instrument, flying onboard Cluster (Rème et al., 2001), but with higher +mass resolution. +Since MIMS was designed for a spinning spacecraft, where it would take advantage of the spacecraft +rotation to obtain a full 3D ion distribution within one spacecraft spin, cMISP on the Gateway, which +is a 3-axis stabilized space station, requires the addition of electrostatic deflection plates at the +instrument entrance to scan the FOV over a 360° ×120° solid angle (±60° with respect to the central +entrance plane). + + + +Lunar Gateway Space Plasma Physics + +14 +This is a provisional file, not the final typeset article +4.3.5 cMESP: Magnetospheric Electron Spectrometer +cMESP is an electron spectrometer that will determine the velocity distribution functions (VDF) of +the solar wind electrons (pristine or reflected from lunar crustal magnetic field anomalies) and of the +plasma sheet electrons, when the Gateway gets into the terrestrial magnetosphere. +A top-hat electrostatic analyzer instrument covering the ~5 eV to ~20 keV energy range is proposed. +As for the cMISP instrument, the addition of electrostatic deflection plates at the instrument entrance +to scan the FOV over a 360° ×120° solid angle is required. +The proposed cMESP instrument is based on the SWEA (Solar Wind Electron Analyzer) instrument, +flying onboard the MAVEN spacecraft (Mitchell et al., 2016). +4.3.6 cENPD: Energetic Particles Detector +The cENPD instrument will detect and measure the fluxes of the energetic charged particles, ions and +electrons: Solar Energetic Particles (SEPs), low-energy Galactic Cosmic Rays (GCRs) and terrestrial +plasma sheet energetic particles. The instrument will also investigate the spectra of the secondary +high energy ions, released from the lunar surface following its irradiation by GCRs and/or SEPs +(albedo energetic particles, cf. section 2.1). It will cover the ~40 keV – ~100 MeV energy range for +ions and ~20 keV – ~30 MeV for electrons. It will provide a ΔE/E ≤ 10 keV energy resolution and +supply, for ions, a measure of the composition (protons to iron nuclei). +In order to cover both pristine and albedo energetic particles, it will consist of two identical detection +heads, each with a 60° × 60° FOV: one pointing to the lunar zenith and the other pointing to the +opposite direction (lunar nadir). Each detection head will be composed of a collimator and a 1 cm2, 1 +mm thick silicon detector. In front of the detector a filter wheel will allow to place either a thick foil, +a pinhole or an obturator to allow the reconfiguration of the detection head to various scientific +modes to measure the combined spectra of electrons and ions, to measure the electron spectrum, to +protect the detector from sunlight or to avoid saturation of the detector. +The proposed cENPD instrument will benefit from the heritage of the IPD instrument, flown onboard +the DEMETER satellite (Sauvaud et al., 2006), and of the IDEE instrument, developed for the +TARANIS satellite (Lefeuvre et al., 2008). +4.3.7 cGCRD: Galactic Cosmic Rays Detector +The cGCRD instrument will measure the spectra and the composition of the Galactic Cosmic Rays +and that of the Solar Energetic Particles, covering the 0.1 to ~ 5 GeV energy range. It will thus be +complementary to the cENPD instrument, covering the higher energies. +The proposed cGCRD instrument is the Mini.PAN penetrating particle analyzer, which is an +approved H2020-FETOPEN project that will build a demonstrator of the Penetrating particle +ANalyzer (PAN) for deep space applications (Wu et al., 2019). +Mini.PAN is based on the particle detection principle of a magnetic spectrometer, with novel layout +and detection concepts to optimize the measurement precision for both high flux and low flux +particles. As above several hundred MeV/nuc standard methods for measuring particle energies +(TOF, dE/dx, ΔE-E) become less efficient, the use of magnetic spectrometry (the charged particle +energy is derived from the degree of bending of its trajectory in the magnetic field) is used as the +principal particle analysis method. In Mini.PAN the bending of the particle in the magnetic field is + + + + Lunar Gateway Space Plasma Physics + +15 +measured by precise silicon strip tracking detectors, while the elemental identity of the particle is +determined by its charge and Z, which is measured with the dE/dx method at multiple points. +Mini.PAN is designed to precisely measure the momentum, the charge, the direction and the time of +energetic particles between 100 MeV/nuc and a few GeV/nuc. +Mini.PAN offers much higher energy resolution (compared to integral measurements), especially in +the > 100 MeV range, and is appropriate for precision energy and species measurements in the 100 +MeV/nuc to low GeV/nuc range, which contains both albedo particles and the low energy part of the +ambient GCR spectrum. This part is not well resolved by past solar wind observatories (e.g. ACE) or +by massive GCR detectors in low Earth orbit (e.g. PAMELA). Mini.PAN is also a new type of +miniaturized, advanced energetic particle detector that can be adapted and adjusted for deep space +missions, where mass limitations exist. +As a bonus, the proposed concept for the cGCRD detector can also detect MeV ENAs (likely of +heliospheric origin, cf. Mewaldt et al., 2009), because the detection method combines a strong +magnet, the ΔE-E technique and particle tracking through successive, pixelated SSDs. Few-MeV +hydrogen ENAs would give the characteristic ΔE-E signal on the SSD stack, but across a straight- +line trajectory, since the magnet does not influence them, i.e. they can be separated from charged +species (which also get detected) and from the very high energy GCRs (which are less detected but +penetrate deeper). +4.3.8 cHENA: High-Energies ENA Imager +cHENA is a high-energies ENA (Energetic Neutral Atoms) imager, for detecting and imaging the +ENAs produced by charge exchange interactions between the terrestrial plasma sheet or ring current +energetic ions and the geocorona neutral hydrogen atoms. It will cover the ~10 – 500 keV energy +range, and will be equipped with a collimator to both delimit the FOV (120° × 90° or narrower) and +reject the charged particles. The transmitted ENAs then go through a TOF system and are detected by +an MCP (64 × 64 pixels). Pointing the instrument optical axis towards the terrestrial inner +magnetosphere requires mounting cHENA on an azimuthal (1-axis) articulation. +The proposed cHENA instrument is based on the MIMI-INCA ENA imager, flown onboard Cassini +(Krimigis et al., 2004) and the HENA ENA imager, flown onboard the IMAGE mission (Mitchell et +al., 2000). +4.3.9 cMENA: Medium-Energies ENA Imager +cMENA is a medium-energies ENA imager, for detecting and imaging the ENAs produced by charge +exchange interactions between the terrestrial plasma sheet ions and the geocorona neutral hydrogen +atoms. It will thus be complementary to the cHENA instrument, extending the coverage to lower +energies (~1 keV – 100 keV). An additional objective for MENA is the detection and imaging of +ENAs produced in the lunar environment, from the charge exchange interactions between the solar +wind protons and the lunar exosphere. This requires flexibility in the instrument pointing (Earth or +Moon pointing), which implies also mounting cMENA on its own azimuthal (1-axis) articulation. +The proposed instrument has a 90° × 10° instantaneous FOV and it uses, as cHENA, a collimator to +both delimit the FOV and reject the charged particles. The collimator includes also a UV filter. The +instrument provides a 5° × 10° angular resolution, i.e. one-dimensional images. It is based on the +heritage of the wide-angle imaging neutral-atom spectrometer onboard the TWINS mission +(McComas et al., 2009b) and of the SERENA-ELENA neutral atom imager onboard the MPO +Mercury Planetary Orbiter of the BepiColombo mission (Orsini et al., 2021). + + + +Lunar Gateway Space Plasma Physics + +16 +This is a provisional file, not the final typeset article +4.3.10 cLENA: Low-Energies ENA Imager +cLENA completes the suite of ENA imagers by covering the lowest energies (down to ~10 eV). +These low-energy ENAs have two main sources: the charge exchange interactions of the solar wind +protons with the lunar exosphere and the charge exchange interactions with the lunar surface. Moon +pointing for its FOV is thus required. +The proposed instrument has a 15° × 15° field-of-view consisting of a single pixel and uses a +conversion surface to ionize incoming ENAs and then feed them into an electrostatic wave system, +which acts as a filter to pass only particles within the proper energy range. The particles then go +through a TOF system. The instrument is capable of high-cadence observations of the solar wind - +lunar surface interaction within the ~10 eV to ~3.3 keV energy range and with a ~50% ΔE / E energy +resolution. It is based on the LNT instrument that has been designed for the Luna-Resurs-Orbiter +(Luna 26) mission. +4.3.11 cUVIS: UV Imaging Spectrometer +cUVIS is a UV / EUV imaging spectrometer, sensitive to specific emission lines for observing the +terrestrial exosphere (H: 121.6 nm, He: 58.4 nm, O: 130.4 nm, and N: 120.0 nm), the terrestrial +plasmasphere (He+: 30.4 nm, O+: 83.6 nm) and the lunar exosphere (He: 58.4 nm, plus emission lines +of other elements). It will thus cover the 30 – 130 nm wavelength range and will have a 0.1° × 7.5° +FOV with a ~5 arcmin angular resolution. This resolution corresponds to about 0.15 RE (Earth radii) +at the plasmasphere, as seen from the Moon. +Earth pointing requires, as for cHENA and for cMENA, mounting the instrument on its own +azimuthal (1-axis) articulation. The articulation allows also pointing the instrument to the Moon, as a +function of the scientific target of each observation session. +The proposed instrument is based on the heritage of the PHEBUS UV / EUV imaging spectrometer +onboard the MPO Mercury Planetary Orbiter of the BepiColombo mission (Chassefière et al., 2010). +4.3.12 cLPEF: Langmuir Probe and E-field +cLPEF is a Langmuir probe instrument for providing ambient plasma diagnostics: a conductive +probe, either biased or floating, is immersed into the plasma and the resulting electron / ion fluxes to +the conducting surface provide electric current or voltage measurements with respect to the +spacecraft. From these measurements the main plasma characteristics can be derived, including the +plasma density, the electric field or the spacecraft floating potential. In order to provide unperturbed +plasma measurements the probe has to be located well outside of the plasma sheath that forms around +the spacecraft with a thickness proportional to the local Debye length. +The proposed cLPEF instrument will employ two spherical probes (~ 8 – 10 cm in diameter), each +placed on the tip of a retractable boom. Since this requirement is identical to the one for the two main +magnetometer sensors of the cMAGF instrument, and in order to optimize the resources and simplify +the interfaces of the whole space plasma package, it is proposed to combine the sensors of these two +instruments and house a tri-axial fluxgate magnetometer sensor within each of the two Langmuir +spherical probes (cf. section 4.4 for the CAD figures). Such a combined Langmuir probe / +magnetometer concept has been originally introduced as a part of an integrated plasma and dust +package study conducted under the ESA Contract No. 4000103352/11/NL/AF in the framework of +the proposed Lunar Lander mission, and it is the approach used on ESA’s Comet Interceptor mission +(Ratti et al., 2022). + + + + Lunar Gateway Space Plasma Physics + +17 +An additional possibility is to mount occasionally a stand-alone Langmuir probe at the edge of the +Gateway external robotic manipulator, so as to use this robotic arm in order to investigate the +properties of the plasma sheath, forming around the different Gateway modules, at various locations. +Langmuir probes benefit from the heritage of instruments that have flown on several space missions, +including the RPWS instrument onboard Cassini (Gurnett et al., 2004), ISL onboard the DEMETER +satellite (Lebreton et al., 2006), DSLP onboard the PROBA-2 satellite, etc. +4.3.13 cWAVE: Waves Radio Instrument +cWAVE is an electromagnetic waves instrument for the study of terrestrial AKR (auroral kilometric +radiation) emissions, occurring in the auroral region. It would then take advantage of the Moon +occultation method, which was first implemented by the Radio Astronomy Explorer‐2 mission +(Kaiser and Alexander, 1976). An additional objective is the study of the radio emissions emitted by +accelerated particles in the solar corona and the solar wind. +The proposed cWAVE instrument will measure the AC electric field (one component: fast electric +waveform at 16.5 MHz, decimated electric waveform at 64.5 kHz) and the AC magnetic field (one +component: magnetic waveform at 64.5 kHz). These three products will be delivered as waveform +(event mode) and / or as averaged spectra (survey mode, with onboard FFT computation). +As for the cMAGF and the cLPEF instruments, the cWAVE sensor needs to be placed at the tip of a +dedicated retractable boom. +Radio waves instruments benefit from the heritage of instruments that have flown on several space +missions, including STAFF onboard Cluster (Cornilleau-Wehrlin et al., 2003), RPWS instrument +onboard Cassini (Gurnett et al., 2004), etc. +4.3.14 Retractable booms +As indicated above, the mounting of the sensors of the combined cMAGF and cLPEF instruments +and of the cWAVE instrument requires a total of three retractable booms, ~3 m each. These booms +need indeed to be stowed at the beginning and at the end of the mission to allow the instruments to +stay within the allocated envelope and to be transferred by the Airlock. We propose to use the +compact deployable and retractable boom that has been developed by Oxford Space Systems: the +Astrotube Boom. It can be deployed to up to 3 m and is TRL 9. +4.3.15 Instruments not included in the conceptual design study +The above-described model payload instruments (cf. also Table 5) cover satisfactorily the +instrumentation requirements, as defined by the topical team (cf. Table 4). However, there are two +instruments that were not included in this conceptual design study: the MeV ENA Imager and the +Soft X-ray Imager. +The MeV ENA Imager was not included due to the absence in Europe, in our knowledge, of a +developed instrument or protype, for ENAs at these very high energies. However, as described +above, the proposed cGCRD instrument (Mini.PAN) will be able to detect few-MeV hydrogen +ENAs, separating them from similar energy protons and providing 1-pixel images of this population. +For X-ray imaging, a soft X-ray imager with a wide field-of-view, using lobster-eye optics and a +position-sensitive MCP detector operating at the 0.1 – 2 keV X-ray bandpass has been considered +(cMXRI instrument). This instrument is based on the DXL/STORM soft X-ray imager protype flown + + + +Lunar Gateway Space Plasma Physics + +18 +This is a provisional file, not the final typeset article +onboard a sounding rocket mission (Collier et al., 2015). However, the size of this instrument (78 cm +length) appears to be incompatible with the Gateway interfaces for mounting external payloads. This +suggests that the X-ray imager could not be accommodated as part of this instruments package. +However, it is recommended to propose cMXRI as a payload for the Large European Lander for the +Moon. +4.4 +Instrument accommodation +The CAD model for the instrument conceptual design and their accommodation on the Gateway was +established in cooperation with the CNES PASO (Plateau d’Architecture des Systèmes Orbitaux), +with the help of its concurrent engineering facilities (CIC : Centre d'Ingénierie Concourante), and +particularly by using the IDM-CIC (Integrated Design Model) and IDM-View tools +(https://idm.virtual-it.fr/). +The instrument accommodation on the Gateway modules has to fulfil several requirements: +− In-situ measurement low-energy plasma instruments have to be placed on areas with low +electrostatic charging (cf. section 4.2). +− Pointing requirements for instruments with a field-of-view (cf. Table 5). +− Unobstructed field-of-view for these instruments. +− cGCRD, which has a strong permanent magnet perturbing the low-energy plasma measurements, +should not be placed close to these instruments. +− Instrument grouping, when possible, to form self-contained “instrument suites”, with instruments +mounted on a common platform, minimizing interfaces with the Gateway and using a single +SORI (external Small ORU Interface) for attachment on the Gateway. +The instrument accommodation configuration we propose, and is compatible with the above +requirements, has the instruments grouped on one main and one secondary platform. Each of these +two platforms, of the order of 0.8 m × 0.8 m, is mounted externally on a SORI attachment and is +double sided, i.e. has instruments mounted on both sides of the platform. This accommodation is of +course notional and could be subject to modifications, depending on Gateway engineering and +programmatic constraints. +Both platforms are on the Logistics Module and they are mounted on two diametrically opposite +positions, on the +X side (Main Instrument Platform) and on the –X side (Secondary Instrument +Platform) of it, cf. Figure 16. +The two sides of the Main Platform are shown in Figure 17. With its positioning on the +X side of +the Logistics Module, the Main Platform provides an unobstructed view to the solar wind arrival +direction (Figures 13C and 13D) and takes advantage of a favorable electrostatic environment +(Figure 15). It is thus well suited for mounting the solar wind instruments (cSWFC and cSWIS), +shown in Figure 18 and Figure 19 respectively. +The Main Instrument Platform hosts also: +− The magnetospheric particle instruments cMESP and cMISP, shown in Figure 18 and Figure 19 +respectively. +− The energetic particle detector cENPD, shown in Figure 20, which is mounted on the –Y edge of +the platform. cENPD has two oppositely directed FOVs, one along the +Z axis and one along the +–Z axis. In this way, during periapsis passes one of the FOVs looks in the zenith direction, to + + + + Lunar Gateway Space Plasma Physics + +19 +monitor the pristine energetic particles precipitating towards the Moon’s surface, whereas the +other looks in the nadir direction, to monitor the albedo energetic particles that are the result of +the interaction of the precipitating energetic particles with the lunar regolith (cf. section 4.5). +Moreover, the +Z / –Z orientation of the two detector heads allows avoiding direct sunlight +entering the detectors (Sun is in the +X direction). +− The two “compact” remote sensing instruments cMENA and cLENA, shown in Figure 18 and +Figure 19 respectively. cMENA uses a dedicated azimuthal (1-axis) articulation. The cLENA +orientation gives access, during the periapsis passes, to the Moon surface and plasma +environment. +− The two booms of the fluxgate magnetometer package (cMAGF), which as described in section +4.3.1 consists of three different types of units. The main type is two pieces of boom-mounted dual +fluxgate magnetometers (one sensor at each of the two ~3 m boom tips and one at the boom root). +These two retractable booms are mounted on the Main Instrument Platform –Z side (Figure 18). +The boom tip mounted cMAGF sensors are integrated together with Langmuir probes (cLPEF +instrument). These units will provide the main measurements. In order to monitor the +perturbations from the station close to the source in more detail, several (~5+) single +magnetometer sensors will be also mounted on various places around the station (not shown). +− The cWAVE instrument, also mounted on a retractable boom, which is on the Main Instrument +Platform +Z side (Figure 19). +The two sides of the Secondary Platform are shown in Figure 21. This platform, mounted on the –X +side of the Logistics Module, is permanently in the shadow. In this way there is no direct sunlight +that could interfere with the measurements of the two instruments mounted on it. On each of its two +sides there is a remote sensing instrument: cUVIS on the one side and cHENA on the other side of +the Secondary Platform. Each of these two instruments in mounted on a dedicated azimuthal (1-axis) +articulation. +The cGCRD instrument, due to the containment of a strong magnet (0.4 Tesla) that would deviate +charged particles to be measured by the other instruments if in close vicinity with them, is not +mounted on any of the two instrument platforms. It is instead mounted as a “standalone” instrument +on the SORI attachment of the +Z side of the Logistics Module (Figure 22). Its FOV, looking +radially out, near periapsis gives access to the albedo energetic particles that are the result of the +interaction of the precipitating galactic cosmic ray particles with the lunar regolith. During the +remaining part of the orbit (most of the time) it points to the open sky. +4.5 +Instruments fields-of-view simulation +The appropriate orientation of the fields-of-view (FOVs) of the remote sensing and of the high- +energy particle instruments, as projected on the sky and on the celestial objects, was analyzed by +simulating the evolution of the FOVs along the Gateway orbit. This simulation was performed in +cooperation with the CNES PASO and by using the VTS software tool +(https://logiciels.cnes.fr/en/content/vts). +The FOVs of the two oppositely directed sensor heads of the cENPD instrument, near a periapsis +pass, are shown in Figure 23. As shown in this figure, one of the two sensor heads is oriented +towards the local zenith, and has an unobstructed view to the pristine energetic particles flux (purple +FOV), whereas the other is oriented towards the nadir and its FOV is dominated by the albedo +energetic particles from the Moon (yellow FOV). Both populations (pristine and albedo high-energy +particles) are thus covered by the cENPD instrument detection capabilities. + + + +Lunar Gateway Space Plasma Physics + +20 +This is a provisional file, not the final typeset article +For the cGCRD instrument, which is a single sensor head GCR detector, the FOV near a periapsis +pass is shown in Figure 24, left panel (light blue FOV). As shown, near periapsis it is dominated by +the albedo GCR particles from the Moon. However, during most of the remaining orbit (right panel) +it has an unobstructed view to the open sky and gives then access to the pristine GCR environment. +The field-of-regard (FOR) of the cMENA instrument, i.e. the total accessible FOV taking into +account the rotation of the 1-axis articulation on which the instrument is mounted, at a given point of +the orbit, is shown in Figure 25A. The azimuthal rotation mechanism gives to the instrument access +to a very large “ribbon” of the sky, which includes the Earth environment and the Moon +environment. The pointing of the instrument to any of these two principal targets, using the flexibility +provided by the 1-axis articulation, can then be programmed as a function of the scientific target of +each observation session. +In Figure 25B is the FOV of the cLENA instrument, close to periapsis, as projected on the sky (no +articulation for this instrument). As shown in this figure, the way the instrument is mounted on the +Gateway gives access, during the periapsis passes, to the Moon surface and to its exosphere and +plasma environment. +The FOR of the cUVIS instrument is shown in Figure 25C. As shown in this figure, the dedicated +articulation allows also for this instrument to point to targets as the Earth space environment +(plasmasphere, exosphere), the Moon space environment (exosphere), or targets in the open sky. The +narrow width of the instantaneous FOV of this instrument (0.1°), in combination with the +articulation, allows also performing altitude profile scans of the lunar exosphere. +5 +Conclusion +The Moon is a unique location to study the deep space plasma environment. The Lunar Orbital +Platform - Gateway, that will be assembled and operated in the vicinity of the Moon starting from the +mid-2020s, is a crewed station that offers new opportunities for fundamental and applied scientific +research in the field of space plasma physics. These have multi-disciplinary dimensions, and they +include: +− Studying the lunar space environment and its interaction with the solar wind and the terrestrial +magnetotail plasma. +− Terrestrial space weather: monitoring, through remote sensing techniques, the response of the +terrestrial magnetosphere and exosphere to solar activity events. +− Planetary space weather: monitoring, through in-situ measurements and through remote sensing, +the response of the lunar space environment to solar activity events. +− Radiation physics: characterizing the lunar high-energy particles environment, including energy +and mass spectrometry of these populations and their variability, particularly in view of the +Artemis human missions to the Moon and the associated radiation risks. +− Studying the heavy ion escape from the terrestrial ionosphere, through in-situ measurements of +the downtail streaming ions, and the role of this escape in the long-term evolution of the +composition of the terrestrial atmosphere (and its habitability). +− Studying the lunar regolith - bounded exosphere - interplanetary space environment as a complex +interacting multi-scale system, and as an archetype of the interaction of an unmagnetized +planetary body with the solar wind. +− Studying the mini-magnetospheres that form above the “swirls” on the Moon, and which +constitute probably the smallest magnetospheres in our solar system. + + + + Lunar Gateway Space Plasma Physics + +21 +− Understanding the surface electric fields that develop on the Moon as a part of a complex and +interacting plasma environment, and their role in electrostatic lunar dust levitation. +− Planetology: understanding the composition of the lunar regolith, and its hydration, through the +spectrometry of the albedo energetic particles. +In preparation of the scientific payload of the Lunar Orbiter Platform - Gateway we first formed a +topical team, under the auspices of ESA, to prepare and to support the definition of payload studies in +the field of space plasma physics. This allowed to identify the scientific objectives that can be +addressed from onboard the Lunar Orbital Platform - Gateway, the physical parameters needed to be +measured in order to address these objectives, and the corresponding instrumentation required to +perform these in-situ measurements and remote-sensing observations. +We then undertook for ESA a conceptual design study for a “Space Plasma Physics Payload Package +onboard the Gateway” (SP4GATEWAY), addressing the objectives identified by the topical team +while remaining compatible with the technical requirements. This conceptual design has considered, +as baseline, a typical “Gateway Phase 2” configuration. +As a first part of this conceptual design study, we simulated the interaction between the Gateway and +its plasma environment, for the case where the Gateway is in the solar wind and also for the case +where the Gateway is in the terrestrial magnetotail. This allowed to identify the Gateway modules on +which the perturbation of the natural plasma environment by the Gateway will be minimal, and are +thus best-suited for placing there the in-situ measurement plasma instruments. +We then defined a model payload consisting of a suite of instruments, for in-situ measurements and +for remote-sensing observations, corresponding to the requirements. These measurement instruments +are largely based, either on existing flight-proven instruments, adapted here for the lunar plasma +environment, or on tested and validated laboratory prototypes. The main characteristics of these +instruments have been defined and CAD conceptual instrument designs elaborated. The instruments’ +measurement characteristics will however have to be refined during a follow-on Phase A study. +The next step was the study for accommodating this model payload on the Gateway modules, taking +into account the various constraints, and in particular the surface and volume charging of the various +Gateway modules, their exposure to the ambient plasma and the pointing and field-of-view +requirements of the instruments. This resulted in an integrated CAD design, including the Gateway +and the instruments, which were grouped into two platforms mounted on two sides of the Logistics +Module. +The fields-of-view of the remote sensing instruments and of the high-energy particle instruments, as +projected on the sky and on the celestial objects, were then analyzed by simulating their evolution +along the Gateway orbit. This allowed to verify the appropriate orientation of the fields-of-view and +the coverage of the observational scientific targets. +Following this conceptual design study for a Space Plasma Physics Payload Package onboard the +Gateway, it results that the Gateway is very well-suited for space plasma physics research and it +allows to address a series of relevant scientific objectives. + +6 +Tables + + + +Lunar Gateway Space Plasma Physics + +22 +This is a provisional file, not the final typeset article + +TABLE 1 | Physical parameter / observable to be monitored from onboard the Gateway + +In-situ +measurements +Solar Wind (particles + fields) +Earth’s foreshock +SEPs, GCRs (pristine + secondary from Moon, at various directions) +Energetic electrons +Magnetotail + magnetosheath plasma (particles + fields) +Outflowing terrestrial ions (ion spectrometry) +Lunar pickup ions (ion spectrometry) +Gateway-induced plasma and fields environment +Lunar Wake + +Imaging +MeV ENAs: produced from SEPs +ENAs: Terrestrial Ring Current and Plasma Sheet +Low-energy ENAs (from Solar Wind and Moon) +SWCX X-rays: Magnetosheath/pause + cusp + planetary targets of opportunity +UV / EUV: Terrestrial Plasmasphere +UV / EUV: Geocorona, Lunar Exosphere, Solar EUV radiometry +Auroral imaging, Planetary imaging +Heliosphere imaging +Lunar surface micrometeorite impacts +Active experiments +Gas release and ionization + + + + + Lunar Gateway Space Plasma Physics + +23 + +TABLE 2 | Physical parameter / observable to be monitored from low lunar orbits + +In-situ +measurements +Crustal Magnetic Anomalies (Plasma + magnetic field + ENAs + electron reflectometry) +Solar wind ions neutralization +Lunar Exosphere / Ionosphere (in-situ measurements) +Dusty plasmas +Imaging +Lunar Exosphere / Ionosphere (imaging) + +TABLE 3 | Physical parameter / observable to be monitored from the lunar surface + +In-situ +measurements +Energetic ion implantation / reflection +Lunar surface electrostatic charging + dust +Crustal Magnetic Anomalies +Magnetosphere radio emissions +Lunar exosphere +Imaging +SWCX X-rays: Magnetosheath/pause + cusp + planetary targets of opportunity + + + + +Lunar Gateway Space Plasma Physics + +24 +This is a provisional file, not the final typeset article +TABLE 4 | Science objectives and corresponding measurement and instrumentation requirements +(from onboard the Gateway) +Science Objective +Measurement Requirement +In-situ Measurements +Instrument +Remote Sensing +Instrument + +Monitor solar wind as a driver +for the dynamics of terrestrial +magnetosphere, terrestrial and +lunar exospheres, lunar surface +sputtering and charging +Solar wind density and +transport velocity +1 – 102 cm-3, 0.1 – 40 keV ions +200 – 1000 km/s, ΔE/E < 17% +Faraday Cup +Electrostatic Analyzer +- +IMF: 100 nT instrument range +1 nT / 0.1 nT absolute / relative +resolution +Magnetometer +- +Monitor and characterize +SEPs and GCRs +for radiation environment +and as lunar surface sputtering +sources +40 keV – 100 MeV ions (SEPs) +up to ~5 GeV (GCRs) +50 MeV / nucleon for +composition +~40 keV – ~30 MeV electrons +Energetic particle +detectors +MeV ENA Imager + +Monitor and characterize the +response of the terrestrial +magnetosphere to the +solar wind with a wide +coverage of geospace +Detect and image solar wind +charge exchange X-rays +0.2 – 2.0 keV +FOV 10° × 10° +angular resolution: +0.3 RE from the Moon +- +Soft X-ray Imager + + +Detect and image terrestrial +magnetosphere ENAs +~1 – 300 keV, FOV ~ 20° × 20° + +ENA Imager +Monitor solar wind interaction +with the lunar exosphere, +regolith and magnetic +anomalies +Detect and image low-energy +ENAs: 0.1 – 10 keV, 30 % ΔE/E, +FOV ~ 20° × 20°, ~5° resolution +Strong UV suppression: 10-8 +- +LENA imager +Reveal the solar wind +ion dynamics in the vicinity of +the lunar magnetic anomalies +Detect and image low-energy +ENAs: 0.01 – 3 keV, 30 % ΔE/E, +FOV ~ 5° × 120°, ~5° resolution +- +LENA imager +Monitor the terrestrial +and lunar exospheres, +plasmasphere +Detect and image +EUV emissions +30.4, 83.6, 121.6 and 130.4 nm +~5 arcmin resolution +Ion mass spectrometer +(lunar pickup ions) +UV / EUV +spectro-imager + +Monitor ambient plasma in +different environments +(solar wind / magnetosheath / +terrestrial magnetotail / +lunar wake) +Plasma density and temperature +~0.01 – 40 keV, 10-3 – 102 cm-3 +Ion composition: m/Δm > 15 +Langmuir probe +Ion mass spectrometer +Electron spectrometer +- +Magnetic field: 1000 nT range +1 nT / 0.1 nT absolute/relative +resolution +Magnetometer +- +Monitor magnetospheric +and planetary radio emissions + + +Radio instrument + + + + Lunar Gateway Space Plasma Physics + +25 + +TABLE 5 | SP4Gateway model payload instruments +Instrument +Acronym +Instrument +Mass +(kg) +Power +(W) +FOV +FOV pointing +cMAGF +Magnetometer(s) +3.5 +4.3 +N/A +N/A +cSWIS +Solar Wind Ion Spectrometer +5 +7 +96° × 48° +Sun +cSWFC +Solar Wind Faraday Cup +5 +4 +45° × 45° (×6) +Sun +cMISP +Magnetospheric Ion +Spectrometer +7 +8 +360° × 120° +N/A +cMESP +Magnetospheric Electron +Spectrometer +3 +6 +360° × 120° +N/A +cENPD +Energetic Particles Detector +3 +6 +60° × 60° (×2) +Moon / Sky +cGCRD +Galactic Cosmic Rays Detector +10 +20 +71° × 71° +Moon / Sky +cHENA +High-Energies ENA Imager +15 +12 +120° × 90°, +articulation +Earth +cMENA +Medium-Energies ENA Imager +5 +15 +90° × 10°, +articulation +Earth / Moon +cLENA +Low-Energies ENA Imager +4 +10 +15° × 15° +Moon +cUVIS +UV Imaging Spectrometer +10 +15 +0.1° × 7.5°, +articulation +Earth / Moon +cLPEF +Langmuir Probe and E-field +1.2 +5 +N/A +N/A +cWAVE +Waves Radio Instrument +5.6 +8.6 +N/A +N/A +Mass and Power: nominal values, without margins, booms and articulation mechanisms included in these values; FOV: +field-of-view; N/A: not applicable + + + + +Lunar Gateway Space Plasma Physics + +26 +This is a provisional file, not the final typeset article +7 +Conflict of Interest +The authors declare that the research was conducted in the absence of any commercial or financial +relationships that could be construed as a potential conflict of interest. +8 +Author Contributions +ID was the coordinator of the topical team “Space Plasma Physics Science Opportunities for the +Lunar Orbital Platform - Gateway”, the coordinator of the SP4GATEWAY project, and is the main +author of this manuscript. MGGT was the ESA support scientist for the topical team and for the +SP4GATEWAY project. JDK, YF, RAB, GBR, JYF, DG, BG, HL, FL, AM, RN, and ER were +members of the topical team and of the SP4GATEWAY project team. PD was the lead engineer for +the SP4GATEWAY project. EDA, JE, ME, PG, DH, LP and ŠŠ were members of the +SP4GATEWAY project team. AL managed the CAD design. JF, AT, SLGH and JCMV performed +the Gateway - plasma environment simulations. JC was the ESA HRE (Human and Robotic +Exploration) correspondent. JW was the ESA manager for the SP4GATEWAY project. +9 +Funding +The topical team “Space Plasma Physics Science Opportunities for the Lunar Orbital Platform - +Gateway” was supported by ESA through contract No. 4000128802 /19/NL/PG/pt. The +SP4GATEWAY project was funded by ESA through contract No. 4000128461/19/NL/FC. 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Acta, 244, 56–85, doi: +10.1016/j.gca.2018.09.017 +Zaman, F. A., Townsend, L. W., de Wet, W. C., Looper, M. D., Brittingham, J. M., Burahmah, N. T., +et al. (2022). Modeling the lunar radiation environment: a comparison among FLUKA, Geant4, +HETC-HEDS, MCNP6, and PHITS. Space Weather, doi: 10.1029/2021SW002895 +Zhang, S., Wimmer-Schweingruber, R. F., Yu, J., Wang, C., et al. (2020). First measurements of the +radiation dose on the lunar surface. Sci. Adv., doi: 10.1126/sciadv.aaz1334 +Zoennchen, J. H., Nass, U., Fahr, H. J., and Goldstein, J. (2017). The response of the H geocorona +between 3 and 8 Re to geomagnetic disturbances studied using TWINS stereo Lyman-α data. +Ann. Geophys., doi: 10.5194/angeo-35-171-2017 +Zoennchen, J. H., Connor, H. K., Jung, J., Nass, U., and Fahr, H. J. (2022). Terrestrial exospheric +dayside H-density profile at 3–15RE from UVIS/HDAC and TWINS Lyman-α data combined. +Ann. Geophys., doi: 10.5194/angeo-40-271-2022 + + + + +Lunar Gateway Space Plasma Physics + +38 +This is a provisional file, not the final typeset article + +1 +Data Availability Statement +N/A. +2 +Figures + + + + + Lunar Gateway Space Plasma Physics + +39 + +FIGURE 1 | Moon’s orbit with respect to the Earth’s magnetosphere. Earth’s and Moon’s sizes are +not on scale. (Adapted from: Tim Stubbs / University of Maryland / GSFC). + +Moon'sOrbit +magneticbowshock +solarwind +Earth's magnetotail +Sun +Earth + +Lunar Gateway Space Plasma Physics + +40 +This is a provisional file, not the final typeset article + +FIGURE 2 | Moon’s environment with the complex interaction between solar radiation, space +plasma, meteoritic flux, dust, exosphere and the surface (Credit: Jasper Halekas). + +Horizontaland/orVertical +EXOSPHERE +DustTransport +Wake Boundary +Plasma +Photoelectrons +Terminator ++10sofVolts +Region +Sheath +Photons(UV) +Photon-Driven +Photon-Stimulated +Solar Wind +Positive Chasging +Desorption +SUN +Spuli +SolarStorms +Magnetosphere +-1000sof Volts +Electron-Driven +Negative Charging +Chemical,Thermal +MeteoriclInflux +Release&Loss +LargeImpacts +inleriorOutgassing + + Lunar Gateway Space Plasma Physics + +41 + + +FIGURE 3 | Typical SEP (Solar Energetic Particles) proton intensities: five-minute averages of +proton intensities measured by GOES-13/EPS/HEPAD during the May 2012 solar events. (From: +Quinn et al., 2017). + +10 +May +MeV)-1 +10 +0.7-4MeV +10° +4-9MeV +102 +9-15MeV +(cm2 +15-40MeV +10 +38-82MeV +Proton Intensity +84-200MeV +110-900MeV +10 +330-420MeV +420-510MeV +10 +510-700MeV +>700MeV +10 +10136 +138 +140 +142 +144 +D0Y2012 + +Lunar Gateway Space Plasma Physics + +42 +This is a provisional file, not the final typeset article + + +FIGURE 4 | Typical GCR (Galactic Cosmic Rays) Hydrogen nuclei (left) and Oxygen nuclei (right) +fluxes. (From: Mrigakshi et al., 2012). + +5000 +Badhwar-0'Neill2010 +O'Neill2010 +50 +CREME2009 +Hydrogen nucleil +1.4 +2009 +Oxygen nuclei +CREME96 +TE96 +4500 +Jsoskin +80 +Burger +ACE/CR +40 +BESS +RelativeNMcountrate +CAPRICE1 +1.2 +4000 +IMAX +30 +PAMELA +60 +Relative NM +Relative +3500 +RelativeNM countrate +20 +sec +Ot +NM +10 +0.8 +xn +Count +Flux +0 +Count Rate +20 +2500 +ted +Integrated I +0.6 +-10 +Rate +20 +1500 +0.4 +[%] +20 +-30 +1000 +0.2 +-40 +a +40 +c +500 +1970 +1975 +1980 +1985 +1990 +1995 +2000 +2005 +2010 +1970 +1975 +1980 +1985 +1990 +1995 +2000 +2005 +2010 + + Lunar Gateway Space Plasma Physics + +43 + + +A + + + + + + + +B + + +FIGURE 5 | (A): Illustration of the effects of a hydrated layer of lunar regolith in the production of +GCR albedo (secondary) protons. The nuclear evaporation process from deep in the regolith produces +abundant secondary particles in all directions. (From: Schwadron et al., 2016). (B): Energy spectra of +pristine GCR species (dashed lines) and of lunar albedo species (continuous lines), calculated with +the Geant4 simulation toolkit. (From: Looper et al., 2013). + +GCR +Enhancedalbedoprotons +Hydrated layer +Nuclear +Proton +evaporation +Neutron-GCRprotons +GCRalphas +10 +GCRheavyions +Albedoelectrons +Albedopositrons +secMeV/nucleon) +Albedoprotons +10 +Albedoneutrons +Albedogammas +Albedo light ions +Albedoheavyions +10-6 +10-1 +100 +101 +102 +103 +104 +105 +Energy,MeVorMeV/nucleon + +Lunar Gateway Space Plasma Physics + +44 +This is a provisional file, not the final typeset article + +FIGURE 6 | Cold O+ beam fluxes, observed by the Geotail spacecraft in the magnetotail lobe and +plasma sheet boundary layer, versus the tailward distance from the Earth (XGSM in RE). The Moon is +at XGSM ≈ -60 RE (Earth radii). (From: Seki et al., 1998). + +(a) St +Observed O+ Flux: St +[cm-2s-1] +in the lobe/mantle +10° +104 +10 +[X(21 : +ON = 1S +102 +2 +0 +-50 +-100 +-150 +-200 + + Lunar Gateway Space Plasma Physics + +45 + +FIGURE 7 | Energy distributions of H+ and O+ ions measured by the IMA sensor onboard the +Kaguya lunar orbiter in the terrestrial magnetotail. During the plasma sheet encounter (top panel) +there is an enhancement of high-energy (1 – 10 keV) O+ ions, in comparison to those measured in the +magnetotail lobe (bottom panel). The calculated density and net flux of these magnetospheric O+ +ions, during the plasma sheet encounter, were 1.2 × 10−3 cm−3 and 2.6 × 104 cm−2 s−1 respectively. +(From: Terada et al., 2017). + +105 +8:00-16:00 on 21 April 2008 +Plasma sheet +104 +●H* +0+ +103 +Flux (eV-1 +Lunar o+ +102 +中 +更更 +101 +Magnetospheric o* +100 +105 +16:00-24:00.on21April2008 +104 +Lobe ++H. +(rSz +103 +Lunar O+ +102 +101 +亚 +中中 +100 +100 +101 +102 +103 +104 +Energy (eV ql) + +Lunar Gateway Space Plasma Physics + +46 +This is a provisional file, not the final typeset article + +FIGURE 8 | MHD Open Global Geospace Circulation Model simulations (backward particle +tracing) suggest how heavy ions, observed in the Moon environment during high geomagnetic +activity events (at XGSE ≈ -60 RE), can be originating from the inner magnetosphere. Earth-to-Moon +transport times are ~2 – 3 hours. (From: Poppe et al., 2016). + +20 +10 +GSEJ +[Re +0 +-10 +-20 +20 +-20 +-40 +-60 +X [Re GSE] + + Lunar Gateway Space Plasma Physics + +47 + +FIGURE 9 | Left: Lunar exosphere density profiles for the atoms and molecules thermally released +from surface; based on the exospheric surface densities from Stern (1999). Right: Lunar exosphere +density profiles for the atoms released through sputtering. Both calculations are done for the sub- +solar point. (From: Wurz et al., 2007). + + +10" +1012 +H Density +f.-2.85.1012ms +H2 Density +SW +ODensity +Na Density +He Density +EV += 670km/s +• CH4 Density +Ai Density +Mg Density +1010 +- co Density +10 to +Kreep soils +Si Density +CO2 Density +- K Density +Ar Density +OH Density +- Ca Density +10° +108 +Ti Density +[gw] +Kr Density +[g-w] +.- Cr Density +Xe Density +Mn Density +Fe Density +Density +106 +Density +10° +104 +10* +100 +100 +10* +105 +1 +10 +100 +1000 +10 +10 +100 +1000 +10* +10° +Altitude[km] +Altitude [km] + +Lunar Gateway Space Plasma Physics + +48 +This is a provisional file, not the final typeset article + +FIGURE 10 | Typical energy spectra of the solar wind ions (right side, open squares) and of the +corresponding reflected from the lunar regolith energetic hydrogen atoms (left side, open circles), +measured by the SARA instrument onboard the Chandrayaan-1 spacecraft. Note the good correlation +between the reflected energetic neutral flux and the solar wind flux variations. (From: Wieser et al., +2009). + + +Solar wind +1X10 +Refle +ev +Flux [cm" +2 +r +1×10 +1: +100 +1000 +Energy [eV, eV/q] + + Lunar Gateway Space Plasma Physics + +49 + +FIGURE 11 | Top: Image of the central region of the Reiner Gamma Formation lunar swirl, taken by +Lunar Reconnaissance Orbiter. Bottom: A slice of the relative solar wind proton density above this +lunar swirl obtained from a 3D simulation, with the initial magnetic field lines corresponding to a +single subsurface dipole. (From: Bamford et al., 2016). + +Observations +Reiner Gamma +~10km +@NASALRO +Simulations +~130c/Wpe +@ E.P. Alves, R. A. Bamford + +Lunar Gateway Space Plasma Physics + +50 +This is a provisional file, not the final typeset article + +FIGURE 12 | Gateway configuration. PPE: Power and Propulsion Element; HLS: Human Landing +System; HALO: Habitation and Logistics Outpost (Credit: NASA). + +Logistics +Module +PPE +DSXR ++X +ESPRIT Refueler +HALO +MPM +International +HLS +Habitat +HLS Descent +Docking +Element +Adapter +US Habitat +HLS Ascent +Element +Orion +HLS Transfer +Element + + Lunar Gateway Space Plasma Physics + +51 + +A + + + + + +B + + + +C + + + + + +D + + + +FIGURE 13 | Top row: SPIS (Spacecraft Plasma Interaction System) simulation of the volume +electrostatic potential distribution (in V) in the solar wind, with the Gateway aligned to the solar +direction. (A): Full potential scale. (B): Potential scale saturated at +5 / -3 V, to highlight the +potential values away from the solar panels. Bottom row: Proton density (in m-3) in the solar wind, +with the Gateway aligned to the solar direction (C) and the Gateway main axis tilted by 20° with +respect to the solar direction (D). Note the plasma wake downstream of the station. + + +9,13 +-4,66 +-18,5 +-32,2 +46,05,00 +3,00 +1,00 +-1,00 +3,006,74e+06 +5,05e+06 +3,37e+06 +1,68e+06 +0,006.74+006 +5.05e+006 +3.37e+006 +1.68e+006 +0.000 + +Lunar Gateway Space Plasma Physics + +52 +This is a provisional file, not the final typeset article + +A + + + + + +B + + + +C + + + + + +D + + + +FIGURE 14 | SPIS simulation of the volume electrostatic potential distribution (in V) in the +terrestrial magnetotail (A), with the Gateway aligned to the solar direction, and photoelectron density +(log scale) under the same conditions (B). H+ ion density (C) and O+ ion density (D) in the terrestrial +magnetotail, both in m-3, showing the plasma wake downstream of the station. + + +13,0 +1,92 +-9,14 +-20,2 +31,38.85 +5.89 +2.93 +-0.0369 +-3.002.00e+006 +1.51e+006 +1.01e+006 +5.17e+005 +2.19e+0047.52e+003 +5.01e+003 +2.5/e+003 +0.000 + + Lunar Gateway Space Plasma Physics + +53 + +FIGURE 15 | Quality of the different positions on the Gateway for placing the space plasmas +instruments. + +Logistics +Module +PPE +DSXR ++X +ESPRITRefueler +HALO +MPM +International +HLSDescent +HLS +Habitat +Docking +Element +Adapter +US Habitat +HLSAscent +Element +Orion +HLSTransfer +Element + +Lunar Gateway Space Plasma Physics + +54 +This is a provisional file, not the final typeset article + +FIGURE 16 | Main and Secondary Instrument Platforms, mounted on the +X side and on the –X side +respectively of the Logistics Module. + +HLS +TransferElement +HLS +DescentElement +I-Hab +MainInstrumentPlatform +Orion +US-Hab +Halo +PPE +Logistics ++Z +Module +SecondaryInstrument Platform + + Lunar Gateway Space Plasma Physics + +55 + +A + + +B + +FIGURE 17 | The two-sided Main Instrument Platform, mounted on the Logistics Module (A), and +in perspective view (B). The “magenta cube”, on the side of the Logistics Module, is the +“standalone” cGCRD instrument. + + +Main Instrument Platform: +X +- Z side +Logistics +Module ++Z +Logistics +Module +. +Main Instrument Platform: ++ Z side + +Lunar Gateway Space Plasma Physics + +56 +This is a provisional file, not the final typeset article + + +FIGURE 18 | Main Instrument Platform –Z side, with all booms deployed. + +CMESP ++X +FOV +Module +cSWFC +0.9m +0.8 m +10.8m +CMAGF +boom-root sensors +(2 retractable booms) +X-Y +Combined +FOV +cLPEF / cMAGF +(turntable) +CMENA +boom-tip sensor + + Lunar Gateway Space Plasma Physics + +57 + + +FIGURE 19 | Main Instrument Platform +Z side. The turquoise solid angle in the cSWIS instrument +inset (upper left) represents the instrument field-of-view (FOV). + + +cSWIS ++Z +X +Module ++X +FOV ++Z +FOV +CLENA +X-Y +cMISP +FOV +Electrostatic +Analyzer +Scattering +MCP +Deviation +Detection +Electrodes +MCP +Front-End +Board +Digital +Board + High-Voltage +Board +Data & +Power +cWAVE + +Lunar Gateway Space Plasma Physics + +58 +This is a provisional file, not the final typeset article + +FIGURE 20 | Main Instrument Platform +X / –Y edges. The cENPD instrument is mounted on the +edge of this platform, to get unobstructed view to both the +Z and –Z directions. + +CLPEFI +CMAGF +CENPD +-Z +FOV +CMESP ++X +CSWFC ++Z +CMENA +(out) +(down) +CMISP +CSWIS ++Z +FOV +CENPD +CWAVE +(mountedonthe-Yedgeoftheplatform) + + Lunar Gateway Space Plasma Physics + +59 + +FIGURE 21 | The two-sided Secondary Instrument Platform, mounted on the Logistics Module. + +Secondary Instrument Platform: +- Z side +cUVIS +Module +X ++Z +FZ +Z +Module +Logistics +CHENA +Secondary Instrument Platform: +7 ++ Z side + +Lunar Gateway Space Plasma Physics + +60 +This is a provisional file, not the final typeset article + +FIGURE 22 | cGCRD mounted as a “standalone” instrument on the Logistics Module. + ++Z +FOV +Module +Logistics +200mm +CGCRD +Secondary +Instrument +Platform +Main +园 +Instrument +Platform +200mm ++2 ++X + + Lunar Gateway Space Plasma Physics + +61 + +FIGURE 23 | cENPD instantaneous FOVs of the two oppositely directed sensor heads, near +periapsis. Purple FOV: pristine energetic particles flux. Yellow FOV: Moon albedo energetic +particles flux. The magenta line is the track of the center of the FOV along the Gateway orbit. + +ORG + +Lunar Gateway Space Plasma Physics + +62 +This is a provisional file, not the final typeset article + +FIGURE 24 | cGCRD FOV. Left: cGCRD FOV near periapsis (light blue cone), dominated by the +albedo GCR particles from the Moon (grid sphere). Right: projection on the sky of the cGCRD FOV +along the Gateway orbit (in magenta). + +?#2 - SkyView +Timestopped +oon +Latitude = -98.0233 ; Longitude = 123.8372 + + Lunar Gateway Space Plasma Physics + +63 + +A + + + + + +B + + + +C + +FIGURE 25 | (A): Field-of-regard (total accessible FOV, taking into account the rotation of the 1- +axis articulation on which the instrument is mounted) of the cMENA instrument, at a given point of +the orbit. The field-of-regard (FOR), as projected on the sky, is shown in magenta. (B): cLENA FOV +(in magenta) near periapsis. The yellow line is the track of the center of the FOV, for the portion of +the orbit close to periapsis, as projected on the sky. (C): Field-of-regard of the cUVIS instrument, in +magenta, as projected on the sky. + +1800 +150° +120° +900 +009 +300 +00 +006 +600 +LO9olXsat +Moon +Earth +LOPG_Ysat +30° +LOPG_Zsat1800 +150° +120° +006 +600 +300 +00 +o06 +600 +0 +Earth* +00 +LoPolXsat +30° +bl +LOPG Zat +Moon1800 +1509 +120° +006 +60° +30° +00 +-300 +-600 +o06- +-120° +006 +Earth +Moon +LoPolXsat +30 +LOPG_Ysat +LOPG Zsat \ No newline at end of file diff --git a/CtE0T4oBgHgl3EQfQQAs/content/tmp_files/load_file.txt b/CtE0T4oBgHgl3EQfQQAs/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..da0eacbdafa71b0b7891b43a032c46e8ddb56b60 --- /dev/null +++ b/CtE0T4oBgHgl3EQfQQAs/content/tmp_files/load_file.txt @@ -0,0 +1,3299 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf,len=3298 +page_content='Space Plasma Physics Science Opportunities for the Lunar Orbital Platform - Gateway Iannis Dandouras1*, Matt G.' 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Oxfordshire,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' UK 6Mullard Space Science Laboratory / UCL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Holmbury St Mary,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' UK 7LATMOS (Laboratoire Atmosphères,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Milieux,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Observations Spatiales) / IPSL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Paris,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' France 8TU-Braunschweig,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Braunschweig,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Germany 9Institute for Space Astrophysics and Planetology / INAF,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Rome,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Italy 10Imperial College,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' London,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' UK 11Institute of Atmospheric Physics / CAS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Prague,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Czechia 12scibit s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', Liberec, Czechia 13Space Research Institute / Austrian Academy of Sciences, Graz, Austria 14CNES, Toulouse, France 15Charles University, Prague, Czechia 16Max Planck Institute for Solar System Research, Göttingen, Germany 17Astronomical Institute / CAS, Prague, Czechia 18Artenum, Ramonville Saint-Agne, France 19ONERA, Toulouse, France Correspondence: Corresponding Author Iannis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='Dandouras@irap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='omp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='eu Keywords: Moon, Gateway, deep space, space plasmas, heliophysics, space weather Submitted to "Frontiers in Astronomy and Space Sciences" 09 Dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' 2022 frontiers Lunar Gateway Space Plasma Physics 2 This is a provisional file, not the final typeset article Abstract The Lunar Orbital Platform - Gateway (LOP - Gateway, or simply Gateway) is a crewed platform that will be assembled and operated in the vicinity of the Moon by NASA and international partner organizations, including ESA, starting from the mid-2020s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' It will offer new opportunities for fundamental and applied scientific research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The Moon is a unique location to study the deep space plasma environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Moreover, the lunar surface and the surface-bounded exosphere are interacting with this environment, constituting a complex multi-scale interacting system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' This paper examines the opportunities provided by externally mounted payloads on the Gateway in the field of space plasma physics, heliophysics and space weather, but also examines the impact of the space environment on an inhabited platform in the vicinity of the Moon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' It then presents the conceptual design of a model payload, required to perform these space plasma measurements and observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' It results that the Gateway is very well-suited for space plasma physics research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' It allows a series of scientific objectives with a multi-disciplinary dimension to be addressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' 1 Introduction The Moon is a unique location to study the deep space plasma environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' During most part of its orbit around the Earth the Moon is directly exposed to the solar wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Due to the absence of a substantial intrinsic magnetic field and of a collisional atmosphere, solar wind and solar energetic particles (SEPs) arrive almost without any deviation or absorption and impact directly on its surface, interacting with the lunar regolith and the tenuous lunar exosphere (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Geiss et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Futaana et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The same phenomenon occurs also with the galactic cosmic rays (GCRs), which present fluxes and energy spectra typical of interplanetary space (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Sohn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Downstream from the Moon, a structured plasma umbra and penumbra region is formed, characterized by the gradual decrease of the ion and electron densities (Bosqued et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Nishino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The Moon’s vicinity is an ideal environment to study galactic cosmic rays, solar wind and solar energetic particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' This environment is typical of deep space (Plainaki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2016), apart from the fact that the Moon itself forms an obstruction to the GCRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' During 5 – 6 days every orbit, however, the Moon crosses the tail of the terrestrial magnetosphere (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' It is then exposed not to the solar wind but to the terrestrial magnetotail plasma environment, offering the possibility to study in-situ magnetotail dynamics and its dependence on solar and geomagnetic activity (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Kallio and Facskó, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Kallio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Phenomena such as plasmoids released from the near-Earth magnetotail and propagating anti-Sunward, bursty bulk flows (BBFs), energetic particle bursts, plasma waves, magnetic reconnection and plasma sheet dynamics can thus be studied in-situ (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Parks et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Nakamura, 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Taylor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Nagai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Du et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Grigorenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Sitnov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Kronberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The Moon is then also very well situated to study atmospheric escape from the Earth into space (Lammer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Harnett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Wei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Dandouras, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' André et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2021), in the form of heavy ions upwelling from the terrestrial ionosphere and transported and lost into the deep magnetotail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The wealth of data supplied from the THEMIS- ARTEMIS and from the Kaguya (SELENE) spacecraft confirmed the observation of such ions, of terrestrial origin, in the lunar environment (Poppe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Terada et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The THEMIS- ARTEMIS data, however, did not include the crucial information on the plasma composition (Angelopoulos, 2011), and the Kaguya plasma measurements where limited to a less than two-year mission and to low-energy plasma (Saito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Far magnetotail studies performed by the Lunar Gateway Space Plasma Physics 3 Geotail spacecraft supplied key information on the dynamics of ion beams streaming downtail (Christon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 1994, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Seki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 1998), but lacked the ion composition measurements at low energies (below ~10 keV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' When the Moon gets again outside of the magnetotail, terrestrial magnetosphere dynamics can be monitored through remote sensing, using a variety of magnetospheric imaging techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' These include Energetic Neutral Atom (ENA) imaging, which conveys information on the interaction between energetic ions and the terrestrial exosphere (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' C:son Brandt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Vallat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2004), solar wind charge exchange X-rays imaging of the interaction between the solar wind / magnetosheath plasma and the terrestrial exosphere (Branduardi-Raymont et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Sibeck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2018), plasmasphere EUV imaging (Sandel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2003), or exosphere Lyman-α imaging (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Zoennchen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' But, most important, the lunar environment offers a unique opportunity to study the Moon surface- bounded exosphere (Figure 2), its production mechanisms, its dynamics, its interaction with the solar wind and with the terrestrial magnetotail plasma, and its escape into space (Potter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Wurz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2007, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Futaana et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Leblanc and Chaufray, 2011, Lammer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The LADEE (Lunar Atmosphere and Dust Environment Explorer) and LRO (Lunar Reconnaissance Orbiter) observations have provided a glimpse of the complexity of the lunar exosphere and of the associated physical mechanisms (Stern et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Elphic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Benna et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Hodges, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Hurley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The lunar surface offers also exciting possibilities for studying energetic ion implantation in the lunar regolith (Ozima et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Ireland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2006), albedo energetic particles produced through the interaction of SEPs and GCRs with the regolith (Schwadron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2016, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2020), solar wind ion implantation or neutralization and reflection from the lunar regolith (Futaana et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2006, 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Vorburger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Tucker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2019), formation of hydrogen bearing molecules (McCord et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Stern et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' McLain et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' 2021) possibly including water (Schörghofer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2021), solar wind interaction with crustal magnetic anomalies (Poppe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Bamford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2016), lunar pickup ion generation (Poppe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2012a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2011), or lunar surface electrostatic charging and dust levitation (Stubbs et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Hess et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Popel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2018), just to mention few examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The analysis of implanted particles on the lunar surface, that originated from the Earth’s atmosphere, will also reveal some knowledge of Earth’s early atmosphere (Marty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Ozima et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Lammer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2018, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' It is expected that early Earth’s atmosphere experienced strong escape of hydrogen, oxygen and carbon, that originated from the dissociation of water and methane molecules, and of nitrogen due to the increased EUV flux from the young Sun (Lammer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Zahnle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Gebauer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Kislyakova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Johnstone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' As suggested by Marty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' (2003), nitrogen originating from the early Earth was implanted on the lunar surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' This is based on the strong variations of N, He, Ne and Ar noble gas isotope implantations into the regolith, of up to 30 % (Ozima et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' According to Marty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' (2003) and Ozima et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' (2005) these enhancements cannot be explained as due to solar wind implantation alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The Moon is also an ideal test case for studying planetary surface weathering resulting from the exposure to energetic particles, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' surface - energetic particle interactions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Hapke, 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Pieters and Noble, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Nénon and Poppe, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Given that the Moon is irradiated by GCRs quasi- uniformly, any differences in the resulting interaction, including the emitted albedo particles, point to the variable properties, physical or chemical, of the surface (Schwadron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2016, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' From the Lunar Gateway Space Plasma Physics 4 This is a provisional file, not the final typeset article perspective of the Gateway, surface - GCR interactions can be mainly probed through the albedo particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Following the legacy of the Apollo missions and of the more recent missions to the Moon (THEMIS- ARTEMIS, Kaguya, LADEE, LRO, Chandrayaan, Chang’e, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' ), a series of lunar missions is in preparation, or already operating, building on their outstanding heritage (Dandouras et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2020a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The Lunar Orbital Platform - Gateway (LOP - Gateway, or simply Gateway) is a crewed platform that will be assembled and operated in the vicinity of the Moon by NASA and international partner organizations, including ESA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Launch of the first modules will start in the mid-2020s (Phase 1), and it will continue with the launch and assembly of additional modules during the late 2020s (Phase 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The Gateway will provide support for all lunar activities, including the Artemis program to return humans to the Moon (Artemis III Science Definition Team Report, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' It will also offer new opportunities for fundamental and applied scientific research (Carpenter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Dandouras et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2020a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' In preparation of its scientific payload, ESA set up international science teams to prepare and to support the definition of payload studies, including a topical team in the field of space plasma physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' In the first part of this article (sections 2 and 3) we report on the outcome of this topical team, which was entitled “Space Plasma Physics Science Opportunities for the Lunar Orbital Platform - Gateway”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' This part focuses on the science objectives identified by the topical team (section 2), and on the corresponding instrumentation required to address them (section 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' In the second part (section 4) we present a conceptual design study for a “Space Plasma Physics Payload Package onboard the Gateway” (SP4GATEWAY) we undertook for ESA, addressing these objectives and compatible with the technical requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' 2 Specific Objectives and Goals The “Space Plasma Physics Science Opportunities for the Lunar Orbital Platform-Gateway” topical team, set up by ESA in 2019, brought together the key expertise required for defining the space plasma parameters to measure from lunar orbit, and the appropriate instrumentation required to perform these observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The science objectives that were identified include: 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='1 Monitoring the solar wind and the lunar energetic particle environment Due to the absence of a substantial intrinsic magnetic field and of a collisional atmosphere, the Moon is directly exposed to: − Solar Wind: ~keV particles − Solar Energetic Particles (SEPs): ~MeV particles − Galactic Cosmic Rays (GCRs): ~GeV particles The monitoring of the solar wind (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' von Steiger, 2008), at the lunar environment, aims to evaluate its role as a driver for the dynamics of the terrestrial and the lunar exospheres, of the dynamics of the terrestrial magnetosphere, and of the lunar surface sputtering and charging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The monitoring and characterization of the SEPs and GCRs, at lunar orbit, aims to evaluate the radiation environment of the Moon and also the role of SEPs and GCRs as lunar surface sputtering sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Since the Moon does not have a substantial magnetic field it is possible, with an appropriate particle detector, to measure the low energy part of the GCR spectrum (< 1 GeV) with high precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=" Lunar Gateway Space Plasma Physics 5 This offers an advantage with respect to low-Earth orbits, where most of the advanced GCR observatories like PAMELA and AMS-02 are located, where this low energy part is filtered out by the Earth's magnetosphere." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Typical SEP proton intensities, measured during a solar event, are shown in Figure 3 (adapted from Quinn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Some of the SEP protons (~MeV energy range) can interact, in the high solar corona, with partially stripped coronal ions, charge exchange with them and produce ~MeV ENAs (Energetic Neutral Atoms) (Mewaldt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' GCR Hydrogen and Oxygen nuclei fluxes are shown in Figure 4, presenting a clear solar cycle modulation (adapted from Mrigakshi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The interaction of these SEPs and GCRs with the lunar regolith produces albedo energetic particles, resolvable with current instruments up to a few ~100 MeV, and with fluxes that are sensitive to the regolith hydration (Looper et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Schwadron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Zaman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2022), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The separation of the pristine energetic particle fluxes from the albedo energetic particles (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' by zenith centered / nadir centered looking directions respectively) appears thus as a requirement, in order to provide information on the deep space SEP and GCR environment and on the interaction of the lunar regolith with this environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='2 Monitoring the terrestrial magnetosphere and exosphere When the Moon is within the terrestrial magnetotail, in-situ measurements of the plasma sheet and plasma sheet boundary layer dynamics are enabled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' These consist of magnetic field and energetic ion and electron monitoring, including the measurement of energetic ions of terrestrial origin streaming downtail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The evolution of the flux of O+ downtail streaming beams, as a function of the tailward distance from the Earth, is shown in Figure 6 (from Seki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' During high geomagnetic activity conditions these beams include heavy atomic and molecular ions (Christon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 1994, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Closer to the Moon, O+ downtail streaming beams have been observed by the Kaguya Lunar Orbiter (Terada et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The spectral characteristics of these streaming O+ ions show a clear distinction between the O+ ions of lunar origin (few 10 eV to ~100 eV) and the terrestrial magnetospheric O+ ions (few keV), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Figure 7 (adapted from Terada et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Particle tracing simulations performed by Harnett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' (2013) using a 3D multi-fluid model, and by Poppe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' (2016) using the MHD Open Global Geospace Circulation Model, show how heavy ions, originating from the Earth’s inner magnetosphere, can be ejected downtail during high geomagnetic activity events, reaching energies of several keV to several 10 keV at lunar distances, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' When the Moon is outside of the magnetotail, terrestrial magnetosphere dynamics and response to solar wind conditions can be monitored through remote sensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' This includes: − Ring current and near-Earth plasma sheet monitoring, by imaging of the ENAs produced by charge exchange between the plasma sheet or ring current energetic ions (few ~keV to few ~10 keV) with the geocorona neutral hydrogen atoms, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Brandt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' (2004), Vallat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' (2004), Goldstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' − Magnetopause and cusps monitoring by detecting and imaging the SWCX (solar wind charge exchange) soft X-rays produced by charge exchange between highly-charged heavy ions, originating from the solar wind, and the exospheric neutral atoms, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Branduardi-Raymont et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' (2012, 2021), Sibeck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' − Plasmasphere imaging, by resonant scattering of the solar EUV (30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='4 nm) by the plasmaspheric He+ ions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Sandel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' (2003), Darrouzet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' (2008), He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Lunar Gateway Space Plasma Physics 6 This is a provisional file, not the final typeset article − Geocorona imaging at Lyman-α (121.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='6 nm), e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Rairden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' (1986), Zoennchen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' (2017, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='3 Monitoring the Moon’s surface-bounded exosphere The Moon surface-bounded exosphere constitutes a complex multi-scale system (Figure 2), characterised by its interactions with the solar radiation, the solar wind and terrestrial magnetotail plasma, the meteoritic flux, dust, and the regolith (Futaana et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The low number densities of this very tenuous atmosphere, particularly of the minority species, and the complexity and multiplicity of the source and loss mechanisms have resulted in a poor understanding of it (Wurz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2007, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Poppe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Figure 9 provides the altitude density profiles of the major species, separately for the atoms and molecules released thermally from the regolith and for the atoms released through sputtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' In addition to atomic and molecular hydrogen, O, OH, CH4, noble gases (He, Ne, Ar, Kr, Xe), metallic atoms (Na, K, Mg, Al) and other elements populate the lunar exosphere (Leblanc and Chaufray, 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Benna et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Grava et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2015, 2016, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Halekas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Hodges, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Hurley and Benna, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Leblanc et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Wurz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' These neutral exospheric atoms and molecules can be subsequently ionized by the solar UV radiation and generate pickup ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' These ions are promptly accelerated from their birthplace by the ambient electric field E and drift across the magnetic field B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The unique orbital characteristics of the pickup ions (cycloidal motion consisting of a combination of E × B drift and a gyration around B) make it possible to infer important details about their sources (Hartle and Killen, 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Such lunar pickup ions have been detected in the terrestrial magnetotail lobes (Poppe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2012a) and in the solar wind (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' As the measurements performed onboard the LADEE and LRO spacecraft have shown, the lunar exosphere can be monitored, either by in-situ measurements, using a neutral mass spectrometer, or by remote sensing using a UV spectrometer (Chin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Elphic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Other techniques for studying the lunar exosphere are: − Remote sensing of the lunar exosphere by detecting and imaging the ENAs produced by charge exchange interactions between the solar wind protons and the exospheric neutral atoms (Futaana et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The energies of these ENAs are comparable to the energies of the parent solar wind protons, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' of the order of ~keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' − Remote sensing of the lunar exosphere by detecting and imaging the SWCX soft X-rays produced by charge exchange between highly-charged heavy solar wind ions and the exospheric neutral atoms (Robertson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' − In-situ measurement of freshly ionized pickup ions, originating from the lunar exosphere neutral species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' (Hartle and Killen, 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Yokota et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Poppe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' At high altitudes above the lunar surface, as those of the Gateway orbit (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='1), this method can provide higher sensitivity in the detection of low number density species than the direct sampling of the parent neutrals (Halekas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Poppe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='4 Monitoring the interaction of the solar wind with the Moon’s surface Solar wind protons, arriving at the Moon’s surface, can be absorbed, or scattered, or can remove another atom from the lunar regolith by sputtering or desorption (Wieser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' McComas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2009a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Futaana et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' It results that a large fraction of the solar wind protons, up to 20%, is reflected back to space as neutral hydrogen atoms (ENAs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' It is noteworthy that backscattering of Lunar Gateway Space Plasma Physics 7 neutralized solar wind protons occurs not only when the Moon is in the pristine solar wind, but also when the Moon enters into the terrestrial magnetosheath and is then exposed to the shocked and thermalized solar wind (Allegrini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Figure 10 shows typical energy spectra of the reflected hydrogen ENAs, compared to the parent solar wind protons energy spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' As shown, the flux of the reflected ENAs closely follows the variations of the flux of the parent proton population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The energies of these ENAs are however a fraction of the parent solar wind protons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Since the solar wind proton trajectories are modulated by the surface electrostatic potential and by the eventual local magnetic field anomalies (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Figure 11), the detection and imaging of these reflected ENAs provides a tool to investigate the lunar surface electric and magnetic fields (Futaana et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Vorburger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2013, 2015, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Bamford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Local crustal magnetic anomalies (or “swirls”) constitute “mini-magnetospheres”, shielding locally the lunar regolith from the solar wind protons and from the resulting space weathering (Wieser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Deca et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Glotch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Hemingway et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Poppe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Pieters and Noble, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Hemingway and Tikoo, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The solar wind protons that do not scatter back, but are absorbed in the lunar regolith (top 20 – 30 nm of the lunar grains), diffuse within the regolith.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' They can then interact with the oxygen atoms in the regolith oxides and form OH (McCord et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Farrell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Tucker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' McLain et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' These solar wind-produced hydroxyl radicals contribute to the formation and release of molecular water, and thus to a solar wind-induced water cycle on the Moon (Crider and Vondrak, 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Futaana et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Honniball et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The exposure of the lunar surface to the solar radiation and to the flux of charged particles results also in an electrostatic surface charging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' An electric potential thus develops between the lunar surface and the ambient plasma, which manifests itself in a near-surface plasma sheath with a scale height of the order of the Debye length (Halekas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Stubbs et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Burinskaya, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Harada et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' This near-surface electric field becomes very complex and highly variable in the vicinity of the terminator, with the surface polarity changing from mostly positive (few 10 V) on the dayside, due to photoelectron emission, to highly negative (of the order of the ambient electron temperature, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' up to several -100 V) on the nightside, and in the trailing lunar wake region (Farrell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Local surface topography is also a factor contributing to a complex near-surface electrostatic and plasma environment, particularly in the vicinity of permanently-shadowed craters (Poppe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2012b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Nénon and Poppe, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' As the THEMIS-ARTEMIS observations have shown, the lunar surface charging can be remotely sensed from a Moon orbiting spacecraft, even several 1000 km away from the lunar surface, through the shifted energy spectra of the detected plasma particles when the spacecraft crosses magnetic field lines connected to the lunar surface (Halekas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Dust is another component of the lunar plasma environment (Stubbs et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Grün et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Horányi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Popel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2018, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Dust grains on (or near) the lunar surface can either be ejected from the regolith, due to the impact of interplanetary micrometeoroids, or be electrostatically levitated due to grain charging, as discussed in the previous paragraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=" This creates a dusty plasma system consisting of neutrals of the lunar exosphere, solar-wind ions and electrons, ions and electrons of the Earth's magnetotail (when the Moon gets inside the terrestrial magnetotail), photoelectrons formed due to the interaction of the solar radiation with the lunar surface, and charged dust grains flying over the lunar surface." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Lunar Gateway Space Plasma Physics 8 This is a provisional file, not the final typeset article 3 Measurement Requirements Following the identification of the scientific objectives in the field of space plasma physics, that can be addressed using instrumentation onboard the Lunar Orbital Platform - Gateway (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' section 2), the ESA topical team identified the physical parameters needed to be measured in order to address these objectives, and the corresponding instrumentation required to perform these observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The topical team addressed thus the following two questions (Dandouras et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2020b): − What plasma physics science questions can be addressed in the vicinity of the Lunar Orbital Platform - Gateway?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' − What are the instrument / payload requirements to achieve such science?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' It identified measurements that can be performed either directly from the Gateway platform (3 200 × 70 000 km altitude lunar orbit), or from instrumented cubesats that could be released from the platform and placed into lower lunar orbits, or directly from the Moon surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Here we will focus on the measurements that can be performed by instrumentation mounted onboard the Gateway, and which can be either in-situ measurements or remote sensing observations, and then we briefly mention the other two possibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Space plasma physics measurements that could be performed directly from the Moon surface will be the object of a dedicated forthcoming paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Table 1 provides an overview of the physical parameters / observables identified, in the field of space plasma physics, that can that be monitored by instrumentation onboard the Gateway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Tables 2 and 3 are for the observations that could be performed, on a longer term, from lower lunar orbits and from the Moon’s surface, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Table 4 focuses then on the science questions that can be addressed from instrumentation onboard the Gateway, and shows how each science objective, identified by the topical team, translates into a measurement requirement, and then to the corresponding instrument / payload requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Additional objectives that could be eventually addressed by remote sensing instrumentation onboard the Gateway, and could point to targets of opportunity, include: aurora imaging, heliosphere imaging (through the ENA imager), and lunar surface imaging (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' meteor impact flashes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' 4 Conceptual design for a Space Plasma Physics Payload Package onboard the Gateway Following the work of the topical team, and the identification of the measurement requirements, ESA issued an Invitation to Tender for a “Deep Space Gateway Plasma Physics Payload Conceptual Design” (ESA AO/1-9789/19/NL/FC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' In response to it we proposed to ESA, were selected and then undertook a conceptual design study for a “Space Plasma Physics Payload Package onboard the Gateway” (SP4GATEWAY), addressing these objectives while being compatible with the technical requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The Gateway modules that are best-suited for hosting the in-situ measurement plasma instruments were first identified, following a simulation we performed of the interaction between the Gateway and its plasma environment (section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The proposed model payload, and its accommodation on the Gateway modules, are presented in sections 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='4 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The fields-of-view (FOVs) of the remote sensing instruments, as projected on the sky and on the celestial objects, were then analyzed by simulating their evolution along the Gateway orbit (section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Lunar Gateway Space Plasma Physics 9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='1 Gateway configuration, orbit and attitude The Gateway will evolve during its lifetime, different modules being added during the successive phases of the project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' For the purpose of this study we considered a typical “Gateway Phase 2” configuration, with the Orion spacecraft attached, shown in Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The Gateway orbit will be a Near Rectilinear Halo Orbit (NRHO) around the Moon, with periapsis × apoapsis altitudes 3 200 × 70 000 km (Whitley and Martinez, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The orbital period is ~6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='5 (Earth) days, and the orbital inclination ~90°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The periapsis will be above the north pole of the Moon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' This orbit provides constant Earth visibility (9:2 resonance with the lunar synodic period).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The Gateway attitude will be with the +X axis (longitudinal axis, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Figure 12) pointed towards the Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The +Z axis will be normal to the Moon orbit plane, pointing southwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The pointing accuracy requirement is that Orion remains in a tail-to-Sun attitude ±20°, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' the +X axis has a ±20° pointing accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='2 Simulation of the Gateway plasma environment The simulation of the interaction between the Gateway and the plasma environment was performed by ONERA and the Artenum company (Hess et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' A 3D mesh model with approximately 64 000 elements was developed to represent the Gateway, and the properties of the surface materials of the different Gateway modules were taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The SPIS (Spacecraft Plasma Interaction System) software tool was then used to simulate the Gateway interaction with its ambient plasma environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' This open-source software, available at https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='spis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='org, computes the potential at the surface of a spacecraft according to its exchange of charges with the space plasma, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' the collection of charge from the plasma and the re-emission of photoelectrons and of secondary electrons due to impacting energetic particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' It also simulates the perturbation induced by this electrostatic charging on the natural plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' This software was further developed to simulate the charging of the regolith and the motion of lunar dust particles (Hess et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2015) and to simulate the perturbation of the measurements by plasma instruments due to the charging (Sarrailh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Two cases were simulated, that correspond to the two situations that will be typically encountered: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Gateway in the solar wind (most frequent case, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' section 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Typical solar wind conditions considered were: solar wind density: 7 cm-3 solar wind velocity: 450 km/s ion and electron temperatures: 10 eV 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Gateway in the terrestrial magnetotail (5 – 6 days per lunar orbit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The plasma environment considered, corresponding to active geomagnetic activity conditions (conditions producing downtail plasma streaming, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='2), was: plasma density: 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='01 cm-3 H+ density: 2 cm-3 O+ density: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='01 cm-3 plasma streaming velocity: 250 km/s (away from the Earth) ion temperature: 200 eV electron temperature: 15 eV Lunar Gateway Space Plasma Physics 10 This is a provisional file, not the final typeset article In each case both a nominal Gateway attitude (Gateway major axis aligned to the solar direction, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='1) and an extreme attitude excursion, with the Gateway major axis tilted by 20° with respect to the solar direction, were considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The simulation runs generated, for each case, maps of the electrostatic potential (volume values in the Gateway environment and surface values on the Gateway modules), and maps of the density values of H+, O+, photoelectrons and secondary electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The details are given in the report by Hess et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The main results of this study are: 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='1 Gateway in the solar wind The volume electrostatic potential distribution, when the Gateway is in the solar wind and the major axis of the station is aligned to the solar direction (nominal attitude), following 600 s of interaction time, is shown in Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The Gateway structure gets to a 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='5 V equilibrium potential, while the major part of the solar panels goes to a 10 V potential on the Sun facing side and -46 V on the rear side (Figure 13A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The PPE (Power and Propulsion Element), bearing the two main solar panels, is thus inappropriate for space plasmas instrumentation for low-energy plasmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The wake effect, due to the solar wind flow, is particularly visible behind the main solar panels, whereas in the front modules of the station the potential perturbation appears to be moderate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' To highlight the potential values away from the solar panels (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' where the plasma instruments should be mounted), the surface and volume potentials are plotted also in a scale saturated between +5 V and -3 V (Figure 13B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' As shown there, the thickness of the sheath formed by the plasma flow around the Gateway, on the station parts exposed to the solar wind and away from the solar panels, is typically ~1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='8 m and the electrostatic potential perturbation is moderate (a few volts).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' This implies that the effect on the ion and electron measurements will be very moderate, and only the lowest energy particles (< ~100 eV) will be affected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Solar wind ions, which have energies of typically ~1 keV, will be almost not affected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' It implies also that a boom of ~2 – 3 m length is adequate for placing sensors as a magnetometer and a wave antenna outside of the sheath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The ambient proton density around the Gateway is shown in Figures 13C and 13D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The left panel (Figure 13C) corresponds to the nominal Gateway attitude (Gateway major axis aligned to the solar direction), whereas the right panel (Figure 13D) corresponds to an extreme attitude excursion of 20° with respect to the solar direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Note, in both cases, the plasma wake downstream of the station.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The tilted axis simulations show a small asymmetry between the illuminated and the shadowed sides of the Gateway and, as expected, a tilted plasma wake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='2 Gateway in the terrestrial magnetotail Here the Gateway is exposed to the terrestrial plasma sheet / magnetosheath plasma streaming downtail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The volume electrostatic potential distribution, under these conditions and when the major axis of the station is aligned to the solar direction (nominal attitude), following 400 s of interaction time, is shown in Figure 14A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The surface equilibrium potential here is 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='5 V, while the major part of the solar panels goes to a 13 V on the Sun facing side and -31 V on the rear side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Due to the lower density of the ambient plasma, the sheath forming around the station is more extended, but the potential barrier is weaker (-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='1 V) and more isotropic, compared to the solar wind case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' However, the overall results are not very different and the conclusions made in the solar wind case apply also here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Figure 14B shows the emitted photoelectron density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Lunar Gateway Space Plasma Physics 11 The ambient H+ and O+ ion densities (terrestrial ions streaming downtail during active geomagnetic conditions, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='2) are shown in Figures 14C and 14D respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The O+ density distribution shows here a high similarity with the proton density distribution in the solar wind, presenting a very clear wake effect due to the higher ion mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='3 Gateway - plasma environment interaction: synthesis The interaction of the Gateway with its plasma environment has been simulated for the two cases that will be encountered: Gateway in the solar wind and Gateway in the terrestrial magnetotail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' In both cases the surface potential of the Gateway away from the solar panels is moderate (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='5 V in the solar wind and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='5 V in the magnetotail).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' A sheath is formed by the plasma flow around the Gateway, which for the solar wind case has a thickness of ~1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='8 m when the Gateway is aligned to the solar direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' However, when the Gateway major axis (X-axis) is tilted by 20° with respect to the solar direction, which corresponds to an extreme excursion from the nominal attitude, this plasma sheath becomes asymmetrical and much thicker in the “shadowed” side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' These results are very encouraging, because they allow to identify the Gateway modules on which the perturbation of the natural plasma environment by the Gateway will be minimal, and are thus well-suited for placing the plasma instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Figure 15 shows the positions identified for instrument mounting in a color code, from green (most favorable) to red (least favorable positions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The US Habitat and the International Habitat present small surface charging, are surrounded by a thin plasma sheath and do not suffer from any plasma wake effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' They are thus suitable for placing the plasma instruments sensitive to electrostatic charging, as the magnetospheric ion and electron spectrometers (green / light green markers in Figure 15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' However, these positions on the cylindrical surfaces of the US Habitat and of the International Habitat are tangent to the solar wind flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' When the Gateway is tilted with respect to the solar direction, the solar wind flow is detached and thus not measurable from these positions (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Figure 13D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Solar wind measurements require, not only limited (less than ~10 V) surface charging and absence of local plasma wake effects, but also a direct “face exposure” to the solar wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The +X side of the Logistics Module (lower light green marker in Figure 15) is thus the most suitable position for the solar wind instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Concerning the wave and field instruments, their positioning on ~2 – 3m booms, on the “green / light green markers”, allows having them outside of the plasma sheath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The remaining positions can be used for energetic particle and magnetospheric imaging instruments, which are not sensitive to plasma charging effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The least favorable positions for placing plasma instruments (positions to avoid) are the PPE (Power and Propulsion Element) and the close to it HALO (Habitation and Logistics Outpost), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' red markers in Figure 15, due to the large solar panels and associated circuitry, their “downstream” positioning (with respect to the solar wind flow), the high surface charging, and the proximity to the ion propulsion engine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='3 Model payload In order to address the scientific objectives identified by the topical team, we first defined a model payload, consisting of a suite of instruments corresponding to the requirements shown in Table 4 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' section 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' These measurement instruments are largely based either on existing flight-proven Lunar Gateway Space Plasma Physics 12 This is a provisional file, not the final typeset article instruments, adapted here for the lunar plasma environment, or on tested and validated laboratory prototypes (TRL (Technology Readiness Level) ≥ 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Table 5 lists these instruments and provides an overview of their main characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The detailed description of the characteristics of the instruments is given in a series of three ESA reports, corresponding respectively to a Requirements Inventory (De Keyser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2020), Conceptual Design Report (Devoto and Dandouras, 2020), and Programmatic Assessment (Futaana et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Here we present their principal characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='1 cMAGF: 3-axis Fluxgate Magnetometer This instrument will provide the ambient vector magnetic field (in solar wind, terrestrial magnetotail, Moon vicinity, lunar wake, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The proposed magnetometers package consists of three different types of units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The main one has three pieces of boom-mounted 3-axis fluxgate magnetometers, on two ~3 m retractable booms: one sensor at the tip of each boom, and a third sensor at the common root of the two booms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' This allows having two main sensors outside the plasma sheath formed by the plasma flow around the Gateway (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='2), while the boom-root sensor provides the possibility for removing eventual Gateway-induced perturbations by using the gradiometer technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The presence of two boom-tip mounted sensors, on two booms, provides for further corrections for eventual perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' These three units will provide the main measurements while supporting the cleaning and processing of the measured data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' In order to monitor the perturbations from the station in more detail, several single magnetometer sensors will be mounted on various places directly on the station (Constantinescu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Additionally, we propose two current monitors, monitoring the currents flowing from the solar panels, which are expected to contribute the largest magnetic field perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The proposed accommodation on the Gateway of this magnetometer package, and the corresponding CAD figures, are shown as also for the other instruments in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Fluxgate magnetometers benefit from a strong heritage, as such instruments have flown on several space missions, including Cluster (Balogh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2001), Cassini (Dougherty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2004), THEMIS- ARTEMIS (Auster et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2008), BepiColombo (Glassmeier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2010), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='2 cSWIS: Solar Wind Ion Spectrometer The cSWIS instrument is a solar wind ion spectrometer that will determine the velocity distribution functions (VDFs) of the solar wind ions and will provide the solar wind density, velocity and temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' A top-hat electrostatic analyzer instrument is considered, covering the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='1 – 40 keV/e energy range and having a field-of-view (FOV) of 96° × 48° aligned with the solar wind arrival direction: 96° angular range in azimuth (+24° to -72° in the ecliptic plane, as the Gateway points to the Sun but it may sometimes drift away from this direction, after which it catches up) and 48° in elevation (between -24° to +24°).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The spatial resolution is 3° in both azimuth and elevation, and the energy resolution is ΔE/E = 8%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' In order to achieve high VDF acquisition cadence, we propose to use solar wind beam tracking, along the lines of the Cold Solar Wind (CSW) instrument (Cara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' De Keyser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2018), that was designed for the THOR (Turbulent Heating ObserveR) mission which was proposed to ESA as a medium-class M4 mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Lunar Gateway Space Plasma Physics 13 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='3 cSWFC: Solar Wind Faraday Cup The cSWFC instrument will be used to determine the solar wind density, velocity and relative alpha- particle content, based on simultaneous measurements of the collector currents provided by six identical Faraday cups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The energy of incoming ions is determined by the high voltages applied onto the control grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The Faraday cups are organized into three units, each of them containing two cups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' One unit serves for the determination of the total ion flux vector, the second unit uses high voltages applied on the control grids and provides two points of energy distribution that are used for the determination of the proton velocity and temperature in the Maxwellian approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The last unit serves for the measurement of the 1D velocity distribution (integral distribution) of protons and alpha particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Each of the six Faraday cups will have a 45° × 45° FOV which, as for the cSWIS instrument, will be aligned with the solar wind arrival direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The energy resolution is 1% (< 50 eV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The proposed cSWFC instrument is based on the BMSW (Bright Monitor of the Solar Wind) Faraday cup instrument, that flew onboard the Spektr-R mission (Šafránková et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='4 cMISP: Magnetospheric Ion Spectrometer cMISP is a mass-discriminating ion spectrometer, that determines the velocity distribution functions of the ambient plasma ions: terrestrial magnetosphere ions, lunar exosphere pickup ions and solar wind ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The proposed instrument is a time-of-flight ion mass spectrometer capable of obtaining ion distributions (about 10 eV/e to 40 keV/e) with a high-resolution mass-per-charge composition determination (m/Δm > 15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Ions are selected as a function of their E/q (energy per charge) ratio, by sweeping the high voltage applied between the two hemispheres of a rotationally symmetric toroidal electrostatic analyzer (360° ×5° instantaneous FOV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Then they go through a post-acceleration of about 5 kV and they subsequently enter into the time-of-flight (TOF) section, where the velocity of the incoming ions is measured, which allows then the calculation of their m/q (mass per charge) ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' A specially designed thin microchannel plate (MCP), through which the ions pass, is used as a conversion surface for the production the “start” TOF signal secondary electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The “stop” TOF signal is provided by the ion detection on another MCP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The instrument provides for a ΔE/E ~7 % energy resolution and a 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='5° angular resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' cMISP is based on the MIMS (MCP Ion Mass Spectrometer) instrument, that was designed for the ESCAPE (European SpaceCraft for the study of Atmospheric Particle Escape) mission, which was proposed to ESA as a medium-class M5 mission (Dandouras et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' MIMS in its turn was based on a successfully tested prototype developed at IRAP (Devoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' MIMS is an evolution of the CIS-CODIF instrument, flying onboard Cluster (Rème et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2001), but with higher mass resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Since MIMS was designed for a spinning spacecraft, where it would take advantage of the spacecraft rotation to obtain a full 3D ion distribution within one spacecraft spin, cMISP on the Gateway, which is a 3-axis stabilized space station, requires the addition of electrostatic deflection plates at the instrument entrance to scan the FOV over a 360° ×120° solid angle (±60° with respect to the central entrance plane).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Lunar Gateway Space Plasma Physics 14 This is a provisional file, not the final typeset article 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='5 cMESP: Magnetospheric Electron Spectrometer cMESP is an electron spectrometer that will determine the velocity distribution functions (VDF) of the solar wind electrons (pristine or reflected from lunar crustal magnetic field anomalies) and of the plasma sheet electrons, when the Gateway gets into the terrestrial magnetosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' A top-hat electrostatic analyzer instrument covering the ~5 eV to ~20 keV energy range is proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' As for the cMISP instrument, the addition of electrostatic deflection plates at the instrument entrance to scan the FOV over a 360° ×120° solid angle is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The proposed cMESP instrument is based on the SWEA (Solar Wind Electron Analyzer) instrument, flying onboard the MAVEN spacecraft (Mitchell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='6 cENPD: Energetic Particles Detector The cENPD instrument will detect and measure the fluxes of the energetic charged particles, ions and electrons: Solar Energetic Particles (SEPs), low-energy Galactic Cosmic Rays (GCRs) and terrestrial plasma sheet energetic particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The instrument will also investigate the spectra of the secondary high energy ions, released from the lunar surface following its irradiation by GCRs and/or SEPs (albedo energetic particles, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' It will cover the ~40 keV – ~100 MeV energy range for ions and ~20 keV – ~30 MeV for electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' It will provide a ΔE/E ≤ 10 keV energy resolution and supply, for ions, a measure of the composition (protons to iron nuclei).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' In order to cover both pristine and albedo energetic particles, it will consist of two identical detection heads, each with a 60° × 60° FOV: one pointing to the lunar zenith and the other pointing to the opposite direction (lunar nadir).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Each detection head will be composed of a collimator and a 1 cm2, 1 mm thick silicon detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' In front of the detector a filter wheel will allow to place either a thick foil, a pinhole or an obturator to allow the reconfiguration of the detection head to various scientific modes to measure the combined spectra of electrons and ions, to measure the electron spectrum, to protect the detector from sunlight or to avoid saturation of the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The proposed cENPD instrument will benefit from the heritage of the IPD instrument, flown onboard the DEMETER satellite (Sauvaud et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2006), and of the IDEE instrument, developed for the TARANIS satellite (Lefeuvre et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='7 cGCRD: Galactic Cosmic Rays Detector The cGCRD instrument will measure the spectra and the composition of the Galactic Cosmic Rays and that of the Solar Energetic Particles, covering the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='1 to ~ 5 GeV energy range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' It will thus be complementary to the cENPD instrument, covering the higher energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The proposed cGCRD instrument is the Mini.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='PAN penetrating particle analyzer, which is an approved H2020-FETOPEN project that will build a demonstrator of the Penetrating particle ANalyzer (PAN) for deep space applications (Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Mini.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='PAN is based on the particle detection principle of a magnetic spectrometer, with novel layout and detection concepts to optimize the measurement precision for both high flux and low flux particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' As above several hundred MeV/nuc standard methods for measuring particle energies (TOF, dE/dx, ΔE-E) become less efficient, the use of magnetic spectrometry (the charged particle energy is derived from the degree of bending of its trajectory in the magnetic field) is used as the principal particle analysis method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' In Mini.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='PAN the bending of the particle in the magnetic field is Lunar Gateway Space Plasma Physics 15 measured by precise silicon strip tracking detectors, while the elemental identity of the particle is determined by its charge and Z, which is measured with the dE/dx method at multiple points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Mini.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='PAN is designed to precisely measure the momentum, the charge, the direction and the time of energetic particles between 100 MeV/nuc and a few GeV/nuc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Mini.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='PAN offers much higher energy resolution (compared to integral measurements), especially in the > 100 MeV range, and is appropriate for precision energy and species measurements in the 100 MeV/nuc to low GeV/nuc range, which contains both albedo particles and the low energy part of the ambient GCR spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' This part is not well resolved by past solar wind observatories (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' ACE) or by massive GCR detectors in low Earth orbit (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' PAMELA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Mini.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='PAN is also a new type of miniaturized, advanced energetic particle detector that can be adapted and adjusted for deep space missions, where mass limitations exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' As a bonus, the proposed concept for the cGCRD detector can also detect MeV ENAs (likely of heliospheric origin, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Mewaldt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2009), because the detection method combines a strong magnet, the ΔE-E technique and particle tracking through successive, pixelated SSDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Few-MeV hydrogen ENAs would give the characteristic ΔE-E signal on the SSD stack, but across a straight- line trajectory, since the magnet does not influence them, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' they can be separated from charged species (which also get detected) and from the very high energy GCRs (which are less detected but penetrate deeper).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='8 cHENA: High-Energies ENA Imager cHENA is a high-energies ENA (Energetic Neutral Atoms) imager, for detecting and imaging the ENAs produced by charge exchange interactions between the terrestrial plasma sheet or ring current energetic ions and the geocorona neutral hydrogen atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' It will cover the ~10 – 500 keV energy range, and will be equipped with a collimator to both delimit the FOV (120° × 90° or narrower) and reject the charged particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The transmitted ENAs then go through a TOF system and are detected by an MCP (64 × 64 pixels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Pointing the instrument optical axis towards the terrestrial inner magnetosphere requires mounting cHENA on an azimuthal (1-axis) articulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The proposed cHENA instrument is based on the MIMI-INCA ENA imager, flown onboard Cassini (Krimigis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2004) and the HENA ENA imager, flown onboard the IMAGE mission (Mitchell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='9 cMENA: Medium-Energies ENA Imager cMENA is a medium-energies ENA imager, for detecting and imaging the ENAs produced by charge exchange interactions between the terrestrial plasma sheet ions and the geocorona neutral hydrogen atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' It will thus be complementary to the cHENA instrument, extending the coverage to lower energies (~1 keV – 100 keV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' An additional objective for MENA is the detection and imaging of ENAs produced in the lunar environment, from the charge exchange interactions between the solar wind protons and the lunar exosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' This requires flexibility in the instrument pointing (Earth or Moon pointing), which implies also mounting cMENA on its own azimuthal (1-axis) articulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The proposed instrument has a 90° × 10° instantaneous FOV and it uses, as cHENA, a collimator to both delimit the FOV and reject the charged particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The collimator includes also a UV filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The instrument provides a 5° × 10° angular resolution, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' one-dimensional images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' It is based on the heritage of the wide-angle imaging neutral-atom spectrometer onboard the TWINS mission (McComas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2009b) and of the SERENA-ELENA neutral atom imager onboard the MPO Mercury Planetary Orbiter of the BepiColombo mission (Orsini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Lunar Gateway Space Plasma Physics 16 This is a provisional file, not the final typeset article 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='10 cLENA: Low-Energies ENA Imager cLENA completes the suite of ENA imagers by covering the lowest energies (down to ~10 eV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' These low-energy ENAs have two main sources: the charge exchange interactions of the solar wind protons with the lunar exosphere and the charge exchange interactions with the lunar surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Moon pointing for its FOV is thus required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The proposed instrument has a 15° × 15° field-of-view consisting of a single pixel and uses a conversion surface to ionize incoming ENAs and then feed them into an electrostatic wave system, which acts as a filter to pass only particles within the proper energy range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The particles then go through a TOF system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The instrument is capable of high-cadence observations of the solar wind - lunar surface interaction within the ~10 eV to ~3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='3 keV energy range and with a ~50% ΔE / E energy resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' It is based on the LNT instrument that has been designed for the Luna-Resurs-Orbiter (Luna 26) mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='11 cUVIS: UV Imaging Spectrometer cUVIS is a UV / EUV imaging spectrometer, sensitive to specific emission lines for observing the terrestrial exosphere (H: 121.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='6 nm, He: 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='4 nm, O: 130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='4 nm, and N: 120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='0 nm), the terrestrial plasmasphere (He+: 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='4 nm, O+: 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='6 nm) and the lunar exosphere (He: 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='4 nm, plus emission lines of other elements).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' It will thus cover the 30 – 130 nm wavelength range and will have a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='1° × 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='5° FOV with a ~5 arcmin angular resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' This resolution corresponds to about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='15 RE (Earth radii) at the plasmasphere, as seen from the Moon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Earth pointing requires, as for cHENA and for cMENA, mounting the instrument on its own azimuthal (1-axis) articulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The articulation allows also pointing the instrument to the Moon, as a function of the scientific target of each observation session.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The proposed instrument is based on the heritage of the PHEBUS UV / EUV imaging spectrometer onboard the MPO Mercury Planetary Orbiter of the BepiColombo mission (Chassefière et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='12 cLPEF: Langmuir Probe and E-field cLPEF is a Langmuir probe instrument for providing ambient plasma diagnostics: a conductive probe, either biased or floating, is immersed into the plasma and the resulting electron / ion fluxes to the conducting surface provide electric current or voltage measurements with respect to the spacecraft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' From these measurements the main plasma characteristics can be derived, including the plasma density, the electric field or the spacecraft floating potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' In order to provide unperturbed plasma measurements the probe has to be located well outside of the plasma sheath that forms around the spacecraft with a thickness proportional to the local Debye length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The proposed cLPEF instrument will employ two spherical probes (~ 8 – 10 cm in diameter), each placed on the tip of a retractable boom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Since this requirement is identical to the one for the two main magnetometer sensors of the cMAGF instrument, and in order to optimize the resources and simplify the interfaces of the whole space plasma package, it is proposed to combine the sensors of these two instruments and house a tri-axial fluxgate magnetometer sensor within each of the two Langmuir spherical probes (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='4 for the CAD figures).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Such a combined Langmuir probe / magnetometer concept has been originally introduced as a part of an integrated plasma and dust package study conducted under the ESA Contract No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' 4000103352/11/NL/AF in the framework of the proposed Lunar Lander mission, and it is the approach used on ESA’s Comet Interceptor mission (Ratti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Lunar Gateway Space Plasma Physics 17 An additional possibility is to mount occasionally a stand-alone Langmuir probe at the edge of the Gateway external robotic manipulator, so as to use this robotic arm in order to investigate the properties of the plasma sheath, forming around the different Gateway modules, at various locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Langmuir probes benefit from the heritage of instruments that have flown on several space missions, including the RPWS instrument onboard Cassini (Gurnett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2004), ISL onboard the DEMETER satellite (Lebreton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2006), DSLP onboard the PROBA-2 satellite, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='13 cWAVE: Waves Radio Instrument cWAVE is an electromagnetic waves instrument for the study of terrestrial AKR (auroral kilometric radiation) emissions, occurring in the auroral region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' It would then take advantage of the Moon occultation method, which was first implemented by the Radio Astronomy Explorer‐2 mission (Kaiser and Alexander, 1976).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' An additional objective is the study of the radio emissions emitted by accelerated particles in the solar corona and the solar wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The proposed cWAVE instrument will measure the AC electric field (one component: fast electric waveform at 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='5 MHz, decimated electric waveform at 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='5 kHz) and the AC magnetic field (one component: magnetic waveform at 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='5 kHz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' These three products will be delivered as waveform (event mode) and / or as averaged spectra (survey mode, with onboard FFT computation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' As for the cMAGF and the cLPEF instruments, the cWAVE sensor needs to be placed at the tip of a dedicated retractable boom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Radio waves instruments benefit from the heritage of instruments that have flown on several space missions, including STAFF onboard Cluster (Cornilleau-Wehrlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2003), RPWS instrument onboard Cassini (Gurnett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2004), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='14 Retractable booms As indicated above, the mounting of the sensors of the combined cMAGF and cLPEF instruments and of the cWAVE instrument requires a total of three retractable booms, ~3 m each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' These booms need indeed to be stowed at the beginning and at the end of the mission to allow the instruments to stay within the allocated envelope and to be transferred by the Airlock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' We propose to use the compact deployable and retractable boom that has been developed by Oxford Space Systems: the Astrotube Boom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' It can be deployed to up to 3 m and is TRL 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='15 Instruments not included in the conceptual design study The above-described model payload instruments (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' also Table 5) cover satisfactorily the instrumentation requirements, as defined by the topical team (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Table 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' However, there are two instruments that were not included in this conceptual design study: the MeV ENA Imager and the Soft X-ray Imager.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The MeV ENA Imager was not included due to the absence in Europe, in our knowledge, of a developed instrument or protype, for ENAs at these very high energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' However, as described above, the proposed cGCRD instrument (Mini.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='PAN) will be able to detect few-MeV hydrogen ENAs, separating them from similar energy protons and providing 1-pixel images of this population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' For X-ray imaging, a soft X-ray imager with a wide field-of-view, using lobster-eye optics and a position-sensitive MCP detector operating at the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='1 – 2 keV X-ray bandpass has been considered (cMXRI instrument).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' This instrument is based on the DXL/STORM soft X-ray imager protype flown Lunar Gateway Space Plasma Physics 18 This is a provisional file, not the final typeset article onboard a sounding rocket mission (Collier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' However, the size of this instrument (78 cm length) appears to be incompatible with the Gateway interfaces for mounting external payloads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' This suggests that the X-ray imager could not be accommodated as part of this instruments package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' However, it is recommended to propose cMXRI as a payload for the Large European Lander for the Moon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content="4 Instrument accommodation The CAD model for the instrument conceptual design and their accommodation on the Gateway was established in cooperation with the CNES PASO (Plateau d’Architecture des Systèmes Orbitaux), with the help of its concurrent engineering facilities (CIC : Centre d'Ingénierie Concourante), and particularly by using the IDM-CIC (Integrated Design Model) and IDM-View tools (https://idm." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='virtual-it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='fr/).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The instrument accommodation on the Gateway modules has to fulfil several requirements: − In-situ measurement low-energy plasma instruments have to be placed on areas with low electrostatic charging (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' − Pointing requirements for instruments with a field-of-view (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Table 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' − Unobstructed field-of-view for these instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' − cGCRD, which has a strong permanent magnet perturbing the low-energy plasma measurements, should not be placed close to these instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' − Instrument grouping, when possible, to form self-contained “instrument suites”, with instruments mounted on a common platform, minimizing interfaces with the Gateway and using a single SORI (external Small ORU Interface) for attachment on the Gateway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The instrument accommodation configuration we propose, and is compatible with the above requirements, has the instruments grouped on one main and one secondary platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Each of these two platforms, of the order of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='8 m × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='8 m, is mounted externally on a SORI attachment and is double sided, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' has instruments mounted on both sides of the platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' This accommodation is of course notional and could be subject to modifications, depending on Gateway engineering and programmatic constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Both platforms are on the Logistics Module and they are mounted on two diametrically opposite positions, on the +X side (Main Instrument Platform) and on the –X side (Secondary Instrument Platform) of it, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Figure 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The two sides of the Main Platform are shown in Figure 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' With its positioning on the +X side of the Logistics Module, the Main Platform provides an unobstructed view to the solar wind arrival direction (Figures 13C and 13D) and takes advantage of a favorable electrostatic environment (Figure 15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' It is thus well suited for mounting the solar wind instruments (cSWFC and cSWIS), shown in Figure 18 and Figure 19 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The Main Instrument Platform hosts also: − The magnetospheric particle instruments cMESP and cMISP, shown in Figure 18 and Figure 19 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' − The energetic particle detector cENPD, shown in Figure 20, which is mounted on the –Y edge of the platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' cENPD has two oppositely directed FOVs, one along the +Z axis and one along the –Z axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' In this way, during periapsis passes one of the FOVs looks in the zenith direction, to Lunar Gateway Space Plasma Physics 19 monitor the pristine energetic particles precipitating towards the Moon’s surface, whereas the other looks in the nadir direction, to monitor the albedo energetic particles that are the result of the interaction of the precipitating energetic particles with the lunar regolith (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Moreover, the +Z / –Z orientation of the two detector heads allows avoiding direct sunlight entering the detectors (Sun is in the +X direction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' − The two “compact” remote sensing instruments cMENA and cLENA, shown in Figure 18 and Figure 19 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' cMENA uses a dedicated azimuthal (1-axis) articulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The cLENA orientation gives access, during the periapsis passes, to the Moon surface and plasma environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' − The two booms of the fluxgate magnetometer package (cMAGF), which as described in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='1 consists of three different types of units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The main type is two pieces of boom-mounted dual fluxgate magnetometers (one sensor at each of the two ~3 m boom tips and one at the boom root).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' These two retractable booms are mounted on the Main Instrument Platform –Z side (Figure 18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The boom tip mounted cMAGF sensors are integrated together with Langmuir probes (cLPEF instrument).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' These units will provide the main measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' In order to monitor the perturbations from the station close to the source in more detail, several (~5+) single magnetometer sensors will be also mounted on various places around the station (not shown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' − The cWAVE instrument, also mounted on a retractable boom, which is on the Main Instrument Platform +Z side (Figure 19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The two sides of the Secondary Platform are shown in Figure 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' This platform, mounted on the –X side of the Logistics Module, is permanently in the shadow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' In this way there is no direct sunlight that could interfere with the measurements of the two instruments mounted on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' On each of its two sides there is a remote sensing instrument: cUVIS on the one side and cHENA on the other side of the Secondary Platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Each of these two instruments in mounted on a dedicated azimuthal (1-axis) articulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The cGCRD instrument, due to the containment of a strong magnet (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='4 Tesla) that would deviate charged particles to be measured by the other instruments if in close vicinity with them, is not mounted on any of the two instrument platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' It is instead mounted as a “standalone” instrument on the SORI attachment of the +Z side of the Logistics Module (Figure 22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Its FOV, looking radially out, near periapsis gives access to the albedo energetic particles that are the result of the interaction of the precipitating galactic cosmic ray particles with the lunar regolith.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' During the remaining part of the orbit (most of the time) it points to the open sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='5 Instruments fields-of-view simulation The appropriate orientation of the fields-of-view (FOVs) of the remote sensing and of the high- energy particle instruments, as projected on the sky and on the celestial objects, was analyzed by simulating the evolution of the FOVs along the Gateway orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' This simulation was performed in cooperation with the CNES PASO and by using the VTS software tool (https://logiciels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='cnes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='fr/en/content/vts).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The FOVs of the two oppositely directed sensor heads of the cENPD instrument, near a periapsis pass, are shown in Figure 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' As shown in this figure, one of the two sensor heads is oriented towards the local zenith, and has an unobstructed view to the pristine energetic particles flux (purple FOV), whereas the other is oriented towards the nadir and its FOV is dominated by the albedo energetic particles from the Moon (yellow FOV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Both populations (pristine and albedo high-energy particles) are thus covered by the cENPD instrument detection capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Lunar Gateway Space Plasma Physics 20 This is a provisional file, not the final typeset article For the cGCRD instrument, which is a single sensor head GCR detector, the FOV near a periapsis pass is shown in Figure 24, left panel (light blue FOV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' As shown, near periapsis it is dominated by the albedo GCR particles from the Moon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' However, during most of the remaining orbit (right panel) it has an unobstructed view to the open sky and gives then access to the pristine GCR environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The field-of-regard (FOR) of the cMENA instrument, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' the total accessible FOV taking into account the rotation of the 1-axis articulation on which the instrument is mounted, at a given point of the orbit, is shown in Figure 25A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The azimuthal rotation mechanism gives to the instrument access to a very large “ribbon” of the sky, which includes the Earth environment and the Moon environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The pointing of the instrument to any of these two principal targets, using the flexibility provided by the 1-axis articulation, can then be programmed as a function of the scientific target of each observation session.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' In Figure 25B is the FOV of the cLENA instrument, close to periapsis, as projected on the sky (no articulation for this instrument).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' As shown in this figure, the way the instrument is mounted on the Gateway gives access, during the periapsis passes, to the Moon surface and to its exosphere and plasma environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The FOR of the cUVIS instrument is shown in Figure 25C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' As shown in this figure, the dedicated articulation allows also for this instrument to point to targets as the Earth space environment (plasmasphere, exosphere), the Moon space environment (exosphere), or targets in the open sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The narrow width of the instantaneous FOV of this instrument (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='1°), in combination with the articulation, allows also performing altitude profile scans of the lunar exosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' 5 Conclusion The Moon is a unique location to study the deep space plasma environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The Lunar Orbital Platform - Gateway, that will be assembled and operated in the vicinity of the Moon starting from the mid-2020s, is a crewed station that offers new opportunities for fundamental and applied scientific research in the field of space plasma physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' These have multi-disciplinary dimensions, and they include: − Studying the lunar space environment and its interaction with the solar wind and the terrestrial magnetotail plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' − Terrestrial space weather: monitoring, through remote sensing techniques, the response of the terrestrial magnetosphere and exosphere to solar activity events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' − Planetary space weather: monitoring, through in-situ measurements and through remote sensing, the response of the lunar space environment to solar activity events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' − Radiation physics: characterizing the lunar high-energy particles environment, including energy and mass spectrometry of these populations and their variability, particularly in view of the Artemis human missions to the Moon and the associated radiation risks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' − Studying the heavy ion escape from the terrestrial ionosphere, through in-situ measurements of the downtail streaming ions, and the role of this escape in the long-term evolution of the composition of the terrestrial atmosphere (and its habitability).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' − Studying the lunar regolith - bounded exosphere - interplanetary space environment as a complex interacting multi-scale system, and as an archetype of the interaction of an unmagnetized planetary body with the solar wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' − Studying the mini-magnetospheres that form above the “swirls” on the Moon, and which constitute probably the smallest magnetospheres in our solar system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Lunar Gateway Space Plasma Physics 21 − Understanding the surface electric fields that develop on the Moon as a part of a complex and interacting plasma environment, and their role in electrostatic lunar dust levitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' − Planetology: understanding the composition of the lunar regolith, and its hydration, through the spectrometry of the albedo energetic particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' In preparation of the scientific payload of the Lunar Orbiter Platform - Gateway we first formed a topical team, under the auspices of ESA, to prepare and to support the definition of payload studies in the field of space plasma physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' This allowed to identify the scientific objectives that can be addressed from onboard the Lunar Orbital Platform - Gateway, the physical parameters needed to be measured in order to address these objectives, and the corresponding instrumentation required to perform these in-situ measurements and remote-sensing observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' We then undertook for ESA a conceptual design study for a “Space Plasma Physics Payload Package onboard the Gateway” (SP4GATEWAY), addressing the objectives identified by the topical team while remaining compatible with the technical requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' This conceptual design has considered, as baseline, a typical “Gateway Phase 2” configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' As a first part of this conceptual design study, we simulated the interaction between the Gateway and its plasma environment, for the case where the Gateway is in the solar wind and also for the case where the Gateway is in the terrestrial magnetotail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' This allowed to identify the Gateway modules on which the perturbation of the natural plasma environment by the Gateway will be minimal, and are thus best-suited for placing there the in-situ measurement plasma instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' We then defined a model payload consisting of a suite of instruments, for in-situ measurements and for remote-sensing observations, corresponding to the requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' These measurement instruments are largely based, either on existing flight-proven instruments, adapted here for the lunar plasma environment, or on tested and validated laboratory prototypes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The main characteristics of these instruments have been defined and CAD conceptual instrument designs elaborated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The instruments’ measurement characteristics will however have to be refined during a follow-on Phase A study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The next step was the study for accommodating this model payload on the Gateway modules, taking into account the various constraints, and in particular the surface and volume charging of the various Gateway modules, their exposure to the ambient plasma and the pointing and field-of-view requirements of the instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' This resulted in an integrated CAD design, including the Gateway and the instruments, which were grouped into two platforms mounted on two sides of the Logistics Module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The fields-of-view of the remote sensing instruments and of the high-energy particle instruments, as projected on the sky and on the celestial objects, were then analyzed by simulating their evolution along the Gateway orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' This allowed to verify the appropriate orientation of the fields-of-view and the coverage of the observational scientific targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Following this conceptual design study for a Space Plasma Physics Payload Package onboard the Gateway, it results that the Gateway is very well-suited for space plasma physics research and it allows to address a series of relevant scientific objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' 6 Tables Lunar Gateway Space Plasma Physics 22 This is a provisional file,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' not the final typeset article TABLE 1 | Physical parameter / observable to be monitored from onboard the Gateway In-situ measurements Solar Wind (particles + fields) Earth’s foreshock SEPs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' GCRs (pristine + secondary from Moon,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' at various directions) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='Energetic electrons ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='Magnetotail + magnetosheath plasma ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='(particles + fields) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='Outflowing terrestrial ions ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='(ion spectrometry) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='Lunar pickup ions ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='(ion spectrometry) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='Gateway-induced plasma and fields environment ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='Lunar Wake ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='Imaging ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='MeV ENAs: produced from SEPs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='ENAs: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='Terrestrial Ring Current and Plasma Sheet ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='Low-energy ENAs (from Solar Wind and Moon) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='SWCX X-rays: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='Magnetosheath/pause + cusp + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='planetary targets of opportunity ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='UV / EUV: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='Terrestrial Plasmasphere ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='UV / EUV: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='Geocorona,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Lunar Exosphere,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Solar EUV radiometry Auroral imaging,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='Planetary imaging ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='Heliosphere imaging ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='Lunar surface micrometeorite impacts ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='Active experiments ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='Gas release and ionization ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='Lunar Gateway Space Plasma Physics ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='23 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='TABLE 2 | Physical parameter / observable to be monitored from low lunar orbits ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='In-situ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='measurements ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='Crustal Magnetic Anomalies ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='(Plasma + magnetic field + ENAs + electron reflectometry) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='Solar wind ions neutralization ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='Lunar Exosphere / Ionosphere ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='(in-situ measurements) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='Dusty plasmas ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='Imaging ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='Lunar Exosphere / Ionosphere ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='(imaging) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='TABLE 3 | Physical parameter / observable to be monitored from the lunar surface ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='In-situ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='measurements ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='Energetic ion implantation / reflection ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='Lunar surface electrostatic charging + dust ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='Crustal Magnetic Anomalies ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='Magnetosphere radio emissions ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='Lunar exosphere ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='Imaging ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='SWCX X-rays: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='Magnetosheath/pause + cusp + planetary targets of opportunity ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='Lunar Gateway Space Plasma Physics ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='24 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='This is a provisional file,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' not the final typeset article TABLE 4 | Science objectives and corresponding measurement and instrumentation requirements (from onboard the Gateway) Science Objective Measurement Requirement In-situ Measurements Instrument Remote Sensing Instrument Monitor solar wind as a driver for the dynamics of terrestrial magnetosphere,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' terrestrial and lunar exospheres,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' lunar surface sputtering and charging Solar wind density and transport velocity 1 – 102 cm-3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='1 – 40 keV ions 200 – 1000 km/s, ΔE/E < 17% Faraday Cup Electrostatic Analyzer IMF: 100 nT instrument range 1 nT / 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='1 nT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='absolute / relative ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='resolution ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='Magnetometer Monitor and characterize ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='SEPs and GCRs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='for radiation environment ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='and as lunar surface sputtering ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='sources ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='40 keV – 100 MeV ions (SEPs) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='up to ~5 GeV (GCRs) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='50 MeV / nucleon for ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='composition ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='~40 keV – ~30 MeV electrons ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='Energetic particle ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='detectors ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='MeV ENA Imager ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='Monitor and characterize the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='response of the terrestrial ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='magnetosphere to the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='solar wind with a wide ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='coverage of geospace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='Detect and image solar wind ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='charge exchange X-rays ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='2 – 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='0 keV FOV 10° × 10° angular resolution: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='3 RE from the Moon Soft X-ray Imager Detect and image terrestrial magnetosphere ENAs ~1 – 300 keV, FOV ~ 20° × 20° ENA Imager Monitor solar wind interaction with the lunar exosphere, regolith and magnetic anomalies Detect and image low-energy ENAs: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='1 – 10 keV, 30 % ΔE/E, FOV ~ 20° × 20°, ~5° resolution Strong UV suppression: 10-8 LENA imager Reveal the solar wind ion dynamics in the vicinity of the lunar magnetic anomalies Detect and image low-energy ENAs: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='01 – 3 keV, 30 % ΔE/E, FOV ~ 5° × 120°, ~5° resolution LENA imager Monitor the terrestrial and lunar exospheres, plasmasphere Detect and image EUV emissions 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='4, 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='6, 121.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='6 and 130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='4 nm ~5 arcmin resolution Ion mass spectrometer (lunar pickup ions) UV / EUV spectro-imager Monitor ambient plasma in different environments (solar wind / magnetosheath / terrestrial magnetotail / lunar wake) Plasma density and temperature ~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='01 – 40 keV, 10-3 – 102 cm-3 Ion composition: m/Δm > 15 Langmuir probe Ion mass spectrometer Electron spectrometer Magnetic field: 1000 nT range 1 nT / 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='1 nT absolute/relative resolution Magnetometer Monitor magnetospheric and planetary radio emissions Radio instrument Lunar Gateway Space Plasma Physics 25 TABLE 5 | SP4Gateway model payload instruments Instrument Acronym Instrument Mass (kg) Power (W) FOV FOV pointing cMAGF Magnetometer(s) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='3 N/A N/A cSWIS Solar Wind Ion Spectrometer 5 7 96° × 48° Sun cSWFC Solar Wind Faraday Cup 5 4 45° × 45° (×6) Sun cMISP Magnetospheric Ion Spectrometer 7 8 360° × 120° N/A cMESP Magnetospheric Electron Spectrometer 3 6 360° × 120° N/A cENPD Energetic Particles Detector 3 6 60° × 60° (×2) Moon / Sky cGCRD Galactic Cosmic Rays Detector 10 20 71° × 71° Moon / Sky cHENA High-Energies ENA Imager 15 12 120° × 90°,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' articulation Earth cMENA Medium-Energies ENA Imager 5 15 90° × 10°,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' articulation Earth / Moon cLENA Low-Energies ENA Imager 4 10 15° × 15° Moon cUVIS UV Imaging Spectrometer 10 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='1° × 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='5°, articulation Earth / Moon cLPEF Langmuir Probe and E-field 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='2 5 N/A N/A cWAVE Waves Radio Instrument 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='6 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='6 N/A N/A Mass and Power: nominal values, without margins, booms and articulation mechanisms included in these values;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' FOV: field-of-view;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' N/A: not applicable Lunar Gateway Space Plasma Physics 26 This is a provisional file, not the final typeset article 7 Conflict of Interest The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' 8 Author Contributions ID was the coordinator of the topical team “Space Plasma Physics Science Opportunities for the Lunar Orbital Platform - Gateway”, the coordinator of the SP4GATEWAY project, and is the main author of this manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' MGGT was the ESA support scientist for the topical team and for the SP4GATEWAY project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' JDK, YF, RAB, GBR, JYF, DG, BG, HL, FL, AM, RN, and ER were members of the topical team and of the SP4GATEWAY project team.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' PD was the lead engineer for the SP4GATEWAY project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' EDA, JE, ME, PG, DH, LP and ŠŠ were members of the SP4GATEWAY project team.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' AL managed the CAD design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' JF, AT, SLGH and JCMV performed the Gateway - plasma environment simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' JC was the ESA HRE (Human and Robotic Exploration) correspondent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' JW was the ESA manager for the SP4GATEWAY project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' 9 Funding The topical team “Space Plasma Physics Science Opportunities for the Lunar Orbital Platform - Gateway” was supported by ESA through contract No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' 4000128802 /19/NL/PG/pt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The SP4GATEWAY project was funded by ESA through contract No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' 4000128461/19/NL/FC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Activities at IRAP were also supported by CNES through order 4500065232.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' 10 Acknowledgments We acknowledge the support of the CNES PASO for the SP4GATEWAYconceptual design study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' 11 References Allegrini, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', Dayeh, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', Desai, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' (2013).' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', Schaufelberger, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', and Asamura , K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Extremely high reflection of solar wind protons as neutral hydrogen atoms from regolith in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Space Sci.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', Townsend, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', de Wet, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', Looper, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', Brittingham, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', Burahmah, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Modeling the lunar radiation environment: a comparison among FLUKA, Geant4, HETC-HEDS, MCNP6, and PHITS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Space Weather, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='1029/2021SW002895 Zhang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', Wimmer-Schweingruber, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', Yu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', Wang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' First measurements of the radiation dose on the lunar surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='1126/sciadv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='aaz1334 Zoennchen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', Nass, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', Fahr, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', and Goldstein, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The response of the H geocorona between 3 and 8 Re to geomagnetic disturbances studied using TWINS stereo Lyman-α data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Geophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='5194/angeo-35-171-2017 Zoennchen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', Connor, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', Jung, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', Nass, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', and Fahr, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Terrestrial exospheric dayside H-density profile at 3–15RE from UVIS/HDAC and TWINS Lyman-α data combined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Geophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='5194/angeo-40-271-2022 Lunar Gateway Space Plasma Physics 38 This is a provisional file, not the final typeset article 1 Data Availability Statement N/A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' 2 Figures Lunar Gateway Space Plasma Physics 39 FIGURE 1 | Moon’s orbit with respect to the Earth’s magnetosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Earth’s and Moon’s sizes are not on scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' (Adapted from: Tim Stubbs / University of Maryland / GSFC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=" Moon'sOrbit magneticbowshock solarwind Earth's magnetotail Sun Earth Lunar Gateway Space Plasma Physics 40 This is a provisional file, not the final typeset article FIGURE 2 | Moon’s environment with the complex interaction between solar radiation, space plasma, meteoritic flux, dust, exosphere and the surface (Credit: Jasper Halekas)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Horizontaland/orVertical EXOSPHERE DustTransport Wake Boundary Plasma Photoelectrons Terminator +10sofVolts Region Sheath Photons(UV) Photon-Driven Photon-Stimulated Solar Wind Positive Chasging Desorption SUN Spuli SolarStorms Magnetosphere 1000sof Volts Electron-Driven Negative Charging Chemical,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='Thermal MeteoriclInflux Release&Loss LargeImpacts inleriorOutgassing Lunar Gateway Space Plasma Physics 41 FIGURE 3 | Typical SEP (Solar Energetic Particles) proton intensities: five-minute averages of proton intensities measured by GOES-13/EPS/HEPAD during the May 2012 solar events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' (From: Quinn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' 10 May MeV)-1 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='7-4MeV 10° 4-9MeV 102 9-15MeV (cm2 15-40MeV 10 38-82MeV Proton Intensity 84-200MeV 110-900MeV 10 330-420MeV 420-510MeV 10 510-700MeV >700MeV 10 10136 138 140 142 144 D0Y2012 Lunar Gateway Space Plasma Physics 42 This is a provisional file, not the final typeset article FIGURE 4 | Typical GCR (Galactic Cosmic Rays) Hydrogen nuclei (left) and Oxygen nuclei (right) fluxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' (From: Mrigakshi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=" 5000 Badhwar-0'Neill2010 O'Neill2010 50 CREME2009 Hydrogen nucleil 1." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='4 2009 Oxygen nuclei CREME96 TE96 4500 Jsoskin 80 Burger ACE/CR 40 BESS RelativeNMcountrate CAPRICE1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='2 4000 IMAX 30 PAMELA 60 Relative NM Relative 3500 RelativeNM countrate 20 sec Ot NM 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='8 xn Count Flux 0 Count Rate 20 2500 ted Integrated I 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='6 10 Rate 20 1500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='4 [%] 20 30 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='2 40 a 40 c 500 1970 1975 1980 1985 1990 1995 2000 2005 2010 1970 1975 1980 1985 1990 1995 2000 2005 2010 Lunar Gateway Space Plasma Physics 43 A B FIGURE 5 | (A): Illustration of the effects of a hydrated layer of lunar regolith in the production of GCR albedo (secondary) protons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The nuclear evaporation process from deep in the regolith produces abundant secondary particles in all directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' (From: Schwadron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' (B): Energy spectra of pristine GCR species (dashed lines) and of lunar albedo species (continuous lines), calculated with the Geant4 simulation toolkit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' (From: Looper et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' GCR Enhancedalbedoprotons Hydrated layer Nuclear Proton evaporation Neutron-GCRprotons GCRalphas 10 GCRheavyions Albedoelectrons Albedopositrons secMeV/nucleon) Albedoprotons 10 Albedoneutrons Albedogammas Albedo light ions Albedoheavyions 10-6 10-1 100 101 102 103 104 105 Energy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='MeVorMeV/nucleon Lunar Gateway Space Plasma Physics 44 This is a provisional file,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' not the final typeset article FIGURE 6 | Cold O+ beam fluxes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' observed by the Geotail spacecraft in the magnetotail lobe and plasma sheet boundary layer,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' versus the tailward distance from the Earth (XGSM in RE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The Moon is at XGSM ≈ -60 RE (Earth radii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' (From: Seki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' (a) St Observed O+ Flux: St [cm-2s-1] in the lobe/mantle 10° 104 10 [X(21 : +ON = 1S 102 2 0 50 100 150 200 Lunar Gateway Space Plasma Physics 45 FIGURE 7 | Energy distributions of H+ and O+ ions measured by the IMA sensor onboard the Kaguya lunar orbiter in the terrestrial magnetotail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' During the plasma sheet encounter (top panel) there is an enhancement of high-energy (1 – 10 keV) O+ ions, in comparison to those measured in the magnetotail lobe (bottom panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The calculated density and net flux of these magnetospheric O+ ions, during the plasma sheet encounter, were 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='2 × 10−3 cm−3 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='6 × 104 cm−2 s−1 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' (From: Terada et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' 105 8:00-16:00 on 21 April 2008 Plasma sheet 104 H* 0+ 103 Flux (eV-1 Lunar o+ 102 中 更更 101 Magnetospheric o* 100 105 16:00-24:00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='on21April2008 104 Lobe +H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' (rSz 103 Lunar O+ 102 101 亚 中中 100 100 101 102 103 104 Energy (eV ql) Lunar Gateway Space Plasma Physics 46 This is a provisional file, not the final typeset article FIGURE 8 | MHD Open Global Geospace Circulation Model simulations (backward particle tracing) suggest how heavy ions, observed in the Moon environment during high geomagnetic activity events (at XGSE ≈ -60 RE), can be originating from the inner magnetosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Earth-to-Moon transport times are ~2 – 3 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' (From: Poppe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' 20 10 GSEJ [Re 0 10 20 20 20 40 60 X [Re GSE] Lunar Gateway Space Plasma Physics 47 FIGURE 9 | Left: Lunar exosphere density profiles for the atoms and molecules thermally released from surface;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' based on the exospheric surface densities from Stern (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Right: Lunar exosphere density profiles for the atoms released through sputtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Both calculations are done for the sub- solar point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' (From: Wurz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' 10" 1012 H Density f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='1012ms H2 Density SW ODensity Na Density He Density EV = 670km/s CH4 Density Ai Density Mg Density 1010 co Density 10 to Kreep soils Si Density CO2 Density K Density Ar Density OH Density Ca Density 10° 108 Ti Density [gw] Kr Density [g-w] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='- Cr Density Xe Density Mn Density Fe Density Density 106 Density 10° 104 10* 100 100 10* 105 1 10 100 1000 10 10 100 1000 10* 10° Altitude[km] Altitude [km] Lunar Gateway Space Plasma Physics 48 This is a provisional file,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' not the final typeset article FIGURE 10 | Typical energy spectra of the solar wind ions (right side,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' open squares) and of the corresponding reflected from the lunar regolith energetic hydrogen atoms (left side,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' open circles),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' measured by the SARA instrument onboard the Chandrayaan-1 spacecraft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Note the good correlation between the reflected energetic neutral flux and the solar wind flux variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' (From: Wieser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Solar wind 1X10 Refle ev Flux [cm" 2 r 1×10 1: 100 1000 Energy [eV, eV/q] Lunar Gateway Space Plasma Physics 49 FIGURE 11 | Top: Image of the central region of the Reiner Gamma Formation lunar swirl, taken by Lunar Reconnaissance Orbiter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Bottom: A slice of the relative solar wind proton density above this lunar swirl obtained from a 3D simulation, with the initial magnetic field lines corresponding to a single subsurface dipole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' (From: Bamford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Observations Reiner Gamma ~10km @NASALRO Simulations ~130c/Wpe @ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Alves, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Bamford Lunar Gateway Space Plasma Physics 50 This is a provisional file, not the final typeset article FIGURE 12 | Gateway configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' PPE: Power and Propulsion Element;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' HLS: Human Landing System;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' HALO: Habitation and Logistics Outpost (Credit: NASA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Logistics Module PPE DSXR +X ESPRIT Refueler HALO MPM International HLS Habitat HLS Descent Docking Element Adapter US Habitat HLS Ascent Element Orion HLS Transfer Element Lunar Gateway Space Plasma Physics 51 A B C D FIGURE 13 | Top row: SPIS (Spacecraft Plasma Interaction System) simulation of the volume electrostatic potential distribution (in V) in the solar wind, with the Gateway aligned to the solar direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' (A): Full potential scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' (B): Potential scale saturated at +5 / -3 V, to highlight the potential values away from the solar panels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Bottom row: Proton density (in m-3) in the solar wind, with the Gateway aligned to the solar direction (C) and the Gateway main axis tilted by 20° with respect to the solar direction (D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Note the plasma wake downstream of the station.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' 9,13 4,66 18,5 32,2 46,05,00 3,00 1,00 1,00 3,006,74e+06 5,05e+06 3,37e+06 1,68e+06 0,006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='74+006 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='05e+006 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='37e+006 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='68e+006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='000 Lunar Gateway Space Plasma Physics 52 This is a provisional file, not the final typeset article A B C D FIGURE 14 | SPIS simulation of the volume electrostatic potential distribution (in V) in the terrestrial magnetotail (A), with the Gateway aligned to the solar direction, and photoelectron density (log scale) under the same conditions (B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' H+ ion density (C) and O+ ion density (D) in the terrestrial magnetotail, both in m-3, showing the plasma wake downstream of the station.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' 13,0 1,92 9,14 20,2 31,38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='85 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='89 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='93 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='0369 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='00e+006 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='51e+006 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='01e+006 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='17e+005 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='19e+0047.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='52e+003 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='01e+003 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='5/e+003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='000 Lunar Gateway Space Plasma Physics 53 FIGURE 15 | Quality of the different positions on the Gateway for placing the space plasmas instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Logistics Module PPE DSXR +X ESPRITRefueler HALO MPM International HLSDescent HLS Habitat Docking Element Adapter US Habitat HLSAscent Element Orion HLSTransfer Element Lunar Gateway Space Plasma Physics 54 This is a provisional file, not the final typeset article FIGURE 16 | Main and Secondary Instrument Platforms, mounted on the +X side and on the –X side respectively of the Logistics Module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' HLS TransferElement HLS DescentElement I-Hab MainInstrumentPlatform Orion US-Hab Halo PPE Logistics +Z Module SecondaryInstrument Platform Lunar Gateway Space Plasma Physics 55 A B FIGURE 17 | The two-sided Main Instrument Platform, mounted on the Logistics Module (A), and in perspective view (B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The “magenta cube”, on the side of the Logistics Module, is the “standalone” cGCRD instrument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Main Instrument Platform: X Z side Logistics Module +Z Logistics Module .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Main Instrument Platform: + Z side Lunar Gateway Space Plasma Physics 56 This is a provisional file, not the final typeset article FIGURE 18 | Main Instrument Platform –Z side, with all booms deployed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' CMESP +X FOV Module cSWFC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='9m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='8 m 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='8m CMAGF boom-root sensors (2 retractable booms) X-Y Combined FOV cLPEF / cMAGF (turntable) CMENA boom-tip sensor Lunar Gateway Space Plasma Physics 57 FIGURE 19 | Main Instrument Platform +Z side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The turquoise solid angle in the cSWIS instrument inset (upper left) represents the instrument field-of-view (FOV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' cSWIS +Z X Module +X FOV +Z FOV CLENA X-Y cMISP FOV Electrostatic Analyzer Scattering MCP Deviation Detection Electrodes MCP Front-End Board Digital Board High-Voltage Board Data & Power cWAVE Lunar Gateway Space Plasma Physics 58 This is a provisional file, not the final typeset article FIGURE 20 | Main Instrument Platform +X / –Y edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The cENPD instrument is mounted on the edge of this platform, to get unobstructed view to both the +Z and –Z directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' CLPEFI CMAGF CENPD Z FOV CMESP +X CSWFC +Z CMENA (out) (down) CMISP CSWIS +Z FOV CENPD CWAVE (mountedonthe-Yedgeoftheplatform) Lunar Gateway Space Plasma Physics 59 FIGURE 21 | The two-sided Secondary Instrument Platform, mounted on the Logistics Module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Secondary Instrument Platform: Z side cUVIS Module X +Z FZ Z Module Logistics CHENA Secondary Instrument Platform: 7 + Z side Lunar Gateway Space Plasma Physics 60 This is a provisional file, not the final typeset article FIGURE 22 | cGCRD mounted as a “standalone” instrument on the Logistics Module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' +Z FOV Module Logistics 200mm CGCRD Secondary Instrument Platform Main 园 Instrument Platform 200mm +2 +X Lunar Gateway Space Plasma Physics 61 FIGURE 23 | cENPD instantaneous FOVs of the two oppositely directed sensor heads, near periapsis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Purple FOV: pristine energetic particles flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Yellow FOV: Moon albedo energetic particles flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The magenta line is the track of the center of the FOV along the Gateway orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' ORG Lunar Gateway Space Plasma Physics 62 This is a provisional file, not the final typeset article FIGURE 24 | cGCRD FOV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Left: cGCRD FOV near periapsis (light blue cone), dominated by the albedo GCR particles from the Moon (grid sphere).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Right: projection on the sky of the cGCRD FOV along the Gateway orbit (in magenta).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='#2 - SkyView Timestopped oon Latitude = -98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='0233 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' Longitude = 123.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content='8372 Lunar Gateway Space Plasma Physics 63 A B C FIGURE 25 | (A): Field-of-regard (total accessible FOV, taking into account the rotation of the 1- axis articulation on which the instrument is mounted) of the cMENA instrument, at a given point of the orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The field-of-regard (FOR), as projected on the sky, is shown in magenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' (B): cLENA FOV (in magenta) near periapsis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' The yellow line is the track of the center of the FOV, for the portion of the orbit close to periapsis, as projected on the sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' (C): Field-of-regard of the cUVIS instrument, in magenta, as projected on the sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} +page_content=' 1800 150° 120° 900 009 300 00 006 600 LO9olXsat Moon Earth LOPG_Ysat 30° LOPG_Zsat1800 150° 120° 006 600 300 00 o06 600 0 Earth* 00 LoPolXsat 30° bl LOPG Zat Moon1800 1509 120° 006 60° 30° 00 300 600 o06- 120° 006 Earth Moon LoPolXsat 30 LOPG_Ysat LOPG Zsat' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf'} diff --git a/D9E1T4oBgHgl3EQfWgQs/vector_store/index.pkl b/D9E1T4oBgHgl3EQfWgQs/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..7f84c0759c68bde9d2e010cdab601c4b8beeb144 --- /dev/null +++ b/D9E1T4oBgHgl3EQfWgQs/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:98dcd39ed321b8b65c7bd781f2593dbde6463b3e2ff41a3144cf10007cc5cbe0 +size 1234162 diff --git a/GNE0T4oBgHgl3EQfhQF8/content/tmp_files/2301.02429v1.pdf.txt b/GNE0T4oBgHgl3EQfhQF8/content/tmp_files/2301.02429v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..d04750d6a8f0fdd2959216889293ee534153d405 --- /dev/null +++ b/GNE0T4oBgHgl3EQfhQF8/content/tmp_files/2301.02429v1.pdf.txt @@ -0,0 +1,2342 @@ +Solar Physics +DOI: 10.1007/•••••-•••-•••-••••-• +Reconstruction of the Sunspot Number Source +Database and the 1947 Zurich Discontinuity +Fr´ed´eric Clette1 · Laure Lef`evre1 · +Sabrina Bechet1 · Renzo Ramelli2 · +Marco Cagnotti3 +© Springer •••• +Abstract The recalibration of the sunspot number series, the primary long- +term record of the solar cycle, requires the recovery of the entire collection of +raw sunspot counts collected by the Zurich Observatory for the production of +this index between 1849 and 1980. +Here, we report about the major progresses accomplished recently in the con- +struction of this global digital sunspot number database, and we derive global +statistics of all the individual observers and professional observatories who pro- +vided sunspot data over more than 130 years. +First, we can announce the full recovery of long-lost source-data tables covering +the last 34 years between 1945 and 1979, and we describe the unique information +available in those tables. We then also retrace the evolution of the core observing +team in Zurich and of the auxiliary stations. In 1947, we find a major disruption +in the composition of both the Zurich team and the international network of +auxiliary stations. +This sharp transition is unique in the history of the Zurich Observatory and +coincides with the main scale-jump found in the original Zurich sunspot number +series, the so-called “Waldmeier” jump. This adds key historical evidence ex- +plaining why methodological changes introduced progressively in the early 20th +century could play a role precisely at that time. We conclude on the remaining +steps needed to fully complete this new sunspot data resource. +Keywords: Sunspots, statistics; Solar Cycle, observations +� F. Clette +frederic.clette@oma.be +1 +Royal Observatory of Belgium, 3 Avenue Circulaire, 1180 Brussels, Belgium +2 +Istituto Ricerche Solari Locarno (IRSOL), Universit`a della Svizzera italiana, Via +Patocchi 57, 6600 Locarno, Switzerland +3 +Specola Solare Ticinese, Via ai Monti 146, 6605 Locarno, Switzerland +SOLA: Clette_SNDB2.tex; 9 January 2023; 1:30; p. 1 +arXiv:2301.02429v1 [astro-ph.SR] 6 Jan 2023 + +Clette et al. +1. Introduction +Our knowledge of the long-term evolution of the solar cycle is largely based on +the historical observations of sunspots since the newly invented telescope was +aimed at the Sun for the first time in 1610. Two main indices were built from +those sunspot observations. The sunspot number (hereafter SN) was initiated by +Rudolf Wolf in 1850 (Wolf, 1856; Friedli, 2016). This daily index combines the +total group count and the total spot count and its goes back to 1700. Much more +recently, Hoyt and Schatten (1998a,b) introduced the sunspot group number +(hereafter GN), which only uses the total group count, but was constructed back +to the very first telescopic observations in 1610. Both indices are abundantly +used by most studies of the long-term evolution of solar activity and Sun-Earth +relations, as constraints for validating physical models of the solar dynamo, and +for calibrating various parameters relevant to space weather and space climate +(geomagnetic and ionospheric indices, cosmogenic radionucleides). +However, significant disagreements between the sunspot number and group +number series over their common time interval indicated that either series or +both suffered from inhomogeneities. This prompted various efforts to identify +flaws and biases in both series, which led to the release of the first revised +versions of the group number (Svalgaard and Schatten, 2016, “backbone” GN) +and of the sunspot number (Clette et al., 2014; Clette and Lef`evre, 2016, SN +Version 2.0). Regarding the GN, further corrections and improvements have been +proposed over recent years, but we will not develop this ongoing work here (see +e.g. Chatzistergos et al., 2017; Willamo, Usoskin and Kovaltsov, 2017; Svalgaard +and Schatten, 2016; Svalgaard, 2020; Usoskin, Kovaltsov and Kiviaho, 2021). +However, a key element that supported this revision effort was the expansion +and correction of the GN database collecting all original observed group counts +(Vaquero et al., 2016). This work, which started from the original database +assembled over many years by Hoyt and Schatten (1998a,b), is still continuing +now, and already allowed new improved reconstructions of the GN directly from +the base source data. As highlighted by Mu˜noz-Jaramillo and Vaquero (2019), +the recovery of all existing historical observations is crucial for future progresses +in such reconstructions of past solar activity. +By contrast, the current revised SN series was reconstructed from source data +only for the recent decades, since 1981, when the production of the SN moved +from the Zurich Observatory to the Royal Observatory of Belgium, where it is +still maintained today (Clette et al., 2007, 2016). Indeed, the data processing was +then computerized, and all collected data from the worldwide network of con- +tributing stations are preserved in digital form (more than 500,000 observations +from 285 stations). On the other hand, for the entire Zurich period before 1981, +the corrected SN series was obtained by deriving and applying correction factors +to the original Zurich SN series, as provided by Wolf and his successors (Clette et +al., 2014; Clette and Lef`evre, 2016). This approach already allowed to correct the +main flaws present in the original SN series and affecting long segments of this +series, in particular a sharp 18% upward jump in 1947 (see Clette and Lef`evre, +2016, for the details), but it faces limitations for finer corrections. +SOLA: Clette_SNDB2.tex; 9 January 2023; 1:30; p. 2 + +Sunspot Number Database and the 1947 Zurich Discontinuity +This more indirect and limited approach was imposed by two main constraints +that are specific to the history of the sunspot number. While the GN was directly +built from the whole set of available observations, the Zurich SN was mostly +based on the sunspot counts from the Zurich Observatory, which acted as pilot +station. The data from auxiliary stations were mostly used to fill in the daily +gaps due, e.g., to bad weather in Zurich, and they thus only played a secondary +role in the production of the early part of the SN (Clette et al., 2014; Dudok de +Wit, Lef`evre and Clette, 2016; Friedli, 2016, 2020). As a consequence, the sources +of inhomogeneity are predominantly associated with a single reference station, +and are thus very different from the GN, which requires other diagnostics. +However, the other major restriction was the absence of a global digital +database of the source data collected by Wolf and his successors. As we will +describe later in this article, only part of those data were published, and none +of those data were converted into digital form. The inaccessibility of the Zurich +source data prevents researchers from getting access to a huge amount of detailed +information and to essential metadata. The recovery of this vast collection can +feed full statistical analyses by current state-of-the-art methods and lead to an +improved index, independent of all assumptions and practices adopted over the +years by Wolf and his successors at the Observatory of Zurich. +This is what motivated a collective effort to recover and digitize all those +original source data. Major progresses have been accomplished over the past +few years. In this article, we report on those major advances. In Section 2, we +first present the global digitization of the published data, available in printed +form, and complemented by deeper archives of hand-written logbooks. Based +on the resulting global chronology of all contributing observers assembled in +Section 3, we summarize the temporal evolution of the sources on which the SN +was founded. In Section 4, we then present the recent recovery of the long-lost +Waldmeier archives, and we describe the contents of those new tables. Based +on the now-continuous historical timeline, we show the occurrence of a double +discontinuity in the composition of the Zurich team of observers (Section 5) and +the network of auxiliary stations (Section 6). In Section 7, we finish by concluding +on the overall Zurich history emerging from this early exploration of the new SN +database, and on the prospects and upcoming tasks. +2. Complete Digitization of Published Tables (1849-1944) +2.1. The Zurich Printed Data: Full Survey of the Mitteilungen +The Zurich sunspot number produced by Wolf and his successors is based on +three types of data: +• +the raw counts from the Zurich staff: essentially, the director and the assis- +tants in Zurich, and also in the course of the 20th century, other assistants +stationed in the Arosa and Locarno observatories in southern Switzerland. +• +the counts sent to the Observatory of Zurich by external auxiliary observers, +either individual solar observers or professional observatories. +SOLA: Clette_SNDB2.tex; 9 January 2023; 1:30; p. 3 + +Clette et al. +• +the historical observations collected by Wolf over the course of his entire +career, which extend the first two sets of data before 1849 and back to +1610. Most of those numbers were recounted by Wolf himself from original +documents (Friedli, 2020). +Most of this material was published on a yearly basis in the bulletins of the Zurich +Observatory, the Astronomische Mitteilungen der Eidgen¨ossischen Sternwarte +Z¨urich (hereafter Mitteilungen). This is a fundamental resource for any future +recomputation of the SN series. As noted in the introduction, a large part of +those data were never directly used for the production of the sunspot number, +as on most days, the SN was simply the raw Wolf number from the Zurich +Observatory. +In each issue of the Mitteilungen, the source data are listed in a series of +numbered rubrics at the end of the issue. The rubric series starts in 1857 (Volume +3, page 126) and ends in 1930 (Volume 122, page 41), at the 1727th entry, +forming all together a very comprehensive census of all data collected by the +Zurich Observatory. Systematic observations by the Zurich observers (with the +director and his assistants listed separately from 1870 onward) and by auxiliary +observers are presented in yearly tables (Figure 1) with, for each observed day, +the number of groups g and number of spots s, in the standard format g.s. +The table is preceded by a brief description of the observer, mainly his/her +name, the general location (city), and in most cases, the kind of telescope used +for the observations (aperture, focal length and magnification). Symbols are +sometimes added in the table to mark changes on a daily basis. The symbol +may identify a specific observer when there are several observers working in the +same observatory. In other cases, it marks a change of location or instrument. A +prominent example involves Wolf himself, who observed either with the standard +83 mm Fraunhofer refractor mounted permanently at the Zurich Observatory or +with smaller portable refractors (Friedli, 2016, 2020). This auxiliary information +can thus prove essential for the proper exploitation of the raw data. +Sometimes, when Wolf includes a new observer who already collected spot +counts over many years, a long multi-year table is published with all those past +observations. Key examples are the tables for Staudacher (Vol. 4, 1857), Schwabe +(Vol. 10 , 1859), Flaugergues (Vol. 13 , 1861), Carrington (Vol. 35, 1873) or +Pastorff (Vol. 36, 1875). Finally, next to the tables, many rubrics mention small +isolated data sets, or even unique spot counts found in old documents during +searches that Wolf did in libraries all over Europe. These are mostly single +sunspot sightings that are embedded in textual descriptions, e.g. spots noticed +at the occasion of solar eclipses. Although they may be individually important, +all together, they form only a tiny fraction of Mitteilungen data (< 1%), and +they are less exploitable because they cannot be calibrated. +2.2. The Digitization: First Milestone +So far, this large collection of data was completely inaccessible in digital form, +by contrast with the GN database, which includes all raw group counts collected +by Hoyt and Schatten (1998a,b) and was recently expanded by Vaquero et al. +SOLA: Clette_SNDB2.tex; 9 January 2023; 1:30; p. 4 + +Sunspot Number Database and the 1947 Zurich Discontinuity +Figure 1. Facsimile of a typical yearly table, as published in the Mitteilungen (first page +going up to early June). This table lists all daily observations from A.Wolfer for the year 1890. +Each column gives the date followed by the total number of groups and total number of spots, +separated by a dot. A star symbol is added for some days, and marks the days when the +observations were made occasionally with a different telescope (On these days, Wolfer used a +small portable “Parisian” telescope with a 40 mm aperture). +(2016). Although there is a rather wide overlap between the GN and SN data and +many observers are common to both data sets, the GN database unfortunately +contains only the number of groups. Therefore, the number of spots can only +be found in the Zurich data, as it was required to compute the SN. This thus +motivates the construction of a complete SN database, equivalent to the existing +GN database. +As a first major step, in 2018, a full encoding of the Mitteilungen data tables +was undertaken at the World Data Center Sunspot Index and Long-term Solar +Observations (SILSO), with the help of students for the bulk encoding work. +By the end of 2019, all the data tables have been digitized, forming the first +version of the SN database, which includes all data published between 1849, +when R. Wolf undertook the production of the sunspot number, and 1944, when +the last director, Max Waldmeier, decided to cease publishing raw data in print. +This database now contains 205,000 individual daily sunspot counts. Isolated +SOLA: Clette_SNDB2.tex; 9 January 2023; 1:30; p. 5 + +625) Alfred Wolfer, Beobachtungen der Sonnenfecken +auf der Sternwarte in Zurich im Jahre 1890. (Fortsetzung +zu 604.) +1890 +1890 +1890 +1890 +1890 +1 +1|1.1 +II +14/1.1 +III +17/0.0 +IV +15|1.2 +V +10/1.11 +21 +1.1 +1610.0* +18|0.0 +161.3 +11/2.11 +41.1 +200.0 +190.0 +170.0 +122.13 +1.1 +210.0 +210.0 +18/0.0 +140.0 +6 +2.5 +2210.0 +22/0.0 +19/0.0 +15/0.0 +18/0.0 +25|0.0 +23|1.3 +20|0.0* +160.0 +19/1.3* +26/0.0 +240.0 +21/0.0 +17|3.11 +20/1.3* +270.0 +26/0.0 +22/0.0 +18|2.11 +24/0.0 +28/1 +1.1 +270.0 +230.0 +192.6 +250.0 +III +11 +1.1 +280.0 +240.0 +203.6 +26|0.0 +2 +0.0 +290.0 +25/1.1 +22|2.5 +270.0 +3|1 +1.1 +300.0 +260.0 +23|1.1 +280.0 +411 +1.6 +31/0.0 +270.0 +24|1.1 +29 +0.0 +1.6 +IV +10.0 +28/1.3 +25/0.0 +30|1.2 +7 +1.5 +20.0 +291.7 +261.10 +31/1.6 +8] +1.10 +40.0* +30/1.11 +270.0 +II +11.3 +9|1 +1.10 +50.0* +V +1|1.1 +290.0 +2|0.0* +10/1.16 +60.0 +2|0.0 +300.0 +30.0 +11/1.11 +70.0 +30.0 +310.0 +40.0 +12|1.6 +9/0.0* +4/0.0 +VI +1/0.0 +50.0 +13|1.3 +10/0.0 +5|0.0 +20.0 +1010.0 +14/1 +1.3 +122.10 +60.0 +30.0 +110.0 +15/1 +1.1 +.13/2.8 +70.0 +一 +4|0.0 +12|0.0 +16/0.0 +141.1 +9|1.2 +一 +5|1.5 +NB. Die mit * bezeichneten Beobachtungen sind mit einem +kleinern Fernrohr gemacht, welchem etwa der Factor 1,5 zukommt. +April1891, +**Clette et al. +numbers mentioned in textual rubrics are not yet included, but we plan to add +them later on, for the sake of historical completeness. +Next to the daily separate counts of spots and groups, the database includes +metadata derived from annotations in the printed tables. When daily symbols +indicated regular changes of observers or instruments and when each subset +included a large number of days, we split the data included in common tables, +and attached the subsets to distinct observers. So, an observer may appear in +different incarnations, corresponding to different instruments and/or locations, +which thus require a different calibration and should not be mixed. +Currently, this first major input to the SN database is subjected to a thorough +quality control, fixing typos, date inconsistencies and occasional ambiguities in +observer names. Meanwhile, we looked for other data sources that can help +recovering information that proved to be missing in the Mitteilungen. One of +the gaps happens in the early part of the SN database. +2.3. Wolf’s Sourcebook and Wolfer’s Global Register +Indeed, before 1870, the information about the core observations made by Wolf +and his assistants is incomplete. A single “master” yearly table contains all the +counts used to produce the sunspot number. It thus consists mainly of the counts +made by Wolf, which are thus largely complete. On the other hand, data from +other observers, assistants or external observers, are only inserted on days when +the primary observer could not observe. As a consequence, between 1864, when +the first assistants were recruited, and 1869, only a small fraction of the data +from the Zurich assistants appear in the Mitteilungen, as Wolf’s own data fill a +majority of days. +Moreover, before 1864, Wolf’s main auxiliary observer was Samuel Heinrich +Schwabe. However, although a significant fraction of the daily counts were from +Schwabe, Wolf did not mark them in the published tables before 1859, as he +first considered Schwabe’s numbers fully equivalent to his own. This now makes +it impossible to distinguish Wolf’s primary counts from rescaled numbers from +Schwabe during the first 10 years of the Wolf series. This important information +about the primary Zurich observers is thus largely incomplete between 1849 and +1870. +Fortunately, two additional sources that provide full tables of the base counts +were preserved, and are archived at the ETH Zurich University Archives of +the Eidgen¨ossische Technische Hochschule (ETH). One of them is the so-called +Wolf’s sourcebook (Wolf, 1878, catalogue entry Hs368:46). Those handwritten +yearly tables gather all daily numbers forming the sunspot number series from +1610 to 1877 (Figure 2). In fact, these are the master tables assembled by Wolf +(Friedli, 2016). Those tables provide two kinds of unique information. Firstly, +right from the start of Wolf’s yearly census in 1849, they include symbols identi- +fying the source observer for each daily number. This additional information will +thus allow to remove the ambiguity in the early Mitteilungen tables. Moreover, +each yearly table indicates the personal k coefficient that was actually used +by Wolf, a precious information that can be crossed with the few occasional +mentions by Wolf of changes in his k calculations. +SOLA: Clette_SNDB2.tex; 9 January 2023; 1:30; p. 6 + +Sunspot Number Database and the 1947 Zurich Discontinuity +Figure 2. Facsimile of the table for 1860 in Wolf’s hand-written sourcebook, which covers the +period 1610 to 1877 (ETH catalogue entry Hs368:46). The layout is similar to the equivalent +yearly table published in the Mitteilungen, but it contains very important additional infor- +mation. Symbols indicate for each day, from which observer the daily sunspot number was +obtained. One can see that most of the observations were from Wolf, as primary observer. For +each auxiliary observer, including here Schwabe and Carrington, a list at bottom left mentions +the personal k coefficient that was used to rescale the raw numbers, to match the scale of +Wolf’s own numbers. +Moreover, as the copying and typesetting process for the publication in the +Mitteilungen most probably led to errors and typos, the original sourcebook +provides the ground truth and will allow fixing those occasional mistakes in the +master database. Thanks to the efforts of the Wolf Gesellschaft (Friedli, 2016), +Wolf’s sourcebook was digitized from 1849 to 1877, when the collection ends. +While the tables can now be consulted online at URL http://www.wolfinstitute. +ch/data-tables.html, this extended information must still be merged with the pri- +mary Mitteilungen database. This work is now in preparation. Finally, the yearly +tables in the sourcebook actually go back to the very first sunspot observations +in the early 17th century. Although this part is less substantial, those data tables +for years before 1849 must still be digitized. +However, like in the Mitteilungen, the sourcebook does not contain the full set +of raw observations collected by Wolf from the auxiliary observers and from his +assistants, between 1849 and 1870, in particular, the observations from Schwabe. +However, a larger set of handwritten tables also exists at the ETH Zurich +University archives (Wolfer, 1909, catalogue entry Hs1050:227). This series is a +standardized compilation of all data and metadata published in the Mitteilungen, +up to 1908 (Figure 3). This huge register was first produced by Wolf, and after +Wolf’s death in 1893, it was continued by A. Wolfer and his assistants until 1909, +as a base for a global verification of the sunspot number series. In this collection, +there is a separate table by observer and by year. Therefore, the full data set is +included, even data that were never used for the calculation of the daily Zurich +sunspot number. In particular, there are also many data series from before 1700, +SOLA: Clette_SNDB2.tex; 9 January 2023; 1:30; p. 7 + +147 +8 +6.32 +92 +.21 +121 +11 114 +10.5 +90 +7 +81 +3.19 +cm +.43 +10.3 +12 +70.9 +34 +52 +132 +.34 +104 +10 +9 +74s +96 +G +52 +35 +94 +97 +.32 +037 +3 +.29 +2.8 +^. +.32 +.22 +115 +10 +23 +121 +116 +.2 +.7 +:8 +14 +73 +h +5 .192 +761 +131 +64 +44 +4 +3.5 +154 +128 +8.36 +11 +5.22 +74 +57 +2.8. +718 +.2. +4. +.9 +36 +C +16 +3.15 +4 5 +.24 +85 +.1c +4.7. +103 +710 +34 +.31 +mm +w +53 +11 +99 +.25. +.21 +14 +112 +7 +5.12 +双 +3 0 +50 +.19 +4.14 +6x +.15 +127 +.61 +114 +4.15 +好 +15 +6 +.9: +.13 +30 +72 +15 +118 +6.33 +94 +.26 +848 +60 +4 +.13 +39 +3 7 +55 +9 +8t: +10 4 +w +5.1 +89529858848 +51 +91 +7 +6.13 +2 4 +104 +8.23 +103 +.4 +5.18 +6.13 +.34 +94 +1 +4.14 +6.22 +85 +.16 +:6 +4 . 7 +83 +19 +25 +4.22 +5.10 +.43 +113 +2.2 +6.1元 +64 +97 +85 +19 +6.41 +104 +89 +93 +93 +5.18 +91 +6.23. +6 +.11 +3h +58 +7.30 +1α1 +6.24 +84 +6.5 +99 +101 +5.25 +5.2 +.14 +w +.135 +94 +8.31 +to, +136 +13 +3k +65 +6.18 +45 +133 +11.34 +14 4 +73 +6 +.13 +双 +.9 +6.8 +s1 +6o +54 +128 +10.65 +9.17 +160 +.33 +93 +.19 +.18 +58 +8.25 +708 +49 +50 +9.13 +160 +G +6 +84 +.21 +712 +46 +.39 +12, +7.28 +6 +g.11 +3.30 +49 +1.61 +.16 +1u 4 +93 +27 +77. +111 +36 +w +10 1 +10 .19, +132 +164 +10. 43 +143 +10.64 +11 .160 +.33 +123 +2 1 +154 +78 +29 +11. 83 +8.16 +. +25 +44 +109 +23 +125 +114 +94 +10.47 +11.72 +-24 +114 +3 18 +6c +19 +86 +2 +8.31 +992 +10.46 +2.11 +33 +4.20 +3 +.17 +59 +33 +3.32. +10 2. +5.16 +4.8 +5 +6.37 +97 +7.44 +114 +10.31 +4. +5.19 +103 +5.13 +13 +3 . +tei +6 +K +6.13 +10 9 +w +44 +M. +116.7 +100,3 +92,2 +107,1 +108,6 +M. +9011 +97·9 +95,6 +M. +M. +81,5 +88.0 +98.9 +71.4 +Bemerkungen : +Bemerkungen: +T,00 += 1,50 +2 +Jehwae += 1,25 +1859 += +(amington += +7, 03 +webs +ht.si. +2h +fmeatmth! +0,47 +4o Vergl.m*tw,k,3 += +Jhon += +7,11 +46Clette et al. +Figure 3. Facsimile of one page from the register of hand-written tables compiled by R. Wolf +and continued by A. Wolfer, and covering the entire period 1610 - 1908 (ETH catalogue entry +Hs1050:227). This page shows the yearly table for Flaugergues in 1796. The layout is similar +to the yearly tables in Wolf’s sourcebook, but here, all daily observations are listed for each +observer. On the right, literal citations and detailed indications are often included to clarify +the interpretation of the tabulated numbers. This series of tables thus gives a complete and +well-standardized view of all data collected by Wolf and Wolfer, including data that were not +used to produce the daily sunspot number, and also data and metadata that were not published +in the Mitteilungen. +which were never used by Wolf, as he decided to compute the sunspot number +only from 1700 onwards. +Still, the tables in this complete register may prove invaluable for crossing +this information collected long ago by Wolf with other recovered observations +of the same observers. They also indicate which data were known by Wolf and +his collaborators at the epoch when they produced the Zurich numbers. The +scanning of this large set of tables is now planned at the ETH Library in Zurich. +When this step will be completed, the encoding into a database will require +substantial additional work. +3. Chronology of the Data +Although series of data and metadata still needs to be added, the database is now +largely complete between 1849 and 1944, and we have now already a complete +chronology of all the observers who provided data to the Zurich Observatory +between 1849 and 1980, i.e. during the entire Zurich era. This allows us to derive +SOLA: Clette_SNDB2.tex; 9 January 2023; 1:30; p. 8 + +1196 +Jlru yuyur I 100 +1796 +区 +X +11] rap 4 110 +1.8 +o.0 +1.s0s +0.0 +00 +0.0 +2 +3. 4 +6.0 +2 · +00 +0.0 +3 +0.0 +0. 0 +0.0 +0.0 +n wlit luhi +4 +2.4 +0·0 +0.0 +2.8 +0.0 +13( Jlg imum un +0 .0 +0.0 +2.3 +0.0 +2.9 +0.0 +0.0 +0.0 +lauhuy sd de ylun ln grrus du +6 +2.5 +H +7 +0.0 +2.3 +0.0 +2. ~ +0.0 +hin +1.1 +0.0 +2. ~ +0.0 +9 +0.0 +1.1 +0.0 +1.1 +0-0 +1.~ +S g' hu nuir 'ni rlri wn +(0 +1.2 +0.0 +1.1 +0.0 +1.( +( +2·3 +1.7 +le g muud ri lesrli; +2.3 +6.0 +2.3 +0.0 +1.1 +0.0 +13 +1.7 +0.0 +1.1 +0-0 +1.s +1-9 +14 +0.0 +1·3 +0.0 +1. G +1.f +0.0 +1·3 +0.0 +duervrd ln us, tuhis +16 +1-3 +0.0 +1.5 +0.0 +0.0 +1、- +1.6 +13 +0.0 +0.0 +0.0 +0.0 +2- +C +2.14 +1. 2 +1.-. +Tvmts +8/ +o.d +1.9 +1.1 +- 3 +mliimms dyuiy ci muir +19 +1.2 +1·2 +0.6 +0.0 +20 +Cro +2 -(2 +1.2 +1.1 +0.0 +21. +00 +nrrt l rcle +0.0 +2 +- 2 +23 +0.0 +1.1 +0.0 +1.2 +Iy) Nn umus d tuulur liyinss +- +24 +0.0 +1. 1 +0.0 +1-1 +25 + 2 +1. 1 +1.1 +0.0 +0.0 +26 +0.0 +1-1 +0.0 +0.d +0.0 +29 +1.4 +0.0 +rry rid rut rnh chs yurli. +1.4 +1.1 +24 +0.. +21 +2-2 +6.0 +0-0 +0.0 +Cmui le misie d dis y. +30 +4-u +1.8 +0.0 +0.0 +0.0 +0.0 +2.2 +1.2 +0.0Sunspot Number Database and the 1947 Zurich Discontinuity +Figure 4. Evolution of the number of stations for each year contained in the data tables +published in the Mitteilungen of the Zurich Observatory (gray curve). After 1919, when the +Zurich Observatory ceased to publish all the data, the total number of contributing stations is +plotted in blue, based on the annual list of stations. Between 1919 and 1944, the data from a +subset of observers were still included, but after 1945, none of the source data were published. +The two vertical shaded bands mark the two world wars, which both definitely left an imprint +on the Zurich sunspot data set. +some global statistics of the observers and the time interval over which they were +active, which provides very interesting new insights in the construction of the +Zurich SN. +Figure 4 shows the number of stations for each year. This illustrates the +evolution of the input data, as published in the Mitteilungen. The number of +stations steadily increased from 1865 to 1896, when it reaches about 20 sta- +tions and then drops slightly, but remaining above 15. This corresponds to the +continuous recruiting of new additional external observers by Wolf and later by +Wolfer. This evolution is completely disrupted in 1919. At the end of World +War I (WWI), Wolfer adds many new observers. The number of stations passes +the 40 mark, doubling the size of what becomes a true international network. +However, probably for financial reasons, Wolfer then decides not to publish all +data anymore (Friedli, 2020). Only the numbers from the Zurich observers and 7 +to 9 primary external observers are still published each year. Although some of +those privileged external observers had been important long-term contributors +by 1919, the selection criteria are unclear and were not explained by Wolfer. +But another drop of the number of tabulated data happens in 1926, when +William Otto Brunner succeeds Wolfer as director of the Zurich Observatory. +Brunner then decides to publish only the data from the Zurich team (Brunner, +1927). None of the data from the network are published after that year. The +only exception is Karl Rapp, a private observer, who observed in Locarno, +Switzerland, from 1940 to 1957. Rapp was actually trained in the same way +as assistants at the main observatory in Zurich, and was thus treated as an +internal observer over his whole observing career. Although Brunner states in +1927 that the external data from auxiliary stations are archived and can be +consulted on request (Brunner, 1927, page 188) (Friedli, 2020, Section 3.3), +searches undertaken over past years failed to recover those archives. So far, only +the data for 1944 were found in a single unpublished manuscript, referenced +SOLA: Clette_SNDB2.tex; 9 January 2023; 1:30; p. 9 + +WWI +WW II +60 +count +40 +Station +20 +Unpublished +Published +1860 +1880 +1900 +1920 +1940 +1960 +1980 +Time (years)Clette et al. +Hs1050:14 in the ETH Zurich University archives, which contains all calculation +sheets for that single year (Friedli, 2020). +Then in 1945, when Max Waldmeier becomes the new director, the publication +of source data ceases completely, as can be seen in Figure 4. By then, the volume +of data collected in Zurich had further increased, with almost 60 contributing +stations (blue curve in Figure 8), making their publication bulky and costly. +The Mitteilungen then switch to a different format. The thick yearly volumes +become a series of shorter thematic issues, with articles about diverse research +topics developed by Waldmeier. The sunspot number gets a more limited space, +compared to the earlier volumes published by Wolfer and Brunner, which were +almost entirely dedicated to sunspots. Again, during this last period of the Zurich +history, all the original data were saved like before in archives at the observatory +in Zurich. +However, since the closing of the Zurich Observatory in 1980, those archives +somehow went lost. This created a major 35-year data gap in the raw data +collection on which the Zurich sunspot number is based. This wide gap falls at +a critical moment, as one of the main scale jumps identified in the Zurich series +falls in 1947, thus precisely within this time interval (Clette et al., 2014; Clette +and Lef`evre, 2016). The raw input data are thus essential to reconstruct the +methodological changes that took place in Zurich at that epoch and may have +caused this inhomogeneity. Moreover, this gap creates a critical missing link +between the early Zurich epoch, up to Brunner, and the modern international +sunspot number produced in Brussels since 1981, for which all data are preserved +in a computer-accessible digital database. +4. The Original Waldmeier Source Tables (1945-1980) +4.1. A Serendipitous and Complete Recovery +Fortunately, in late 2018 and early 2019, a serendipitous finding by the staff of the +Specola Solare Ticinese Observatory in Locarno (https://www.specola.ch/e/), +followed by subsequent searches, allowed to recover the entire Waldmeier data +archive (1945 – 1979), which was in fact dispersed over three locations: the Specola +Observatory (26 years, 1945 – 1970), the Royal Observatory of Belgium in Brus- +sels (4 years, 1971 – 1974), and in the deep storage of the ETH Zurich University +archives in Zurich (5 years, 1975 – 1979). This dispersion seems to be due to +the rather tumultuous closure of the Zurich Observatory (for an evocation of +that transition, see Stenflo, 2016). Except for copies of the years 1975 – 1979 +on microfiches at the ETH archives, the fragmented original collection was also +stored without inclusion in any inventory or catalogue. +This recovery is a breakthrough, and given the amount of data collected over +those 35 years, it will keep researchers busy for many years. Indeed, we estimate +that those tables contain about 350,000 individual daily numbers, thus more +than in all published tables from 1849 to 1944. In a first step, all the elements of +this archive were brought together again at the ETH Zurich University archives. +They are now fully cataloged (Waldmeier, 1980), and the ETH archives have +SOLA: Clette_SNDB2.tex; 9 January 2023; 1:30; p. 10 + +Sunspot Number Database and the 1947 Zurich Discontinuity +Figure 5. Facsimile of a typical handwritten yearly table from the complete 1945 – 1980 +collection of source tables that was recovered in 2018 – 2019. This table lists the data from +H. M¨uller, one of the assistants observing at the Zurich Observatory with the standard 8-cm +Fraunhofer refractor, for the year 1960 (ETH catalogue entry Hs1304.8:16.3; DOI: 10.7891/e– +manuscripta-87290). For each day, the table gives the number of groups, the total number +of spots, the calculated personal k value relative to the primary observer (Waldmeier), and +a sky quality index. For each column, monthly sums and the mean k coefficient are given at +the bottom. The yearly totals and the mean k coefficient for the whole year are appended +at the lower right. Here, k equals 0.52 and thus differs by more than 15% from Waldmeier’s +target value of 0.6, revealing a significant dispersion of the Wolf numbers from the assistants, +although they were expected to be closely aligned on the primary observer. +completed the digitization of the whole collection in 2020. The scans of all tables +are now accessible online on the digital platform for manuscript material from +Swiss libraries and archives at https://www.e-manuscripta.ch/ (ETH catalogue +entry Hs 1304.8). Now, in order to make all the data computer-readable, all +those tables need to be encoded. This work has now just started at the Royal +Observatory of Belgium. +4.2. The Waldmeier Yearly Tables: a Key to the Zurich Method +The Waldmeier archive consists in yearly handwritten tables, one per observer, +and each one on a separate sheet. Over the period 1945 – 1980, there was an +average of 50 stations each year. All tables adopt the same standard format, +with one column per month. Figure 5 illustrates the typical layout of one sheet, +here with the table for H. M¨uller, one of the Zurich observers, for the year 1960. +External auxiliary stations are presented with exactly the same layout. Each +table lists all daily observations provided by the observer. The number of spots +SOLA: Clette_SNDB2.tex; 9 January 2023; 1:30; p. 11 + +Hs +1304.8:16.3 +Methode: +Beobachter: +Jahr... +V +VI +VII +IX +X +IX +IV +VIII +IIX +II +III +I +1960 +g, f +g, f +g, f +g, f +k +k +g, f +k +f +k +g, f +J‘8 +g, f +k +k +k +g, +f +k +k +k +k +ots +C.82#0.S2 +8.1024 +11.2532130.46 +2.93 +0.46 +13.2082-30-516623 +￥.182 +1 +8.14 +1-21230.935 +0.55 +3.193 +0.69 +8.1 +0.53 +3.21420.49 +0.50 +15.4420 +M.MiUn... +2 +12.82+30.531 +15.0.51 +0.50 +3.M +3.340.46 +3.242-0.50 +.60240.5310.1220.53 + Bemerkungen: +14.195130.55 +n-190230.51 +8.1202 +0.51 +19.141- +0.4 +3 +9.592-3 +8.1061-20.52 +M.291230.so3.210.63 +11.168-4 +M.21923 +0.433 +16.155, +4 +M.1692-3 +1-26050483.1530.55 +8.612-s0.50 +8.662-3 +0.5111.19520.51 +.1082130.52 +5 + Objektivoffnung: +14.228,0.4+13.1542 +8.14130.1731323 +4.843 +m.153420546.121230-51 +6 +8.432-30.52 +9443243 +14:1992130.4919.1363* +o.106r.0.52m.112o.s5 +8.1622-s0.4915.342-0.65 +< +10136.518.16530.515.51 +8 +8.1310.5510.1180.5111.104.0.52 +9.834-5 +10.1930.4812.110 +8.1542+3 +8.900.45 +1.1180.50 +1.14056 +9 +1.139 +M.100.5110.942-0.56 +8.106元 +10.1850.5213.1680.49 +1.10320.54 +6.11430.53 +10 +12-140.508.1413-4 +12-13230.53 +7.10210.49 +M.1920.491-161 +11 +9.142 +6.803 +13.301元 +11.1400.5014.1602 +1件.132 +6.2003 +12 +10.125,0.48 +0.52 +82051 +1.1182 +M.1410.S411.144420.517.89 +1Y31804815.159 +13 +0.45 +h4.3230.5116.1482 +4.2112-3 +13.992-30.54 +9.11242 +8.911-2 +1610.58 +14 +13.902+3 +3887050 +.200元3 +4.1432 +4.93元 +12180.481-133 +：3520.5111.1313 +15 +10.729 +8.10213 +3.1452 +9.4092 +Y.3850.441.10150.50 +16 +4.992-5 +14359元 +12.1043 +8.1500.50M.M50.4 +17 +Vergr.: +8.12230.52 +0.1233 +7.14250.4515 +210.8020.50 +5.10930.52 +trinih +0.50 +18 +13.14143 +￥1212-50.50 +L.lb-y0.S6 +8.1202 +￥.1300.53 +12.321 +7.8330.52 +19 +9.693 +.558.148,0.50 +L.622-3o.n.143 +13.182 +893130541 +5.45元 +810230.52 +16.150元 +20 +5.413 +L.482- 0.52 13.1342 +15.18+ +64. +21 +3.38- +8.1331-20.50 +552150.49 13.13025 +15.21330.52 +3.893-± 0.50 +22 +3.1523 +10.168,0.544.1212 +14-1912-30.S1 +23 +10.19120.49.1830.4113.81 +14.1712 +3.632-30.53 +5.6230.5Y +11.145 +0.52 +8.163/0.51 +24 +10.18843.53 +1.14510.18 +1188230.5069 +12.1412 +936.4 +25 +t86 +4.9325 +C.54-4 +0.52 +￥.12523 +10.MLL0.53 +.1950.53 +1.81, +01.1621420.49 + Sonnendurchmesser: +26 +8.83元 +10.81 +6.4030.52 +16.543-4 +8hoFstrw +8.14 +0.58 +0.53 +M.2080.41.1590.44 +27 +M.185r-2 +10.1930.481.81 +tsh +9.2224 +13.182130.468.105240.54 +88 +m.023 +9.2240.44 +1-18:7 +S.SS.0.50 +016.1432+3 +8.4030.52 +13.2430.5z +1.962 +0.49 +29 +8.110 +5.822 +tso +138 +0.55121L230.54 +13.2112-3 +S.130.50 +3149-5L0.94 +4.12 +8.2042-3 +30 +$·3 +9.1200.53 +6.48 +3.200 +31 +1.69 +8.34 +3.84 +6.33 +6-66 +9.33 +2-61 +6.43 +4:1 +件 +M +6.89 +Nr.: +2 +5 +13 +8 +1 +19 +19 +件 +9 +4 +N +21 +13 +19 +26 +1 +21 +14 +26 +11 +25 +19 +25 +0.52 +0.52 +0.54 +0.52 +0.53 +0.54 +0.49 +0-18 +0.54J +0.53 +0.52 +0.51 +M +45.20 +1960 +N +11462%3 +k= 0.52 (0.515)Clette et al. +and groups are given separately, exactly like in the tables published earlier in +the Mitteilungen. +This essential piece of information, which was so far entirely lost, will allow +to determine for each day exactly how the observers were separating sunspot +groups, on the one hand, and counting sunspots on the other hand. In partic- +ular, it will help clarifying and quantifying the use of weighted sunspot counts, +an alternate counting method adopted by the Zurich observers, in particular +by Waldmeier himself. This alternate counting rule, in which large spots with +extended penumbra are counted as more than 1, is suspected to be the cause of +the 18% upward jump that affected the original SN series in 1947 (Clette et al., +2014; Clette and Lef`evre, 2016; Svalgaard, Cagnotti and Cortesi, 2017). Indeed, +recent double counts, using the regular Wolf formula or weighted counts, were +made at the Specola Observatory during several years, between 2003 and 2015, +and led exactly to the same inflation of the sunspot number as the one found in +the Zurich series after 1947 (Clette et al., 2014; Svalgaard, Cagnotti and Cortesi, +2017). The recovered tables are thus providing the same kind of evidence, but +over 35 years, including the epoch when the jump occurred. +The tables also include the monthly and yearly mean k personal coefficients +computed by the Zurich Observatory, a very important piece of metadata to +understand how Zurich was treating the source observations. In particular, k +coefficients are given for all Zurich assistants, and also the associated observers +of the Specola station in Locarno. As all internal observers were assumed to align +themselves on the primary observer (Waldmeier during that period), without +applying any rescaling by a personal k coefficient, those internal yearly k values +can bring invaluable insights on how and to what extent assistants managed to +actually align themselves on the primary reference in their daily raw observa- +tions. As this internal practice was introduced by Wolf, as soon as 1870, when he +started to combine his own counts with those of his first assistants, this can thus +help in the understanding of the Zurich number production well before 1945. +In this collection, the most important tables are the yearly tables for the +primary observer, Max Waldmeier (Figure 6). They provide unique information +about three key aspects of the resulting Zurich sunspot numbers. Firstly, those +tables were the master tables from which the daily sunspot number was derived +for each day of the year. Therefore, they include raw counts and the resulting +Wolf number for each day of the year. They thus provide a complete day-by-day +census of how each daily SN was derived. +Secondly, most of the days contain the personal counts by Waldmeier, who +had the role of base reference. Therefore, this is the yearly table of raw group +and sunspot counts by the primary observer, which allows tracking changes in +Waldmeier’s own daily observations. For instance, Waldmeier was sometimes on +mission at the coronagraph of the astronomical station in Arosa, then observing +from high altitude with an alternate telescope. The counts for those days may de- +viate from the base reference scale defined by the standard Fraunhofer refractor +used on the front terrace of the observatory in downtown Zurich. Fortunately, +Waldmeier marked the days when he observed from Arosa, which will allow +analyzing the consequences of this site alternation. +SOLA: Clette_SNDB2.tex; 9 January 2023; 1:30; p. 12 + +Sunspot Number Database and the 1947 Zurich Discontinuity +Figure 6. Facsimile of the primary table for Max Waldmeier in 1957, extracted from the +complete 1945 – 1980 collection of source tables (ETH catalogue entry Hs1304.8:13.2; DOI: +10.7891/e-manuscripta-87246). Such tables are particularly important, as Waldmeier was +the pilot observer of the Zurich sunspot number over that 35-year interval. They include +various annotations that allow retracing day-by-day, how Waldmeier himself was observing, +and which alternate number was used on days when he could not observe. They thus contain +essential information about the Zurich data processing that cannot be found in any other +Zurich document. +Thirdly, the days in which Waldmeier could not observe are filled with num- +bers from local assistants or from the stations in Arosa or Locarno (Karl Rapp +until 1 April 1957 and the Specola Observatory starting on 1 October 1957). +As can be seen in Figure 6, those days are also marked in the tables with a +symbol identifying which alternate observer was used. Finally, as those tables +record the provisional values issued immediately at the end of each month, on the +remaining missing days when none of the local stations had managed to observe +the Sun, the numbers were simply interpolated between adjacent days, and those +dates are marked as “interpolated”. These are the few days which were later +replaced by definitive values calculated using k-normalized Wolf numbers from +the auxiliary stations, according to a standard method, of which the principle +can be reconstructed from a few reference documents (Friedli, 2020). +Those master tables thus provide almost all the keys that were badly miss- +ing to reconstruct the method and practices implemented in Zurich, and most +probably, to retrace persisting changes or local inconsistencies in the Zurich +processing. +SOLA: Clette_SNDB2.tex; 9 January 2023; 1:30; p. 13 + +Methode: +Beobachter: +Jahr: ... +T +II +III +IV +V +VI +VII +VIII +IX +X +XI +XII +Tmnisiris he kdaf'vzah lm. +R +g, f +R +R +g, f +R +g, f +R +g,f +g,f +R +R +g, f +R +g, f +R +g, f +R +R +1954 +g, f +A +g, f +doc +244/20.242 +150 +105 +14.145, +153 +140 +118 +13.134), +[M,202 ] +187 +15.2M +216 +[9.160,] +[12.114, +dr. +is. +(180) +13.6oz +11.1642 +164 +13.130 +(121) +14.2002 +204 +2 +[12.152,] +20.200 +240 +3-4 +13,244, +206 + Bemerkungen : ...mumliri. +[13.209,] +9.1382 +131 +3 +12.85, +123 +12.226, +20.215, +Ghz +4５ +10.1 +121 +3.130, +106 +18.268, +33 +(210) +10.362 +10.182 +19.144 +12.882 +12.110, +h8't +13.230 +230 +200 +18.250 +.Objektivoffnung: +[20.20b.] +６7 +16.83 +10.80 +14.183 +18.248 +daz +17.189 +1313, +9.136, +da + Sonnenfleckenbeobachtungen +1128, +8.160, +8 +15.176, +150 +12.122 +10.145 +250 +13130 +18 +1M.130 +9 +. +10.132 +163 +8.190, +162 +6.120, +12.81, +214 +13.149 +158 +1313 +195 +1 +15.1602 +9.160, +9.230, +[2.144 +14.98 +10 +150 +152471 +Its 1304.8:13.2 +11 +13112 +145 +10.88 +113 +2 +10.235, +1.124 +140 +110 +12.130 +12845678022230 +(16) +(160) +9702 +17.263, +260 +20.16b +13.1 +11.87, +NA +160 +忆忆忆 +22.1 +14 +[9.70,] +140 +13.204] +10.165, +M.1152 +1 +169 +44.192] +10 +1b,239, +yr +139 +11.134 +181 +1948 +M448 +121 +46.287 +100 +.Vergr.... +126 +(120 +13.116 +8h +15.1602 +186 +[18.184] +150 +14.102 +. +150 +8.102 +10 +13,296 +6.90 +19.283 +10 +149 +16.315 +18.184 +15.175 +13,288 +183 +15.204* +12.138 +6yx. +13.155 +dn +18.197 +16.1463 +dn +4.195, +(126) +12.122 +104 +15.249 +20.235 +11.282, +22 +35 +12.232 +20.3754 +150 +.. mdue. hidgnelhitit... Nr.: +170 +138 +127 +13.136, +15.333 +10 +200 +24.355 +17.160 +864 +1.134 +146/10.113 +128 +17.202 +21.307 +(Otl) +hhzhl +42.224,1 +3 +10 + Sonnendurchmesser: +12.10% +20.277 +11.1463 +(1%0) +14,256 +238 +171 +19.259, +19.1203 +14.683 +125 +12b +12.1234 +14+) +12.196h +20.234, +186 +29 +1215, +117 +16.135 +154 +[12.184] +142 +50 +23.228, +30 +9.513 +88 +15136 +155 +14.143,] +142 +M2,244,1 +213 +171 +21340 +33011.1bog +162 +24.217, +11.43, +92 +12.1221 +M45 +15.40,] +152 +22,290 +306 +21.215, +31 +4320 +5255 +$108 +6164 +6023 +5041 +330 +4867 +8149 +62.18 +7251 +32.69 +N +28 +31 +30 +[] +8 +31 +31 +31 +[13]+4+2* +30 +[2] +3 +30 +30 +31 +B] +16++* +10 +[28] +31 +[4] +13 +1a7+12 +3 +[10]+ 1 +34 +152.3 +145.2 +164.8 +105.6 +194.3 +12.6 +M +157.0 +944.3 +116.8 +262.9 +233.9 +1954 +69318 +355 +180 +29,6.02 184) -182 +ponoredus fora mMd +30.b[12.244,]-200Clette et al. +Figure 7. Timelines of the active observing periods of all Zurich observers. In red (top group), +the primary observers and in orange (bottom group), the assistants. In purple, the observers +of the auxiliary station in Locarno, who were considered as members of the Zurich core group. +The vertical shaded band marks World War II and the vertical dashed line indicates the time +when the 1947 scale jump occurs in the original SN series. The bottom plot gives the number +of active Zurich observers for each year. +5. A Major Disruption: Zurich Observers +Although the above data still need to be digitized, we now have the full list +of observers who contributed year-by-year to the Zurich sunspot number up to +1980. By assembling the timelines of each individual observer, we could map +how their observing period overlaps with other observers. Figure 7 retraces the +observing periods for all Zurich primary observers and all the assistants, between +1850 and 1960. +In this figure, Schwabe is included among the assistants (orange group) al- +though he was an external observer. Indeed, before Wolf could recruit his first +assistants in the newly founded Zurich Observatory in 1865, he used Schwabe’s +numbers as primary alternate source for filling the gaps in his own observations, +and even initially considered Schwabe’s numbers as fully equivalent to his own +(personal k = 1) before 1859. We also included K. Rapp in the associated Lo- +carno station (purple group) although he contributed before the establishment +of the Specola Observatory by Waldmeier in 1957, starting in 1940. Indeed, +both Brunner and Waldmeier always included Rapps’s data together with the +SOLA: Clette_SNDB2.tex; 9 January 2023; 1:30; p. 14 + +Wolf ST +WolfPR +Wolfer +Brunner,wlo. +Waldmeier +Schwabe +Fretz +Weilenmann +Mever +Billwiller +Fauquez +Hoffler +Broger +Observers +Biske +Buser Arosa +Brunner Ass +Muller +Beck +Muller,E +Wile +Lemans +Scheidlegger +Frick +Hermes +Riesen +Zelenka +Durst +Pfister +Rapp +Keller +Schmidt +ilszak +Cbrtesi +Pittini +1840 +1860 +1880 +1900 +1920 +1940 +1960 +1980 +Time (years)Sunspot Number Database and the 1947 Zurich Discontinuity +Zurich data in the Mitteilungen, even when the data of all the other external +stations were not published anymore. Rapp was also trained to follow the Zurich +observing methods, and can thus be considered as an internal member of the +Zurich group of stations. Finally, although Waldmeier, the last primary observer +(red group), started observing as an assistant in 1936, his participation was +partly interrupted, as explained below. +For the period before 1944, the resulting chronology reveals a few interesting +facts. In particular, one of Wolfer’s assistants, Max Broger, had a very long +observing career (40 years, 1896 to 1935). He actually observed over more years +than several primary observers. As he observed in parallel with Wolfer and then +with Brunner, his observations can provide an essential link to check the Wolfer- +Brunner homogeneity. +This touches the fundamental issue of the weighted sunspot counts used by +the Zurich Observatory, as mentioned in the previous section. Indeed, Clette +et al. (2014), Clette and Lef`evre (2016), and Svalgaard, Cagnotti and Cortesi +(2017) conclude that this alternate counting method is the most likely cause of +the 1947 scale jump in the original SN series. However, the timing and sharpness +of the jump seem to be contradicted by the fact that this weighting practice was +introduced progressively well before 1947, in the early 20th century by Wolfer +(Cortesi et al., 2016; Svalgaard, Cagnotti and Cortesi, 2017). Although Wolfer +himself never used it for his own counts (Svalgaard, Cagnotti and Cortesi, 2017), +this practice was implemented to help assistants aligning their raw counts on the +reference of Wolfer, the primary observer. This could be verified by taking the +counts on occasional days when only a single big spot was visible on the Sun. +Then, when one of Wolfer’s assistants, W.O. Brunner, took over as director and +as primary observer in 1926, he continued to use weighted counts, but this time as +primary observer. Although this marks the moment when the break with Wolf’s +original methodology occurred, Brunner managed to maintain the stability of his +counts, as found by Svalgaard, Cagnotti and Cortesi (2017). When Waldmeier +took his succession in 1945, after being assistant for a few years, he thus just +continued an established practice. So, apparently, this chronology does not match +at all the abrupt occurrence of a jump in 1947, two years after Waldmeier became +the new reference observer, a status that he kept for the next 35 years without +any other noticeable transition. +Now, by retracing the composition of the network of collaborating observers, +we found evidence of a major transition that occurred between 1945 and 1947. +The change was twofold. Firstly, at the Zurich Observatory, although Waldmeier +became director in 1945, the former director, W.O. Brunner, actually continued +observing during one year until December 1945 (see Figure 7). Moreover, his +primary assistant, W. Brunner-Hagger, who was part of the team since 1928, +continued until August 1946. This actually marks the moment when the link +with the former Zurich core team is broken. As shown in Figure 7, in 1945, +Waldmeier starts to recruit new assistants. However, the first one, Beck, worked +in parallel with Brunner only during a few months, when solar activity was +rather low, and he left the observatory already in 1949. Then follows a succession +of other assistants who also leave after only a few years. This means that the +overlap between the old and new team was extremely limited and that for several +SOLA: Clette_SNDB2.tex; 9 January 2023; 1:30; p. 15 + +Clette et al. +years the Zurich team was very unstable, contrary to the Brunner team that had +remained unchanged for nearly 20 years. +So, the internal stability of the Zurich system during the 1945 Brunner- +Waldmeier transition rested only on Waldmeier himself. This is unprecedented +in the entire Zurich history. Indeed, the stability of the Wolf-Wolfer transition +benefited from a 17-year period, during which Wolf and Wolfer observed jointly. +Although the Wolfer-Brunner joint period was shorter (3 years, 1926-1928), +another assistant, Broger brought a solid reference to bridge the Wolfer-Brunner +transition, as he had worked jointly with Wolfer for 30 years, since 1896, and +then continued for 10 years together with Brunner, until 1935. +Finally, although Waldmeier started collaborating with the Zurich Obser- +vatory in 1936, he did not contribute during three years, from 1939 to 1941 +because of the onset of World War II. Moreover, as he was strongly involved in +coronagraph observations, he worked for a large part of his time at the Arosa +station, rather than as an ordinary assistant observing side by side with Brunner +in Zurich. We also note that the last years before 1946 fell in a minimum of +the solar cycle, when the low sunspot activity makes mutual comparisons less +accurate. Therefore, all those circumstances reduced the effective overlap period +between Brunner and Waldmeier. +6. A Major Disruption: Auxiliary Stations +In parallel with the Zurich internal transition, another major and unprecedented +disruption also occurred just after 1945, but now for the Zurich auxiliary sta- +tions. Although those external data were not at the core of published sunspot +numbers, they definitely provided a wide ensemble of independent data series +against which the Zurich numbers were continuously compared. Moreover, all +external stations derived their counts using Wolf’s original definition, without +any weighting. Therefore, the auxiliary data were not affected by the introduc- +tion of Zurich’s internal weighting practice, and in that sense, they provided +the only base against which the Zurich team could infer that their weighted +numbers remained coherent with the unwheighted Wolf numbers that formed +the original SN series until Wolfer’s retirement in 1926 (Clette et al., 2014; +Svalgaard, Cagnotti and Cortesi, 2017). This continuous bench-marking could +only work if at any given time, there was a large number of active auxiliary +stations which had already contributed data during many past years, preferably +over one or more full solar cycles. +Figure 8 shows the timelines of all auxiliary stations that contributed obser- +vations to the Zurich Observatory since the mid-19th, over a duration longer +than 11 years, i.e. a full solar cycle. This subset of long-duration stations is +indeed the most important for the long-term calibration and stability of the +series. We distinguished the professional observatories from the individual ama- +teur observers, which reveals a deep evolution in the composition of the Zurich +observing network. While a large majority of stations were individual observers +before World War II (WWII), professional observatories dominate the network +after WWII. +SOLA: Clette_SNDB2.tex; 9 January 2023; 1:30; p. 16 + +Sunspot Number Database and the 1947 Zurich Discontinuity +Figure 8. Timelines of the active observing periods of all external stations that sent data +to Zurich until the observatory was closed in 1980. The stations are ordered according to the +starting date of their series. The top series (dark blue) gathers the professional observatories +and the bottom group (light blue) gathers the individual amateur observers. The vertical +shaded band marks World War II and the vertical dashed line indicates the 1947 scale jump +in the original Zurich series. The bottom plot gives the total number of active stations per +year. The light-blue section indicates unpublished data that have not been recovered yet in +the Zurich archives. +However, a much more drastic change is also caused by WWII. In Figure 8, +we see that, starting in 1938, long-time contributing stations cease to send data, +one after the other. When WWII ended, none of those stations, which gave an +external benchmark for the earlier Zurich SN, had survived. During the war, +given the steep drop of contributing stations, Brunner and Waldmeier called to +the rescue a large number of local Swiss amateur astronomers, but this local +network was quickly changing, as most observers contributed only for one year +or at best a few years (therefore, they do not appear in Figure 8). None of those +observers were long-term observers in the preceding Zurich network established +by Wolfer and Brunner. +SOLA: Clette_SNDB2.tex; 9 January 2023; 1:30; p. 17 + +Observers +Obs/year +40 +20 +0 +1860 +1880 +1900 +1920 +1940 +1960 +1980 +Time (years)Clette et al. +Then, just after the war, Waldmeier quickly undertakes the construction of +a new international network. The number of stations grows steeply and reaches +about 50 stations (see Figure 4), a number that will remain rather stable until +1980. As noted before, this new network includes many professional observa- +tories, which since then, have delivered observations over very long durations. +In fact, some of them are still contributing nowadays to the worldwide SILSO +network, and thus provide an invaluable long-term reference spanning up to 75 +years, since 1945. However, none of those new stations were part of the pre- +1940 long-term network. Therefore, the context in which the sunspot number +was produced after 1945 was largely disconnected from the context surrounding +this production before 1940. This further weakened the thin internal continuity +within the Zurich Observatory. +In order to give a more quantitative measure of this second disruption, we +summed the number of past observed years already accumulated by all stations +that were active on a given year. Figure 9 (top plot) shows the temporal evolution +of this total number, which gives a measure of the total amount of past informa- +tion that the Zurich Observatory had at its disposal for past comparisons and the +verification of their stability relative to independent observers. As expected, the +evolution is characterized by a steady increase in the total amount of available +data. The only interruption in this trend is the steep drop during WWII, when +the count suddenly drops back to the values of the early 20th century. After +WWII, there is a recovery, but it takes about 15 years before the amount of +past reference data comes back to the value just before WWII. Afterwards, the +amount of past data from active stations continues to grow and finally stabilizes +in the 1970’s. +If we divide this total number of past observed years by the number of active +stations, we obtain the mean past duration over which stations active at a given +time have been observing before that time (Figure 9; bottom plot). This mean +duration quantifies the past memory built into the SN system. Between 1860 and +1890, this mean duration increases. This marks the progressive recruiting of the +first auxiliary observers by Wolf. Then, the mean duration largely stabilizes until +1926, i.e. the Wolfer-Brunner transition. The only feature is a temporary peak +associated with WWI, which thus left only a minor imprint in this evolution. +Thanks to the many new observers recruited after WWI, and who continue +observing until WWII, the mean duration grows to almost 15 years in 1938. +Then, WWII again produces a steep drop, by a factor of two. In 1945 and the +decade that follows, the mean memory range falls back to about 7 years, a level +that was not encountered since 1880, i.e. the epoch when Wolf was still recruiting +his first associated observers. After this dramatic shortening of the past memory, +there was a steady recovery. However, it is only around 1965, 20 years after +WWII, that the pre-WWII mean memory range is recovered. It continues to +rise until 1980, when the Zurich Observatory was closed. This continuous trend +largely rests on the long-term contribution from the professional observatories +that entered the network just after WWII. Figure 9 thus illustrates that the +years immediately following WWII were abruptly affected by a major loss of +past references, and that this loss had no equivalent in the history of the Zurich +SN number. +SOLA: Clette_SNDB2.tex; 9 January 2023; 1:30; p. 18 + +Sunspot Number Database and the 1947 Zurich Discontinuity +Figure 9. Evolution of the amount of past data available for each year at Zurich. The upper +plot gives the total number of preceding observed years by all the stations active on a given +year. After an almost continuous increase, a sharp drop occurs just after WWII. The lower +plot shows the mean number of preceding observed years per station, for all stations active on +a given year. The rise after 1925 indicates the growing participation of stations with very long +duration, but a drop to 19th century levels marks the late 1940’s and early 1950’s. +Although the above indicators are indirect contextual elements, the fact that +this unique double discontinuity in the history of the Zurich sunspot number +production coincides with the jump revealed by the SN series itself is a very +strong indication that the sharp SN scale jump was a consequence of this abrupt +and radical change in the base data input. Until 1946, the potential biasing effect, +which was present since the weighted counting method had been introduced, +had been kept under control thanks to the double stabilizing effect of long-term +internal and external observers who did not change their counting practices. +This stabilizing continuity was clearly broken between 1946 and 1947, which +suddenly opened the way for the biasing effect inherent to the weighted counts, as +evidenced by the 1947 upward jump. This new contextual evidence thus explains +SOLA: Clette_SNDB2.tex; 9 January 2023; 1:30; p. 19 + +800 +Amount of past data +600 +400 +200 +0 +1860 +1880 +1900 +1920 +1940 +1960 +1980 +Time (years) +25 +duration (years) +20 +15 +past +10 +Mean +5 +0 +1860 +1880 +1900 +1920 +1940 +1960 +1980 +Time (years)Clette et al. +simultaneously the delayed effect of the weighting practice and the abruptness +of the jump. +7. Conclusion +Over just a few years, we thus achieved major progress in the construction of +the SN database. Now, about two thirds of the existing source data are recorded +in digital form. We can now also report on the recovery of a major missing part +of this collection, the yearly source tables of the Waldmeier era from 1945 to +1980. This fills the main gap in the SN database and provides the missing link +between the contemporary index and the rest of this long series before 1945 +and back to 1700. While significant work is still needed to digitize those newly +recovered documents, the global panorama that the SN database now offers +made it possible to establish the complete chronology of contributing stations +and observers. We found that the two world wars had deep consequences on +the production of the SN by the Zurich Observatory. WWI brought a major +expansion of the network of auxiliary observers, but without disrupting the +internal practices and organization of the Zurich sunspot observers. +On the other hand, after WWII, we find a double disruption in the Zurich +system. A complete renewal of the Zurich observing team occurred between 1946 +and 1947, with almost no overlap between the old team, which had remained +mostly unchanged for more than 20 years, and the new team progressively built +by Waldmeier between 1946 and 1950. Moreover, after the loss of most of the +external observers active over the decades preceding WWII, between 1938 and +1945, an entirely new worldwide network is established after the war with entirely +different stations. The narrow correspondence of this drastic and unprecedented +structural change with the 18% SN scale-jump diagnosed in the SN series pro- +vides strong historical evidence that a sharp jump in the SN exactly at that +moment is a real and logical consequence. Although the suspected cause, i.e. the +introduction of the size-based weighting of the spot counts, was introduced much +earlier in the practice of Zurich assistants, our now-complete timeline explains +why it only led to actual consequences when this sharp and unprecedented +discontinuity in the Zurich system took place. +All together, those recovered tables open the way to future major steps in the +end-to-end calibration of the sunspot number series. Full statistical diagnostics of +the actual stability of each separate Zurich observer, which was simply postulated +since the epoch of Wolf, will allow disentangling in detail the causes of anomalies +found in the heritage series. Much more importantly, those data open the way for +a full recalculation of the sunspot number, starting again from the full set of raw +input data. This recalculation will use new advanced computer-based processing +methods, which exploit the entire set of data instead of mostly using the numbers +of the single primary observer, as was the case in the original Zurich series. This +should improve further the stability and accuracy of the sunspot number in the +interval 1945-1980, where so far, SN Version 2 consisted only in a correction +factor applied to the original Zurich SN series. This would also finally bridge the +gap separating the current international sunspot number from the early epoch +before 1945. +SOLA: Clette_SNDB2.tex; 9 January 2023; 1:30; p. 20 + +Sunspot Number Database and the 1947 Zurich Discontinuity +However, a partial gap still remains. Although all observations made in Zurich +from Wolf in 1849 to Waldmeier in 1980 now finally form a complete and unin- +terrupted thread, we still miss the unpublished archives from the Brunner era. +Therefore, efforts are still continuing to try recovering the last missing data from +the network of the auxiliary stations between 1919 and 1944. Hopefully, this will +finally bring the last touch to this digital database that will feed sunspot science +and long-term solar-cycle studies for many years. +Acknowledgments +This work and the team of the World Data Center SILSO (http://www. +sidc.be/silso/), which produces the international sunspot number and maintains the sunspot +database used in this study, are supported by Belgian Solar-Terrestrial Center of Excellence +(STCE, http://www.stce.be) funded by the Belgian Science Policy Office (BelSPo). This work +was also supported by the International Space Science Institute (ISSI, Bern, Switzerland) via +the International Team 417 “Recalibration of the Sunspot Number Series”, chaired by M. +Owens and F. Clette (https://www.issibern.ch/teams/sunspotnoser/). Specola Solare Ticinese +acknowledges the financial support provided by Canton Ticino through the Swisslos fund +and by the Federal Office of Meteorology and Climatology MeteoSwiss, in the framework of +GCOS. We would like to thank Thomas Friedli for digitizing and making available the original +sourcebook by R. Wolf via the web site of the Rudolf Wolf Society (http://www.wolfinstitute. +ch). We also thank the ETH Library (https://library.ethz.ch/en/), and in particular Evelyn +Boesch, of the Hochschularchiv, for the deep searches in the catalogues and archives, and for +giving us access to original historical documents from the Zurich Observatory. We also thank +Olivier Lemaˆıtre for developing the software and computer database, Stephen Fay and Shreya +Bhattasharya for the quality control, and last but not least, we are also grateful to the summer- +job students who patiently and carefully encoded all numbers tabulated in the original paper +documents: Elfaniel Hermel, Esther-Lauren M’Bilo and Mael Panouillot. +Disclosure of Potential Conflicts of Interest +The authors declare that they have no conflicts of interest. +References +Brunner, W., 1927. 2. Die Sonnenfleckenstatistik f¨ur das Jahr 1926, Astron. Mitteil. Eidgn. +Sterw. Z¨urich, CXVI, 179-194, Sept. 1927. +Chatzistergos, T., Usoskin, I.G., Kovaltsov, G.A., Krivova, N.A., Solanki, S.K., 2017. +New reconstruction of the sunspot group numbers since 1739 using direct calibration +and “backbone” methods, Astron. Astrophys., 602, id.A69, 18 pp., DOI 10.1051/0004- +6361/201630045 +Clette, F. and Lef`evre, L., 2016. 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Sonnenflecken – Statistik 1600 – 1900, ETH Bibliothek, Hochschularchiv, Hs +1050:227, 13 Dossiers, unpublished manuscript, Zurich. +SOLA: Clette_SNDB2.tex; 9 January 2023; 1:30; p. 22 + diff --git a/GNE0T4oBgHgl3EQfhQF8/content/tmp_files/load_file.txt b/GNE0T4oBgHgl3EQfhQF8/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6513721699b8e5befd66501c88abdb80d1a31abc --- /dev/null +++ b/GNE0T4oBgHgl3EQfhQF8/content/tmp_files/load_file.txt @@ -0,0 +1,1658 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf,len=1657 +page_content='Solar Physics DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='1007/•••••-•••-•••-••••-• Reconstruction of the Sunspot Number Source Database and the 1947 Zurich Discontinuity Fr´ed´eric Clette1 · Laure Lef`evre1 · Sabrina Bechet1 · Renzo Ramelli2 · Marco Cagnotti3 © Springer •••• Abstract The recalibration of the sunspot number series, the primary long- term record of the solar cycle, requires the recovery of the entire collection of raw sunspot counts collected by the Zurich Observatory for the production of this index between 1849 and 1980.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Here, we report about the major progresses accomplished recently in the con- struction of this global digital sunspot number database, and we derive global statistics of all the individual observers and professional observatories who pro- vided sunspot data over more than 130 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' First, we can announce the full recovery of long-lost source-data tables covering the last 34 years between 1945 and 1979, and we describe the unique information available in those tables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' We then also retrace the evolution of the core observing team in Zurich and of the auxiliary stations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' In 1947, we find a major disruption in the composition of both the Zurich team and the international network of auxiliary stations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' This sharp transition is unique in the history of the Zurich Observatory and coincides with the main scale-jump found in the original Zurich sunspot number series, the so-called “Waldmeier” jump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' This adds key historical evidence ex- plaining why methodological changes introduced progressively in the early 20th century could play a role precisely at that time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' We conclude on the remaining steps needed to fully complete this new sunspot data resource.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Keywords: Sunspots, statistics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Solar Cycle, observations � F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Clette frederic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='clette@oma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='be 1 Royal Observatory of Belgium, 3 Avenue Circulaire, 1180 Brussels, Belgium 2 Istituto Ricerche Solari Locarno (IRSOL), Universit`a della Svizzera italiana, Via Patocchi 57, 6600 Locarno, Switzerland 3 Specola Solare Ticinese, Via ai Monti 146, 6605 Locarno, Switzerland SOLA: Clette_SNDB2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='tex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' 9 January 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' 1:30;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='02429v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='SR] 6 Jan 2023 Clette et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Introduction Our knowledge of the long-term evolution of the solar cycle is largely based on the historical observations of sunspots since the newly invented telescope was aimed at the Sun for the first time in 1610.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Two main indices were built from those sunspot observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' The sunspot number (hereafter SN) was initiated by Rudolf Wolf in 1850 (Wolf, 1856;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Friedli, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' This daily index combines the total group count and the total spot count and its goes back to 1700.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Much more recently, Hoyt and Schatten (1998a,b) introduced the sunspot group number (hereafter GN), which only uses the total group count, but was constructed back to the very first telescopic observations in 1610.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Both indices are abundantly used by most studies of the long-term evolution of solar activity and Sun-Earth relations, as constraints for validating physical models of the solar dynamo, and for calibrating various parameters relevant to space weather and space climate (geomagnetic and ionospheric indices, cosmogenic radionucleides).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' However, significant disagreements between the sunspot number and group number series over their common time interval indicated that either series or both suffered from inhomogeneities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' This prompted various efforts to identify flaws and biases in both series, which led to the release of the first revised versions of the group number (Svalgaard and Schatten, 2016, “backbone” GN) and of the sunspot number (Clette et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Clette and Lef`evre, 2016, SN Version 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Regarding the GN, further corrections and improvements have been proposed over recent years, but we will not develop this ongoing work here (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Chatzistergos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Willamo, Usoskin and Kovaltsov, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Svalgaard and Schatten, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Svalgaard, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Usoskin, Kovaltsov and Kiviaho, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' However, a key element that supported this revision effort was the expansion and correction of the GN database collecting all original observed group counts (Vaquero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' This work, which started from the original database assembled over many years by Hoyt and Schatten (1998a,b), is still continuing now, and already allowed new improved reconstructions of the GN directly from the base source data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' As highlighted by Mu˜noz-Jaramillo and Vaquero (2019), the recovery of all existing historical observations is crucial for future progresses in such reconstructions of past solar activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' By contrast, the current revised SN series was reconstructed from source data only for the recent decades, since 1981, when the production of the SN moved from the Zurich Observatory to the Royal Observatory of Belgium, where it is still maintained today (Clette et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=', 2007, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Indeed, the data processing was then computerized, and all collected data from the worldwide network of con- tributing stations are preserved in digital form (more than 500,000 observations from 285 stations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' On the other hand, for the entire Zurich period before 1981, the corrected SN series was obtained by deriving and applying correction factors to the original Zurich SN series, as provided by Wolf and his successors (Clette et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Clette and Lef`evre, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' This approach already allowed to correct the main flaws present in the original SN series and affecting long segments of this series, in particular a sharp 18% upward jump in 1947 (see Clette and Lef`evre, 2016, for the details), but it faces limitations for finer corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' SOLA: Clette_SNDB2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='tex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' 9 January 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' 1:30;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' 2 Sunspot Number Database and the 1947 Zurich Discontinuity This more indirect and limited approach was imposed by two main constraints that are specific to the history of the sunspot number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' While the GN was directly built from the whole set of available observations, the Zurich SN was mostly based on the sunspot counts from the Zurich Observatory, which acted as pilot station.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' The data from auxiliary stations were mostly used to fill in the daily gaps due, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=', to bad weather in Zurich, and they thus only played a secondary role in the production of the early part of the SN (Clette et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Dudok de Wit, Lef`evre and Clette, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Friedli, 2016, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' As a consequence, the sources of inhomogeneity are predominantly associated with a single reference station, and are thus very different from the GN, which requires other diagnostics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' However, the other major restriction was the absence of a global digital database of the source data collected by Wolf and his successors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' As we will describe later in this article, only part of those data were published, and none of those data were converted into digital form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' The inaccessibility of the Zurich source data prevents researchers from getting access to a huge amount of detailed information and to essential metadata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' The recovery of this vast collection can feed full statistical analyses by current state-of-the-art methods and lead to an improved index, independent of all assumptions and practices adopted over the years by Wolf and his successors at the Observatory of Zurich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' This is what motivated a collective effort to recover and digitize all those original source data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Major progresses have been accomplished over the past few years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' In this article, we report on those major advances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' In Section 2, we first present the global digitization of the published data, available in printed form, and complemented by deeper archives of hand-written logbooks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Based on the resulting global chronology of all contributing observers assembled in Section 3, we summarize the temporal evolution of the sources on which the SN was founded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' In Section 4, we then present the recent recovery of the long-lost Waldmeier archives, and we describe the contents of those new tables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Based on the now-continuous historical timeline, we show the occurrence of a double discontinuity in the composition of the Zurich team of observers (Section 5) and the network of auxiliary stations (Section 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' In Section 7, we finish by concluding on the overall Zurich history emerging from this early exploration of the new SN database, and on the prospects and upcoming tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Complete Digitization of Published Tables (1849-1944) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' The Zurich Printed Data: Full Survey of the Mitteilungen The Zurich sunspot number produced by Wolf and his successors is based on three types of data: the raw counts from the Zurich staff: essentially, the director and the assis- tants in Zurich, and also in the course of the 20th century, other assistants stationed in the Arosa and Locarno observatories in southern Switzerland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' the counts sent to the Observatory of Zurich by external auxiliary observers, either individual solar observers or professional observatories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' SOLA: Clette_SNDB2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='tex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' 9 January 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' 1:30;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' 3 Clette et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' the historical observations collected by Wolf over the course of his entire career, which extend the first two sets of data before 1849 and back to 1610.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Most of those numbers were recounted by Wolf himself from original documents (Friedli, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Most of this material was published on a yearly basis in the bulletins of the Zurich Observatory, the Astronomische Mitteilungen der Eidgen¨ossischen Sternwarte Z¨urich (hereafter Mitteilungen).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' This is a fundamental resource for any future recomputation of the SN series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' As noted in the introduction, a large part of those data were never directly used for the production of the sunspot number, as on most days, the SN was simply the raw Wolf number from the Zurich Observatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' In each issue of the Mitteilungen, the source data are listed in a series of numbered rubrics at the end of the issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' The rubric series starts in 1857 (Volume 3, page 126) and ends in 1930 (Volume 122, page 41), at the 1727th entry, forming all together a very comprehensive census of all data collected by the Zurich Observatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Systematic observations by the Zurich observers (with the director and his assistants listed separately from 1870 onward) and by auxiliary observers are presented in yearly tables (Figure 1) with, for each observed day, the number of groups g and number of spots s, in the standard format g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' The table is preceded by a brief description of the observer, mainly his/her name, the general location (city), and in most cases, the kind of telescope used for the observations (aperture, focal length and magnification).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Symbols are sometimes added in the table to mark changes on a daily basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' The symbol may identify a specific observer when there are several observers working in the same observatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' In other cases, it marks a change of location or instrument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' A prominent example involves Wolf himself, who observed either with the standard 83 mm Fraunhofer refractor mounted permanently at the Zurich Observatory or with smaller portable refractors (Friedli, 2016, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' This auxiliary information can thus prove essential for the proper exploitation of the raw data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Sometimes, when Wolf includes a new observer who already collected spot counts over many years, a long multi-year table is published with all those past observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Key examples are the tables for Staudacher (Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' 4, 1857), Schwabe (Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' 10 , 1859), Flaugergues (Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' 13 , 1861), Carrington (Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' 35, 1873) or Pastorff (Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' 36, 1875).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Finally, next to the tables, many rubrics mention small isolated data sets, or even unique spot counts found in old documents during searches that Wolf did in libraries all over Europe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' These are mostly single sunspot sightings that are embedded in textual descriptions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' spots noticed at the occasion of solar eclipses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Although they may be individually important, all together, they form only a tiny fraction of Mitteilungen data (< 1%), and they are less exploitable because they cannot be calibrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' The Digitization: First Milestone So far, this large collection of data was completely inaccessible in digital form, by contrast with the GN database, which includes all raw group counts collected by Hoyt and Schatten (1998a,b) and was recently expanded by Vaquero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' SOLA: Clette_SNDB2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='tex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' 9 January 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' 1:30;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' 4 Sunspot Number Database and the 1947 Zurich Discontinuity Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Facsimile of a typical yearly table, as published in the Mitteilungen (first page going up to early June).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' This table lists all daily observations from A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='Wolfer for the year 1890.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Each column gives the date followed by the total number of groups and total number of spots, separated by a dot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' A star symbol is added for some days, and marks the days when the observations were made occasionally with a different telescope (On these days, Wolfer used a small portable “Parisian” telescope with a 40 mm aperture).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Although there is a rather wide overlap between the GN and SN data and many observers are common to both data sets, the GN database unfortunately contains only the number of groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Therefore, the number of spots can only be found in the Zurich data, as it was required to compute the SN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' This thus motivates the construction of a complete SN database, equivalent to the existing GN database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' As a first major step, in 2018, a full encoding of the Mitteilungen data tables was undertaken at the World Data Center Sunspot Index and Long-term Solar Observations (SILSO), with the help of students for the bulk encoding work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' By the end of 2019, all the data tables have been digitized, forming the first version of the SN database, which includes all data published between 1849, when R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Wolf undertook the production of the sunspot number, and 1944, when the last director, Max Waldmeier, decided to cease publishing raw data in print.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' This database now contains 205,000 individual daily sunspot counts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Isolated SOLA: Clette_SNDB2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='tex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' 9 January 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' 1:30;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' 5 625) Alfred Wolfer, Beobachtungen der Sonnenfecken auf der Sternwarte in Zurich im Jahre 1890.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' (Fortsetzung zu 604.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=') 1890 1890 1890 1890 1890 1 1|1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='2 一 5|1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='5 NB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Die mit * bezeichneten Beobachtungen sind mit einem kleinern Fernrohr gemacht, welchem etwa der Factor 1,5 zukommt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' April1891, **Clette et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' numbers mentioned in textual rubrics are not yet included, but we plan to add them later on, for the sake of historical completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Next to the daily separate counts of spots and groups, the database includes metadata derived from annotations in the printed tables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' When daily symbols indicated regular changes of observers or instruments and when each subset included a large number of days, we split the data included in common tables, and attached the subsets to distinct observers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' So, an observer may appear in different incarnations, corresponding to different instruments and/or locations, which thus require a different calibration and should not be mixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Currently, this first major input to the SN database is subjected to a thorough quality control, fixing typos, date inconsistencies and occasional ambiguities in observer names.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Meanwhile, we looked for other data sources that can help recovering information that proved to be missing in the Mitteilungen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' One of the gaps happens in the early part of the SN database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Wolf’s Sourcebook and Wolfer’s Global Register Indeed, before 1870, the information about the core observations made by Wolf and his assistants is incomplete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' A single “master” yearly table contains all the counts used to produce the sunspot number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' It thus consists mainly of the counts made by Wolf, which are thus largely complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' On the other hand, data from other observers, assistants or external observers, are only inserted on days when the primary observer could not observe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' As a consequence, between 1864, when the first assistants were recruited, and 1869, only a small fraction of the data from the Zurich assistants appear in the Mitteilungen, as Wolf’s own data fill a majority of days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Moreover, before 1864, Wolf’s main auxiliary observer was Samuel Heinrich Schwabe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' However, although a significant fraction of the daily counts were from Schwabe, Wolf did not mark them in the published tables before 1859, as he first considered Schwabe’s numbers fully equivalent to his own.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' This now makes it impossible to distinguish Wolf’s primary counts from rescaled numbers from Schwabe during the first 10 years of the Wolf series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' This important information about the primary Zurich observers is thus largely incomplete between 1849 and 1870.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Fortunately, two additional sources that provide full tables of the base counts were preserved, and are archived at the ETH Zurich University Archives of the Eidgen¨ossische Technische Hochschule (ETH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' One of them is the so-called Wolf’s sourcebook (Wolf, 1878, catalogue entry Hs368:46).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Those handwritten yearly tables gather all daily numbers forming the sunspot number series from 1610 to 1877 (Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' In fact, these are the master tables assembled by Wolf (Friedli, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Those tables provide two kinds of unique information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Firstly, right from the start of Wolf’s yearly census in 1849, they include symbols identi- fying the source observer for each daily number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' This additional information will thus allow to remove the ambiguity in the early Mitteilungen tables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Moreover, each yearly table indicates the personal k coefficient that was actually used by Wolf, a precious information that can be crossed with the few occasional mentions by Wolf of changes in his k calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' SOLA: Clette_SNDB2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='tex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' 9 January 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' 1:30;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' 6 Sunspot Number Database and the 1947 Zurich Discontinuity Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Facsimile of the table for 1860 in Wolf’s hand-written sourcebook, which covers the period 1610 to 1877 (ETH catalogue entry Hs368:46).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' The layout is similar to the equivalent yearly table published in the Mitteilungen, but it contains very important additional infor- mation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Symbols indicate for each day, from which observer the daily sunspot number was obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' One can see that most of the observations were from Wolf, as primary observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' For each auxiliary observer, including here Schwabe and Carrington, a list at bottom left mentions the personal k coefficient that was used to rescale the raw numbers, to match the scale of Wolf’s own numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Moreover, as the copying and typesetting process for the publication in the Mitteilungen most probably led to errors and typos, the original sourcebook provides the ground truth and will allow fixing those occasional mistakes in the master database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Thanks to the efforts of the Wolf Gesellschaft (Friedli, 2016), Wolf’s sourcebook was digitized from 1849 to 1877, when the collection ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' While the tables can now be consulted online at URL http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='wolfinstitute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' ch/data-tables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='html, this extended information must still be merged with the pri- mary Mitteilungen database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' This work is now in preparation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Finally, the yearly tables in the sourcebook actually go back to the very first sunspot observations in the early 17th century.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Although this part is less substantial, those data tables for years before 1849 must still be digitized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' However, like in the Mitteilungen, the sourcebook does not contain the full set of raw observations collected by Wolf from the auxiliary observers and from his assistants, between 1849 and 1870, in particular, the observations from Schwabe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' However, a larger set of handwritten tables also exists at the ETH Zurich University archives (Wolfer, 1909, catalogue entry Hs1050:227).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' This series is a standardized compilation of all data and metadata published in the Mitteilungen, up to 1908 (Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' This huge register was first produced by Wolf, and after Wolf’s death in 1893, it was continued by A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Wolfer and his assistants until 1909, as a base for a global verification of the sunspot number series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' In this collection, there is a separate table by observer and by year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Therefore, the full data set is included, even data that were never used for the calculation of the daily Zurich sunspot number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' In particular, there are also many data series from before 1700, SOLA: Clette_SNDB2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='tex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' 9 January 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' 1:30;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' 7 147 8 6.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='13 10 9 w 44 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' 116.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='7 100,3 92,2 107,1 108,6 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' 9011 97·9 95,6 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' 81,5 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='0 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='9 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='4 Bemerkungen : Bemerkungen: T,00 = 1,50 2 Jehwae = 1,25 1859 = (amington = 7, 03 webs ht.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' 2h fmeatmth!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' 0,47 4o Vergl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='m*tw,k,3 = Jhon = 7,11 46Clette et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Facsimile of one page from the register of hand-written tables compiled by R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Wolf and continued by A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Wolfer, and covering the entire period 1610 - 1908 (ETH catalogue entry Hs1050:227).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' This page shows the yearly table for Flaugergues in 1796.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' The layout is similar to the yearly tables in Wolf’s sourcebook, but here, all daily observations are listed for each observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' On the right, literal citations and detailed indications are often included to clarify the interpretation of the tabulated numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' This series of tables thus gives a complete and well-standardized view of all data collected by Wolf and Wolfer, including data that were not used to produce the daily sunspot number, and also data and metadata that were not published in the Mitteilungen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' which were never used by Wolf, as he decided to compute the sunspot number only from 1700 onwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Still, the tables in this complete register may prove invaluable for crossing this information collected long ago by Wolf with other recovered observations of the same observers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' They also indicate which data were known by Wolf and his collaborators at the epoch when they produced the Zurich numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' The scanning of this large set of tables is now planned at the ETH Library in Zurich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' When this step will be completed, the encoding into a database will require substantial additional work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Chronology of the Data Although series of data and metadata still needs to be added, the database is now largely complete between 1849 and 1944, and we have now already a complete chronology of all the observers who provided data to the Zurich Observatory between 1849 and 1980, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' during the entire Zurich era.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' This allows us to derive SOLA: Clette_SNDB2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='tex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' 9 January 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' 1:30;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' p.' metadata={'source': 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+page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='0Sunspot Number Database and the 1947 Zurich Discontinuity Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Evolution of the number of stations for each year contained in the data tables published in the Mitteilungen of the Zurich Observatory (gray curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' After 1919, when the Zurich Observatory ceased to publish all the data, the total number of contributing stations is plotted in blue, based on the annual list of stations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Between 1919 and 1944, the data from a subset of observers were still included, but after 1945, none of the source data were published.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' The two vertical shaded bands mark the two world wars, which both definitely left an imprint on the Zurich sunspot data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' some global statistics of the observers and the time interval over which they were active, which provides very interesting new insights in the construction of the Zurich SN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Figure 4 shows the number of stations for each year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' This illustrates the evolution of the input data, as published in the Mitteilungen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' The number of stations steadily increased from 1865 to 1896, when it reaches about 20 sta- tions and then drops slightly, but remaining above 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' This corresponds to the continuous recruiting of new additional external observers by Wolf and later by Wolfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' This evolution is completely disrupted in 1919.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' At the end of World War I (WWI), Wolfer adds many new observers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' The number of stations passes the 40 mark, doubling the size of what becomes a true international network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' However, probably for financial reasons, Wolfer then decides not to publish all data anymore (Friedli, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Only the numbers from the Zurich observers and 7 to 9 primary external observers are still published each year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Although some of those privileged external observers had been important long-term contributors by 1919, the selection criteria are unclear and were not explained by Wolfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' But another drop of the number of tabulated data happens in 1926, when William Otto Brunner succeeds Wolfer as director of the Zurich Observatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Brunner then decides to publish only the data from the Zurich team (Brunner, 1927).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' None of the data from the network are published after that year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' The only exception is Karl Rapp, a private observer, who observed in Locarno, Switzerland, from 1940 to 1957.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Rapp was actually trained in the same way as assistants at the main observatory in Zurich, and was thus treated as an internal observer over his whole observing career.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Although Brunner states in 1927 that the external data from auxiliary stations are archived and can be consulted on request (Brunner, 1927, page 188) (Friedli, 2020, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='3), searches undertaken over past years failed to recover those archives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' So far, only the data for 1944 were found in a single unpublished manuscript, referenced SOLA: Clette_SNDB2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='tex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' 9 January 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' 1:30;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' 9 WWI WW II 60 count 40 Station 20 Unpublished Published 1860 1880 1900 1920 1940 1960 1980 Time (years)Clette et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Hs1050:14 in the ETH Zurich University archives, which contains all calculation sheets for that single year (Friedli, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Then in 1945, when Max Waldmeier becomes the new director, the publication of source data ceases completely, as can be seen in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' By then, the volume of data collected in Zurich had further increased, with almost 60 contributing stations (blue curve in Figure 8), making their publication bulky and costly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' The Mitteilungen then switch to a different format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' The thick yearly volumes become a series of shorter thematic issues, with articles about diverse research topics developed by Waldmeier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' The sunspot number gets a more limited space, compared to the earlier volumes published by Wolfer and Brunner, which were almost entirely dedicated to sunspots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Again, during this last period of the Zurich history, all the original data were saved like before in archives at the observatory in Zurich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' However, since the closing of the Zurich Observatory in 1980, those archives somehow went lost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' This created a major 35-year data gap in the raw data collection on which the Zurich sunspot number is based.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' This wide gap falls at a critical moment, as one of the main scale jumps identified in the Zurich series falls in 1947, thus precisely within this time interval (Clette et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Clette and Lef`evre, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' The raw input data are thus essential to reconstruct the methodological changes that took place in Zurich at that epoch and may have caused this inhomogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Moreover, this gap creates a critical missing link between the early Zurich epoch, up to Brunner, and the modern international sunspot number produced in Brussels since 1981, for which all data are preserved in a computer-accessible digital database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' The Original Waldmeier Source Tables (1945-1980) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' A Serendipitous and Complete Recovery Fortunately, in late 2018 and early 2019, a serendipitous finding by the staff of the Specola Solare Ticinese Observatory in Locarno (https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='specola.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='ch/e/), followed by subsequent searches, allowed to recover the entire Waldmeier data archive (1945 – 1979), which was in fact dispersed over three locations: the Specola Observatory (26 years, 1945 – 1970), the Royal Observatory of Belgium in Brus- sels (4 years, 1971 – 1974), and in the deep storage of the ETH Zurich University archives in Zurich (5 years, 1975 – 1979).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' This dispersion seems to be due to the rather tumultuous closure of the Zurich Observatory (for an evocation of that transition, see Stenflo, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Except for copies of the years 1975 – 1979 on microfiches at the ETH archives, the fragmented original collection was also stored without inclusion in any inventory or catalogue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' This recovery is a breakthrough, and given the amount of data collected over those 35 years, it will keep researchers busy for many years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Indeed, we estimate that those tables contain about 350,000 individual daily numbers, thus more than in all published tables from 1849 to 1944.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' In a first step, all the elements of this archive were brought together again at the ETH Zurich University archives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' They are now fully cataloged (Waldmeier, 1980), and the ETH archives have SOLA: Clette_SNDB2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='tex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' 9 January 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' 1:30;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' 10 Sunspot Number Database and the 1947 Zurich Discontinuity Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Facsimile of a typical handwritten yearly table from the complete 1945 – 1980 collection of source tables that was recovered in 2018 – 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' This table lists the data from H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' M¨uller, one of the assistants observing at the Zurich Observatory with the standard 8-cm Fraunhofer refractor, for the year 1960 (ETH catalogue entry Hs1304.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='8:16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='7891/e– manuscripta-87290).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' For each day, the table gives the number of groups, the total number of spots, the calculated personal k value relative to the primary observer (Waldmeier), and a sky quality index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' For each column, monthly sums and the mean k coefficient are given at the bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' The yearly totals and the mean k coefficient for the whole year are appended at the lower right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Here, k equals 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='52 and thus differs by more than 15% from Waldmeier’s target value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='6, revealing a significant dispersion of the Wolf numbers from the assistants, although they were expected to be closely aligned on the primary observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' completed the digitization of the whole collection in 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' The scans of all tables are now accessible online on the digital platform for manuscript material from Swiss libraries and archives at https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='e-manuscripta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='ch/ (ETH catalogue entry Hs 1304.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Now, in order to make all the data computer-readable, all those tables need to be encoded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' This work has now just started at the Royal Observatory of Belgium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' The Waldmeier Yearly Tables: a Key to the Zurich Method The Waldmeier archive consists in yearly handwritten tables, one per observer, and each one on a separate sheet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Over the period 1945 – 1980, there was an average of 50 stations each year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' All tables adopt the same standard format, with one column per month.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Figure 5 illustrates the typical layout of one sheet, here with the table for H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' M¨uller, one of the Zurich observers, for the year 1960.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' External auxiliary stations are presented with exactly the same layout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Each table lists all daily observations provided by the observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' The number of spots SOLA: Clette_SNDB2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='tex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' 9 January 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' 1:30;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' 11 Hs 1304.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='8:16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='3 Methode: Beobachter: Jahr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' V VI VII IX X IX IV VIII IIX II III I 1960 g, f g, f g, f g, f k k g, f k f k g, f J‘8 g, f k k k g, f k k k k ots C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='82#0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='S2 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='1024 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='2532130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='46 2.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='53 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='52 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='51 M 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='20 1960 N 11462%3 k= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='52 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='515)Clette et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' and groups are given separately, exactly like in the tables published earlier in the Mitteilungen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' This essential piece of information, which was so far entirely lost, will allow to determine for each day exactly how the observers were separating sunspot groups, on the one hand, and counting sunspots on the other hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' In partic- ular, it will help clarifying and quantifying the use of weighted sunspot counts, an alternate counting method adopted by the Zurich observers, in particular by Waldmeier himself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' This alternate counting rule, in which large spots with extended penumbra are counted as more than 1, is suspected to be the cause of the 18% upward jump that affected the original SN series in 1947 (Clette et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Clette and Lef`evre, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Svalgaard, Cagnotti and Cortesi, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Indeed, recent double counts, using the regular Wolf formula or weighted counts, were made at the Specola Observatory during several years, between 2003 and 2015, and led exactly to the same inflation of the sunspot number as the one found in the Zurich series after 1947 (Clette et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Svalgaard, Cagnotti and Cortesi, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' The recovered tables are thus providing the same kind of evidence, but over 35 years, including the epoch when the jump occurred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' The tables also include the monthly and yearly mean k personal coefficients computed by the Zurich Observatory, a very important piece of metadata to understand how Zurich was treating the source observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' In particular, k coefficients are given for all Zurich assistants, and also the associated observers of the Specola station in Locarno.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' As all internal observers were assumed to align themselves on the primary observer (Waldmeier during that period), without applying any rescaling by a personal k coefficient, those internal yearly k values can bring invaluable insights on how and to what extent assistants managed to actually align themselves on the primary reference in their daily raw observa- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' As this internal practice was introduced by Wolf, as soon as 1870, when he started to combine his own counts with those of his first assistants, this can thus help in the understanding of the Zurich number production well before 1945.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' In this collection, the most important tables are the yearly tables for the primary observer, Max Waldmeier (Figure 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' They provide unique information about three key aspects of the resulting Zurich sunspot numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Firstly, those tables were the master tables from which the daily sunspot number was derived for each day of the year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Therefore, they include raw counts and the resulting Wolf number for each day of the year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' They thus provide a complete day-by-day census of how each daily SN was derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Secondly, most of the days contain the personal counts by Waldmeier, who had the role of base reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Therefore, this is the yearly table of raw group and sunspot counts by the primary observer, which allows tracking changes in Waldmeier’s own daily observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' For instance, Waldmeier was sometimes on mission at the coronagraph of the astronomical station in Arosa, then observing from high altitude with an alternate telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' The counts for those days may de- viate from the base reference scale defined by the standard Fraunhofer refractor used on the front terrace of the observatory in downtown Zurich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Fortunately, Waldmeier marked the days when he observed from Arosa, which will allow analyzing the consequences of this site alternation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' SOLA: Clette_SNDB2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='tex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' 9 January 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' 1:30;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' 12 Sunspot Number Database and the 1947 Zurich Discontinuity Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Facsimile of the primary table for Max Waldmeier in 1957, extracted from the complete 1945 – 1980 collection of source tables (ETH catalogue entry Hs1304.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='8:13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='7891/e-manuscripta-87246).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Such tables are particularly important, as Waldmeier was the pilot observer of the Zurich sunspot number over that 35-year interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' They include various annotations that allow retracing day-by-day, how Waldmeier himself was observing, and which alternate number was used on days when he could not observe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' They thus contain essential information about the Zurich data processing that cannot be found in any other Zurich document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Thirdly, the days in which Waldmeier could not observe are filled with num- bers from local assistants or from the stations in Arosa or Locarno (Karl Rapp until 1 April 1957 and the Specola Observatory starting on 1 October 1957).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' As can be seen in Figure 6, those days are also marked in the tables with a symbol identifying which alternate observer was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Finally, as those tables record the provisional values issued immediately at the end of each month, on the remaining missing days when none of the local stations had managed to observe the Sun, the numbers were simply interpolated between adjacent days, and those dates are marked as “interpolated”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' These are the few days which were later replaced by definitive values calculated using k-normalized Wolf numbers from the auxiliary stations, according to a standard method, of which the principle can be reconstructed from a few reference documents (Friedli, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Those master tables thus provide almost all the keys that were badly miss- ing to reconstruct the method and practices implemented in Zurich, and most probably, to retrace persisting changes or local inconsistencies in the Zurich processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' SOLA: Clette_SNDB2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='tex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' 9 January 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' 1:30;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' 13 Methode: Beobachter: Jahr: .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=" T II III IV V VI VII VIII IX X XI XII Tmnisiris he kdaf'vzah lm." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' R g, f R R g, f R g, f R g,f g,f R R g, f R g, f R g, f R R 1954 g, f A g, f doc 244/20.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='8 262.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='9 233.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='9 1954 69318 355 180 29,6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='02 184) -182 ponoredus fora mMd 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='b[12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='244,]-200Clette et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Timelines of the active observing periods of all Zurich observers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' In red (top group), the primary observers and in orange (bottom group), the assistants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' In purple, the observers of the auxiliary station in Locarno, who were considered as members of the Zurich core group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' The vertical shaded band marks World War II and the vertical dashed line indicates the time when the 1947 scale jump occurs in the original SN series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' The bottom plot gives the number of active Zurich observers for each year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' A Major Disruption: Zurich Observers Although the above data still need to be digitized, we now have the full list of observers who contributed year-by-year to the Zurich sunspot number up to 1980.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' By assembling the timelines of each individual observer, we could map how their observing period overlaps with other observers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Figure 7 retraces the observing periods for all Zurich primary observers and all the assistants, between 1850 and 1960.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' In this figure, Schwabe is included among the assistants (orange group) al- though he was an external observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Indeed, before Wolf could recruit his first assistants in the newly founded Zurich Observatory in 1865, he used Schwabe’s numbers as primary alternate source for filling the gaps in his own observations, and even initially considered Schwabe’s numbers as fully equivalent to his own (personal k = 1) before 1859.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' We also included K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Rapp in the associated Lo- carno station (purple group) although he contributed before the establishment of the Specola Observatory by Waldmeier in 1957, starting in 1940.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Indeed, both Brunner and Waldmeier always included Rapps’s data together with the SOLA: Clette_SNDB2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='tex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' 9 January 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' 1:30;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' 14 Wolf ST WolfPR Wolfer Brunner,wlo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Waldmeier Schwabe Fretz Weilenmann Mever Billwiller Fauquez Hoffler Broger Observers Biske Buser Arosa Brunner Ass Muller Beck Muller,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='E Wile Lemans Scheidlegger Frick Hermes Riesen Zelenka Durst Pfister Rapp Keller Schmidt ilszak Cbrtesi Pittini 1840 1860 1880 1900 1920 1940 1960 1980 Time (years)Sunspot Number Database and the 1947 Zurich Discontinuity Zurich data in the Mitteilungen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' even when the data of all the other external stations were not published anymore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Rapp was also trained to follow the Zurich observing methods, and can thus be considered as an internal member of the Zurich group of stations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Finally, although Waldmeier, the last primary observer (red group), started observing as an assistant in 1936, his participation was partly interrupted, as explained below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' For the period before 1944, the resulting chronology reveals a few interesting facts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' In particular, one of Wolfer’s assistants, Max Broger, had a very long observing career (40 years, 1896 to 1935).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' He actually observed over more years than several primary observers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' As he observed in parallel with Wolfer and then with Brunner, his observations can provide an essential link to check the Wolfer- Brunner homogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' This touches the fundamental issue of the weighted sunspot counts used by the Zurich Observatory, as mentioned in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Indeed, Clette et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' (2014), Clette and Lef`evre (2016), and Svalgaard, Cagnotti and Cortesi (2017) conclude that this alternate counting method is the most likely cause of the 1947 scale jump in the original SN series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' However, the timing and sharpness of the jump seem to be contradicted by the fact that this weighting practice was introduced progressively well before 1947, in the early 20th century by Wolfer (Cortesi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Svalgaard, Cagnotti and Cortesi, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Although Wolfer himself never used it for his own counts (Svalgaard, Cagnotti and Cortesi, 2017), this practice was implemented to help assistants aligning their raw counts on the reference of Wolfer, the primary observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' This could be verified by taking the counts on occasional days when only a single big spot was visible on the Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Then, when one of Wolfer’s assistants, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Brunner, took over as director and as primary observer in 1926, he continued to use weighted counts, but this time as primary observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Although this marks the moment when the break with Wolf’s original methodology occurred, Brunner managed to maintain the stability of his counts, as found by Svalgaard, Cagnotti and Cortesi (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' When Waldmeier took his succession in 1945, after being assistant for a few years, he thus just continued an established practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' So, apparently, this chronology does not match at all the abrupt occurrence of a jump in 1947, two years after Waldmeier became the new reference observer, a status that he kept for the next 35 years without any other noticeable transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Now, by retracing the composition of the network of collaborating observers, we found evidence of a major transition that occurred between 1945 and 1947.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' The change was twofold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Firstly, at the Zurich Observatory, although Waldmeier became director in 1945, the former director, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Brunner, actually continued observing during one year until December 1945 (see Figure 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Moreover, his primary assistant, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Brunner-Hagger, who was part of the team since 1928, continued until August 1946.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' This actually marks the moment when the link with the former Zurich core team is broken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' As shown in Figure 7, in 1945, Waldmeier starts to recruit new assistants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' However, the first one, Beck, worked in parallel with Brunner only during a few months, when solar activity was rather low, and he left the observatory already in 1949.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Then follows a succession of other assistants who also leave after only a few years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' This means that the overlap between the old and new team was extremely limited and that for several SOLA: Clette_SNDB2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='tex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' 9 January 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' 1:30;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' 15 Clette et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' years the Zurich team was very unstable, contrary to the Brunner team that had remained unchanged for nearly 20 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' So, the internal stability of the Zurich system during the 1945 Brunner- Waldmeier transition rested only on Waldmeier himself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' This is unprecedented in the entire Zurich history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Indeed, the stability of the Wolf-Wolfer transition benefited from a 17-year period, during which Wolf and Wolfer observed jointly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Although the Wolfer-Brunner joint period was shorter (3 years, 1926-1928), another assistant, Broger brought a solid reference to bridge the Wolfer-Brunner transition, as he had worked jointly with Wolfer for 30 years, since 1896, and then continued for 10 years together with Brunner, until 1935.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Finally, although Waldmeier started collaborating with the Zurich Obser- vatory in 1936, he did not contribute during three years, from 1939 to 1941 because of the onset of World War II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Moreover, as he was strongly involved in coronagraph observations, he worked for a large part of his time at the Arosa station, rather than as an ordinary assistant observing side by side with Brunner in Zurich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' We also note that the last years before 1946 fell in a minimum of the solar cycle, when the low sunspot activity makes mutual comparisons less accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Therefore, all those circumstances reduced the effective overlap period between Brunner and Waldmeier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' A Major Disruption: Auxiliary Stations In parallel with the Zurich internal transition, another major and unprecedented disruption also occurred just after 1945, but now for the Zurich auxiliary sta- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Although those external data were not at the core of published sunspot numbers, they definitely provided a wide ensemble of independent data series against which the Zurich numbers were continuously compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Moreover, all external stations derived their counts using Wolf’s original definition, without any weighting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Therefore, the auxiliary data were not affected by the introduc- tion of Zurich’s internal weighting practice, and in that sense, they provided the only base against which the Zurich team could infer that their weighted numbers remained coherent with the unwheighted Wolf numbers that formed the original SN series until Wolfer’s retirement in 1926 (Clette et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Svalgaard, Cagnotti and Cortesi, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' This continuous bench-marking could only work if at any given time, there was a large number of active auxiliary stations which had already contributed data during many past years, preferably over one or more full solar cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Figure 8 shows the timelines of all auxiliary stations that contributed obser- vations to the Zurich Observatory since the mid-19th, over a duration longer than 11 years, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' a full solar cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' This subset of long-duration stations is indeed the most important for the long-term calibration and stability of the series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' We distinguished the professional observatories from the individual ama- teur observers, which reveals a deep evolution in the composition of the Zurich observing network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' While a large majority of stations were individual observers before World War II (WWII), professional observatories dominate the network after WWII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' SOLA: Clette_SNDB2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='tex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' 9 January 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' 1:30;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' 16 Sunspot Number Database and the 1947 Zurich Discontinuity Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Timelines of the active observing periods of all external stations that sent data to Zurich until the observatory was closed in 1980.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' The stations are ordered according to the starting date of their series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' The top series (dark blue) gathers the professional observatories and the bottom group (light blue) gathers the individual amateur observers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' The vertical shaded band marks World War II and the vertical dashed line indicates the 1947 scale jump in the original Zurich series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' The bottom plot gives the total number of active stations per year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' The light-blue section indicates unpublished data that have not been recovered yet in the Zurich archives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' However, a much more drastic change is also caused by WWII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' In Figure 8, we see that, starting in 1938, long-time contributing stations cease to send data, one after the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' When WWII ended, none of those stations, which gave an external benchmark for the earlier Zurich SN, had survived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' During the war, given the steep drop of contributing stations, Brunner and Waldmeier called to the rescue a large number of local Swiss amateur astronomers, but this local network was quickly changing, as most observers contributed only for one year or at best a few years (therefore, they do not appear in Figure 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' None of those observers were long-term observers in the preceding Zurich network established by Wolfer and Brunner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' SOLA: Clette_SNDB2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='tex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' 9 January 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' 1:30;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' 17 Observers Obs/year 40 20 0 1860 1880 1900 1920 1940 1960 1980 Time (years)Clette et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Then, just after the war, Waldmeier quickly undertakes the construction of a new international network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' The number of stations grows steeply and reaches about 50 stations (see Figure 4), a number that will remain rather stable until 1980.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' As noted before, this new network includes many professional observa- tories, which since then, have delivered observations over very long durations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' In fact, some of them are still contributing nowadays to the worldwide SILSO network, and thus provide an invaluable long-term reference spanning up to 75 years, since 1945.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' However, none of those new stations were part of the pre- 1940 long-term network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Therefore, the context in which the sunspot number was produced after 1945 was largely disconnected from the context surrounding this production before 1940.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' This further weakened the thin internal continuity within the Zurich Observatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' In order to give a more quantitative measure of this second disruption, we summed the number of past observed years already accumulated by all stations that were active on a given year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Figure 9 (top plot) shows the temporal evolution of this total number, which gives a measure of the total amount of past informa- tion that the Zurich Observatory had at its disposal for past comparisons and the verification of their stability relative to independent observers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' As expected, the evolution is characterized by a steady increase in the total amount of available data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' The only interruption in this trend is the steep drop during WWII, when the count suddenly drops back to the values of the early 20th century.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' After WWII, there is a recovery, but it takes about 15 years before the amount of past reference data comes back to the value just before WWII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Afterwards, the amount of past data from active stations continues to grow and finally stabilizes in the 1970’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' If we divide this total number of past observed years by the number of active stations, we obtain the mean past duration over which stations active at a given time have been observing before that time (Figure 9;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' bottom plot).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' This mean duration quantifies the past memory built into the SN system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Between 1860 and 1890, this mean duration increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' This marks the progressive recruiting of the first auxiliary observers by Wolf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Then, the mean duration largely stabilizes until 1926, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' the Wolfer-Brunner transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' The only feature is a temporary peak associated with WWI, which thus left only a minor imprint in this evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Thanks to the many new observers recruited after WWI, and who continue observing until WWII, the mean duration grows to almost 15 years in 1938.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Then, WWII again produces a steep drop, by a factor of two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' In 1945 and the decade that follows, the mean memory range falls back to about 7 years, a level that was not encountered since 1880, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' the epoch when Wolf was still recruiting his first associated observers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' After this dramatic shortening of the past memory, there was a steady recovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' However, it is only around 1965, 20 years after WWII, that the pre-WWII mean memory range is recovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' It continues to rise until 1980, when the Zurich Observatory was closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' This continuous trend largely rests on the long-term contribution from the professional observatories that entered the network just after WWII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Figure 9 thus illustrates that the years immediately following WWII were abruptly affected by a major loss of past references, and that this loss had no equivalent in the history of the Zurich SN number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' SOLA: Clette_SNDB2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='tex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' 9 January 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' 1:30;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' 18 Sunspot Number Database and the 1947 Zurich Discontinuity Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Evolution of the amount of past data available for each year at Zurich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' The upper plot gives the total number of preceding observed years by all the stations active on a given year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' After an almost continuous increase, a sharp drop occurs just after WWII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' The lower plot shows the mean number of preceding observed years per station, for all stations active on a given year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' The rise after 1925 indicates the growing participation of stations with very long duration, but a drop to 19th century levels marks the late 1940’s and early 1950’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Although the above indicators are indirect contextual elements, the fact that this unique double discontinuity in the history of the Zurich sunspot number production coincides with the jump revealed by the SN series itself is a very strong indication that the sharp SN scale jump was a consequence of this abrupt and radical change in the base data input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Until 1946, the potential biasing effect, which was present since the weighted counting method had been introduced, had been kept under control thanks to the double stabilizing effect of long-term internal and external observers who did not change their counting practices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' This stabilizing continuity was clearly broken between 1946 and 1947, which suddenly opened the way for the biasing effect inherent to the weighted counts, as evidenced by the 1947 upward jump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' This new contextual evidence thus explains SOLA: Clette_SNDB2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='tex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' 9 January 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' 1:30;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' 19 800 Amount of past data 600 400 200 0 1860 1880 1900 1920 1940 1960 1980 Time (years) 25 duration (years) 20 15 past 10 Mean 5 0 1860 1880 1900 1920 1940 1960 1980 Time (years)Clette et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' simultaneously the delayed effect of the weighting practice and the abruptness of the jump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Conclusion Over just a few years, we thus achieved major progress in the construction of the SN database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Now, about two thirds of the existing source data are recorded in digital form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' We can now also report on the recovery of a major missing part of this collection, the yearly source tables of the Waldmeier era from 1945 to 1980.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' This fills the main gap in the SN database and provides the missing link between the contemporary index and the rest of this long series before 1945 and back to 1700.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' While significant work is still needed to digitize those newly recovered documents, the global panorama that the SN database now offers made it possible to establish the complete chronology of contributing stations and observers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' We found that the two world wars had deep consequences on the production of the SN by the Zurich Observatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' WWI brought a major expansion of the network of auxiliary observers, but without disrupting the internal practices and organization of the Zurich sunspot observers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' On the other hand, after WWII, we find a double disruption in the Zurich system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' A complete renewal of the Zurich observing team occurred between 1946 and 1947, with almost no overlap between the old team, which had remained mostly unchanged for more than 20 years, and the new team progressively built by Waldmeier between 1946 and 1950.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Moreover, after the loss of most of the external observers active over the decades preceding WWII, between 1938 and 1945, an entirely new worldwide network is established after the war with entirely different stations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' The narrow correspondence of this drastic and unprecedented structural change with the 18% SN scale-jump diagnosed in the SN series pro- vides strong historical evidence that a sharp jump in the SN exactly at that moment is a real and logical consequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Although the suspected cause, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' the introduction of the size-based weighting of the spot counts, was introduced much earlier in the practice of Zurich assistants, our now-complete timeline explains why it only led to actual consequences when this sharp and unprecedented discontinuity in the Zurich system took place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' All together, those recovered tables open the way to future major steps in the end-to-end calibration of the sunspot number series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Full statistical diagnostics of the actual stability of each separate Zurich observer, which was simply postulated since the epoch of Wolf, will allow disentangling in detail the causes of anomalies found in the heritage series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Much more importantly, those data open the way for a full recalculation of the sunspot number, starting again from the full set of raw input data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' This recalculation will use new advanced computer-based processing methods, which exploit the entire set of data instead of mostly using the numbers of the single primary observer, as was the case in the original Zurich series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' This should improve further the stability and accuracy of the sunspot number in the interval 1945-1980, where so far, SN Version 2 consisted only in a correction factor applied to the original Zurich SN series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' This would also finally bridge the gap separating the current international sunspot number from the early epoch before 1945.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' SOLA: Clette_SNDB2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='tex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' 9 January 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' 1:30;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' 20 Sunspot Number Database and the 1947 Zurich Discontinuity However, a partial gap still remains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Although all observations made in Zurich from Wolf in 1849 to Waldmeier in 1980 now finally form a complete and unin- terrupted thread, we still miss the unpublished archives from the Brunner era.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Therefore, efforts are still continuing to try recovering the last missing data from the network of the auxiliary stations between 1919 and 1944.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Hopefully, this will finally bring the last touch to this digital database that will feed sunspot science and long-term solar-cycle studies for many years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Acknowledgments This work and the team of the World Data Center SILSO (http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' sidc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='be/silso/), which produces the international sunspot number and maintains the sunspot database used in this study, are supported by Belgian Solar-Terrestrial Center of Excellence (STCE, http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='stce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='be) funded by the Belgian Science Policy Office (BelSPo).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' This work was also supported by the International Space Science Institute (ISSI, Bern, Switzerland) via the International Team 417 “Recalibration of the Sunspot Number Series”, chaired by M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Owens and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Clette (https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='issibern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='ch/teams/sunspotnoser/).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Specola Solare Ticinese acknowledges the financial support provided by Canton Ticino through the Swisslos fund and by the Federal Office of Meteorology and Climatology MeteoSwiss, in the framework of GCOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' We would like to thank Thomas Friedli for digitizing and making available the original sourcebook by R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Wolf via the web site of the Rudolf Wolf Society (http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='wolfinstitute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' ch).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' We also thank the ETH Library (https://library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='ethz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content='ch/en/), and in particular Evelyn Boesch, of the Hochschularchiv, for the deep searches in the catalogues and archives, and for giving us access to original historical documents from the Zurich Observatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' We also thank Olivier Lemaˆıtre for developing the software and computer database, Stephen Fay and Shreya Bhattasharya for the quality control, and last but not least, we are also grateful to the summer- job students who patiently and carefully encoded all numbers tabulated in the original paper documents: Elfaniel Hermel, Esther-Lauren M’Bilo and Mael Panouillot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' Disclosure of Potential Conflicts of Interest The authors declare that they have no conflicts of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf'} +page_content=' References Brunner, W.' metadata={'source': 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Diffuse polarized +foregrounds from component separation with +QUIJOTE-MFI +E. de la Hoz,1,2⋆R. B. Barreiro,1 P. Vielva,1 E. Mart´ınez-Gonz´alez,1 +J. A. Rubi˜no-Mart´ın,3,4 B. Casaponsa,1 F. Guidi,3,4,5 M. Ashdown,6,7 +R. T. G´enova-Santos,3,4 E. Artal,8 F. J. Casas,1 R. Fern´andez-Cobos,9 +M. Fern´andez-Torreiro,3,4 D. Herranz,1 R. J. Hoyland,3,4 A. N. Lasenby,6,7 +M. L´opez-Caniego,10,11 C. H. L´opez-Caraballo,3,4 M. W. Peel,3,4 L. Piccirillo,12 +F. Poidevin,3,4 R. Rebolo,3,4,13 B. Ruiz-Granados,3,4,14 D. Tramonte,15,16,3,4 +F. Vansyngel,3,4 R. A. Watson.12 +Affiliations are listed at the end of the paper +Accepted 2022 October 2014. Received 2022 October 14; in original form 2022 July 27 +ABSTRACT +We derive linearly polarized astrophysical component maps in the Northern Sky from +the QUIJOTE-MFI data at 11 and 13 GHz in combination with the WMAP K and Ka +bands (23 and 33 GHz) and all Planck polarized channels (30-353 GHz), using the para- +metric component separation method B-SeCRET. The addition of QUIJOTE-MFI +data significantly improves the parameter estimation of the low-frequency foregrounds, +especially the estimation of the synchrotron spectral index, βs. We present the first +detailed βs map of the Northern Celestial Hemisphere at a smoothing scale of 2◦. We +find statistically significant spatial variability across the sky. We obtain an average +value of −3.08 and a dispersion of 0.13, considering only pixels with reliable goodness- +of-fit. The power law model of the synchrotron emission provides a good fit to the +data outside the Galactic plane but fails to track the complexity within this region. +Moreover, when we assume a synchrotron model with uniform curvature, cs, we find +a value of cs = −0.0797 ± 0.0012. However, there is insufficient statistical significance +to determine which model is favoured, either the power law or the power law with +uniform curvature. Furthermore, we estimate the thermal dust spectral parameters +in polarization. Our CMB, synchrotron, and thermal dust maps are highly correlated +with the corresponding products of the PR4 Planck release, although some large-scale +differences are observed in the synchrotron emission. Finally, we find that the βs es- +timation in the high signal-to-noise synchrotron emission areas is prior-independent +while, outside these regions, the prior governs the βs estimation. +Key words: cosmology: observations – methods: data analysis – polarization – cosmic +microwave background +1 +INTRODUCTION +Currently, most of the efforts of the CMB community are +devoted to the search for primordial B-modes. These pre- +dicted B-modes at large scales can only be produced by +tensor modes, and their detection would constitute com- +⋆ e-mail:delahoz@ifca.unican.es +pelling evidence of an inflationary phase. The intensity of +this primordial signal is determined by the tensor-to-scalar +ratio r, the relative amplitude between the tensor and scalar +modes at a given pivot scale. The current best upper bound +on the tensor-to-scalar ratio is: r < 0.032 at 95% CL, set by +the combination of Planck, BICEP2/KeckArray and baryon- +acoustic-oscillation data (Tristram et al. 2022). +The weakness of the primordial B-modes makes its de- +© 0000 The Authors +arXiv:2301.05117v1 [astro-ph.CO] 12 Jan 2023 + +2 +E. de la Hoz +tection a tremendous experimental challenge, requiring high- +sensitivity experiments as well as an exquisite control of sys- +tematics. Indeed, a large effort is currently on-going with +the aim to detect, or at least to constrain, r with a sen- +sitivity σr(r = 0) ⩽ 10−3. This includes many planned +ground-based experiments, e.g., GroundBIRD (Lee et al. +2020), LSPE-Strip (Lamagna et al. 2020), CMB-S4 (Abaza- +jian et al. 2016), Simons Observatory (Ade et al. 2019) and +BICEP array (Hui et al. 2018), as well as satellite missions, +e.g., LiteBIRD (LiteBIRD Collaboration et al. 2022) and +PICO (Hanany et al. 2019). +The detectability of the primordial B-modes could be +improved by removing the secondary B-mode component in- +duced by weak gravitational lensing. Several delensing pro- +cedures have been proposed in the literature (Planck Collab- +oration et al. 2016; Millea et al. 2019) and have been applied +to data from current CMB experiments (Planck Collabora- +tion et al. 2016; Carron et al. 2017; BICEP/Keck Collabora- +tion et al. 2021), and in forecasts of future CMB experiments +(Diego-Palazuelos et al. 2020; Namikawa et al. 2022). +It is necessary to disentangle the CMB polarization +signal from those coming from other microwave emissions, +such as Galactic synchrotron, thermal dust and extragalactic +point sources. Thus, the problem of component separation +is a crucial step in order to detect the primordial B-mode +of CMB polarization. This process benefits from the char- +acterization of foreground emissions using complementary +frequency ranges that provide unique information about the +contaminants. +The main diffuse polarized contaminants are the syn- +chrotron emission (at low-frequencies) and the thermal dust +emission (at high-frequencies). The best characterization of +these diffuse foregrounds has been done by Planck (Planck +Collaboration 2020d) using a data set covering frequencies +from 30 to 353 GHz. This frequency range limited strongly +the estimation of the synchrotron spectral parameters. In +Planck Collaboration (2020d) it is shown that, with Planck +data only, one cannot test the spatial variability of the syn- +chrotron spectral index due to limited sensitivity and fre- +quency coverage. The data only allows a measurement of a +global spectral index of βs = −3.1 ± 0.1. The synchrotron +spectral index has also been estimated using other datasets, +e.g., Fuskeland et al. (2014); Krachmalnicoff et al. (2018); +Fuskeland et al. (2021). +The Q-U-I JOint Tenerife Experiment (QUIJOTE) +(Rubi˜no-Mart´ın et al. 2010) is a polarimetric ground-based +CMB experiment whose main scientific goal is the charac- +terization of the polarization of the cosmic microwave back- +ground (CMB) and other Galactic and extragalactic phys- +ical processes in the frequency range 10–40 GHz and at +large angular scales (≳ 1◦). The experiment is located at +the Teide Observatory (at ∼ 2400 m above sea level) in +Tenerife. It is composed of two telescopes equipped with +three instruments: the Multi-Frequency Instrument (MFI), +the Thirty-GHz Instrument (TGI), and the Forty-GHz In- +strument (FGI), operating at 10–20 GHz, 26–36 GHz and +39–49 GHz respectively. +The MFI instrument has been operating from Novem- +ber 2012 to October 2018. It conducted two different sur- +veys: i) a shallow Galactic survey (called “wide survey”) +covering all the visible sky from Tenerife at elevations larger +than 30◦, and ii) a deep cosmological survey covering ap- +proximately 3000 deg2 in three separated sky patches in the +northern sky. In this work we use the QUIJOTE-MFI wide +survey maps. This survey provides an average sensitivity in +polarization of ∼ 35–40 µK per 1-degree beam in four bands +centred around 11, 13, 17 and 19 GHz (Rubi˜no-Mart´ın et al. +2023). Those frequencies are crucial to achieving a better +characterization of the low-frequency foregrounds. In inten- +sity, this additional information helps breaking degenera- +cies between the synchrotron, free-free and anomalous mi- +crowave emissions while, in polarization, the QUIJOTE-MFI +channels are key to characterize the synchrotron spectral de- +pendence. +In the present work we perform a component separa- +tion analysis to obtain more information about the polarized +sky using the QUIJOTE-MFI data1 (Rubi˜no-Mart´ın et al. +2023) in combination with the publicly available Planck +(Planck Collaboration 2020f,a) and Nine-Year WMAP (Ben- +nett et al. 2013) data. To perform component separa- +tion analysis we use B-SeCRET (Bayesian-Separation of +Components and Residual Estimation Tool), a parametric +maximum-likelihood method described in de la Hoz et al. +(2020). +The paper is organized as follows: in Section 2 we pro- +vide details of the main components in the polarized mi- +crowave sky and the corresponding parametric models used +to characterize them. Section 3 describes briefly the B- +SeCRET method. The data used in the analysis are pre- +sented in Section 4. Then, the main component separation +results obtained are shown in Section 5. Finally, the main +conclusions from the analysis are given in Section 6. In Ap- +pendix A we provide maps of the synchrotron spectral index +obtained from independent fits in linear Stokes parameters +Q and U. Appendix B compares the variations on the syn- +chrotron spectral index due to rotations of the polarized +angle with Faraday rotation. +2 +THE MICROWAVE SKY MODEL +The polarized microwave sky is composed primarily of pho- +tons from the CMB, synchrotron and thermal dust. As +stated before, the synchrotron emission dominates at low- +frequencies while the thermal dust is the principal compo- +nent at higher frequencies. The contribution from other com- +ponents, discussed in Section 2.5, is expected to be insignif- +icant and not taken into account. Apart from these astro- +nomical signals, the measured sky signal maps have another +contribution from the instrumental noise. The characteris- +tics of this noise depend on the specifications of the exper- +iment. Furthermore, contaminants such as the atmosphere +and artificial signals from satellites also contribute to the +microwave sky, see Rubi˜no-Mart´ın et al. (2023) for more de- +tails. Thus, the measured polarized sky signal for a given ν +channel can be expressed as the following sum: +� +Q +U +� +ν += +� +Qcmb +Ucmb +� +ν ++ +� +Qs +Us +� +ν ++ +� +Qd +Ud +� +ν ++ +� +Qn +Un +� +ν +, +(1) +1 This is one of the papers which are part of the MFI wide survey +data release. +MNRAS 000, 000–000 (0000) + +QUIJOTE-MFI Diffuse polarized foregrounds +3 +where Xcmb, Xs, and Xd are the CMB, synchrotron and +thermal dust signals respectively, and Xn is the instrumen- +tal noise (X ∈ {Q, U}). In the subsequent subsections we +describe the main physical components that encompass the +sky signal as well as some effects that alter this signal. More- +over, we present the parametric models that we use in the +component separation analysis for each polarized astronom- +ical component. +2.1 +Synchrotron +The synchrotron emission arises from relativistic particles +(cosmic rays) passing through the Galactic magnetic field. +Its emissivity depends both on the magnetic field strength +and energy distribution of the relativistic particles (generally +electrons). These quantities are not uniform in the Galactic +disc. For instance, the free electrons are more predominant +in compact regions as supernovae remnants. On the other +hand, the magnetic field is amplified in some compact re- +gions and can have different strength and direction across +the sky. +The synchrotron spectral energy distribution (SED) is +generally described as a power law (Rybicki & Lightman +2008): +� +Qs +Us +� +ν += +� +AQ +s +AU +s +� � ν +νs +�βs +, +(2) +where As is the amplitude in brightness temperature at the +pivot frequency νs = 30 GHz and βs is the spectral index +which is assumed to be equal for both Q and U Stokes pa- +rameters. +Previous works dedicated to the estimation of the spec- +tral index, found values around βs ≃ −3.1 (Planck Col- +laboration 2020d). However, the spectral index is expected +to vary spatially due to its dependence on the energy dis- +tribution of the cosmic rays N(E). Studies such as Fuske- +land et al. (2014); Vidal et al. (2015); Krachmalnicoff et al. +(2018); Martire et al. (2022); Weiland et al. (2022) indicate +that different polarized regions present different spectral in- +dices. Here, we conduct a more detailed analysis of the βs +spatial variations in the Northern Hemisphere by perform- +ing a pixel-by-pixel component separation analysis using the +QUIJOTE MFI polarized maps. +The S-PASS survey (Carretti et al. 2019), has provided +the most sensitive reconstruction of the βs variations of the +South Celestial Hemisphere (Krachmalnicoff et al. 2018). +They found large variability over the sky, and a mean value +of −3.22 ± 0.08. Those results were further confirmed in the +analysis of Fuskeland et al. (2021) that estimated the spec- +tral index taking into account the Faraday Rotation effect. +They also studied the Galactic plane and found compatible +results to those where only WMAP data were used, finding +a flatter index in the Galactic plane than at high Galactic +latitudes. +We have also considered an extension of equation (2) +where we include a possible curvature in the synchrotron’s +SED: +� +Qs +Us +� +ν += +� +AQ +s +AU +s +� +ν +� ν +νs +�βs+cs log +� +ν +νs +� +, +(3) +where cs is the parameter that represents the curvature. +This extension is worth studying since a curved spectrum +can account for steepening or flattening of the SED due +to different effects, e.g., cosmic ray aging effect or multiple +synchrotron components along the line of sight. This model +could also account for the presence of polarized anomalous +microwave emission (AME). +2.2 +Thermal Dust +The thermal dust radiation comes from dust grains present +in the interstellar medium. Those grains absorb ultraviolet +light and re-emit as a grey body. In general, these dust grains +are not perfectly spherical and typically have their minor +axis aligned with the direction of the local magnetic field. +This effect yields polarized thermal dust emission. The SED +of this radiation is often described as a modified black body +with emissivity index βd and dust temperature Td: +� +Qd +Ud +� +ν += +� +AQ +d +AU +d +� � ν +νd +�βd+1 eγνd − 1 +eγν − 1 , +(4) +where Ad is the amplitude of the dust in brightness tem- +perature evaluated at the pivot frequency νd = 143 GHz +and γ = +h +kBTd +2. The amplitude is well characterized by +the higher frequency channels where the other components +are clearly sub-dominant. The current temperature map of +the dust grains (Td) is obtained from temperature analy- +sis and has values mostly between 14 K and 26 K. The +polarized dust emissivity evaluated with Planck data is +βd = 1.55 ± 0.05 (Planck Collaboration 2020d). +Several works support the idea that a single component +dust model is too simplistic and more components might be +required to fully characterize this emission (e.g., McBride +et al. 2022; Ritacco et al. 2022). Nonetheless, since this pa- +per is focused on the low frequency foregrounds, we keep the +model used in Planck Collaboration (2020d) which seems to +provide a good description at the Planck polarized frequen- +cies (30 GHz < ν < 353 GHz). +2.3 +CMB +The CMB radiation has a thermal black body spectrum with +a temperature of To = 2.7255 ± 0.0006 K (Fixsen 2009). +CMB photons are linearly polarized due to the Thomson +scattering experienced with the hot electron gas at the last +scattering surface. Unlike in intensity, where the CMB can +be the dominant contribution at intermediate frequencies +(70-150 GHz) and high Galactic latitudes, in polarization, +the foreground contribution cannot be overlooked. Thus, in +order to detect the primordial B-mode, experiments with +very high sensitivity, exquisite control of systematics and a +careful removal of foregrounds are mandatory. +The CMB signal at each pixel is given by its amplitude +Acmb, which is the only free parameter for this component. +Since the rest of the components are given in brightness +temperature we convert the thermodynamic temperature of +the CMB to the same units: +� +Qcmb +Ucmb +� +ν += +� +AQ +cmb +AU +cmb +� +x2ex +(ex − 1)2 , +(5) +2 h and kB are Planck and Boltzmann constants respectively. +MNRAS 000, 000–000 (0000) + +4 +E. de la Hoz +where x = +hν +kBTo . +2.4 +Faraday Rotation +Another issue intrinsic to the polarization signal is the Fara- +day rotation effect, i.e., the rotation of the plane of polariza- +tion that occurs when light passes through the interstellar +medium in the presence of a magnetic field. The magnitude +of this effect scales with the square of the wavelength, hence +its repercussions are more significant at low-frequencies. To +properly account for this effect we require a broad knowl- +edge of the Galactic magnetic field as well as the interstellar +medium, in order to recognise the regions where the effect is +more significant. Moreover, since the instrumental beam has +a finite size, the measured signal is an average of the emis- +sion from various directions within the beam with slightly +different rotation angles. This results in a “beam depolar- +ization” of the signal. +Hutschenreuter et al. (2022) show that the possible +Faraday Rotation effects at the QUIJOTE-MFI frequencies +(10-20/,GHz) are very small in most of the sky, and partic- +ularly at high Galactic latitudes. Thus, in this work we have +not considered any Faraday Rotation effect. Nevertheless, in +Appendix B we study variations on the synchrotron spectral +index due to rotations of the polarized angle and compare +it to Faraday Rotation models such as the one proposed in +Hutschenreuter et al. (2022). +2.5 +Other contributions +It is well known that there are other foreground components +whose emissions are important for intensity analyses. In par- +ticular, at low frequencies, one needs to consider two ad- +ditional Galactic emission components: the bremsstrahlung +radiation generated from electron–ion scattering in interstel- +lar plasma (free-free), and AME, whose physical origin still +is not fully clear. At high frequencies, in addition to ther- +mal dust, we find an isotropic extragalactic emission called +the cosmic infrared background (CIB), coming from differ- +ent sources, e.g., dusty star-forming galaxies, quasars, in- +tergalactic stars, inter-cluster dust in the Local group, etc. +We also have other contributions such as CO line emission +or Sunyaev-Zeldovich effect (SZ) from clusters of galaxies +(Sunyaev & Zeldovich 1972) that should be taken into ac- +count in intensity analyses (Planck Collaboration 2016). In +addition, emission from extragalactic point sources, both at +radio and infrared frequencies is an important contaminant +at small scales. In polarization the problem is simplified since +several of these emissions (free-free, CIB, SZ, etc.) are not +expected to be polarized (at least significantly), therefore we +do not consider them. +The polarization of the anomalous microwave emission +is still under study because its nature is still uncertain +(Dickinson et al. 2018). Several models have been proposed +such as spinning dust particles (Ali-Ha¨ımoud 2013), mag- +netic dipole emission (Draine & Lazarian 1999) or more +recently the proposal of spinning nano-diamonds (Greaves +et al. 2018). The predicted polarization fraction of the AME +emission for most of these models is below 5%. From the data +analysis point of view, no evidence of polarization has been +found in compact region studies (the most stringent con- +straints on the polarization fraction, Π, have been provided +by G´enova-Santos et al. (2017), Π < 0.22% at 41 GHz). Due +to this lack of evidence, we do not take into account the +AME component in this work. +On the other hand, point sources present some degree +of polarization, which is in general small (a few percent). +However, at the resolutions considered in this work, they are +subdominant with respect to Galactic foregrounds. Thus, we +do not include them in our analysis. We note however that in +the data, a few polarized point sources are present that are +not taken into account in the component separation analysis, +see Herranz et al. (2023). +3 +COMPONENT SEPARATION +METHODOLOGY +In this work, we apply the parametric component separation +method B-SeCRET to extract the polarized astrophysical +signals. Parametric methods are very powerful since they +provide physical information of each sky component. How- +ever, they require a profound theoretical understanding of +the nature of the foregrounds and accurate knowledge of the +experiment’s characteristics to avoid biases in the analysis. +Below, in Section 3.1, we outline the component separa- +tion technique applied in this work. Then, in Section 3.2, we +describe the prior information that is used in the Bayesian +analyses. +3.1 +Bayesian analyses +The B-SeCRET methodology is a parametric pixel-based +maximum-likelihood method, which relies on an Affine- +Invariant Markov Chain Monte Carlo Ensemble sampler +to draw samples from a posterior distribution (Foreman- +Mackey et al. 2013). This methodology has already been +applied in previous studies, e.g., de la Hoz et al. (2020, 2022). +B-SeCRET applies Bayesian inference to determine the +best-fit model parameters given some prior information. In +Bayesian statistics, the probability of the model parameters +θp given the signal data dp at the pixel p is proportional to +the probability of the dp given θp times the probability of +θp, i.e., +P(θp|dp) ∝ P(dp|θp)P(θp) . +(6) +P(θp) is commonly known as the prior information, whereas +P(dp|θp) is usually referred to as the likelihood. Assuming +Gaussian noise, the likelihood of the data can be expressed +as +P(dp|θp) = +exp +� +−1 +2(dp − Sp)T C−1(dp − Sp) +� +� +(2π)N det(C) +, +(7) +where C is the noise covariance matrix, N is the number of +elements in the dp array, and Sp is the parametric model +considered, which has been described in detail in Section 2. +To draw samples from the posterior probability we use +the Python implementation emcee (Foreman-Mackey et al. +2013) of an affine-invariant ensemble sampler for MCMC +(Goodman & Weare 2010). In each pixel, the best-fit pa- +rameters and their uncertainties are obtained as the median +and the standard deviation of their respective marginalized +posterior probability. +MNRAS 000, 000–000 (0000) + +QUIJOTE-MFI Diffuse polarized foregrounds +5 +Figure 1. QUIJOTE observed sky after removing the geosta- +tionary satellite band and the region around the north celestial +pole, which is affected by high atmospheric air-mass (fsky = 51%, +Galactic coordinates centred on (0,0)). +3.2 +Priors +In this work we benefit from prior information about astro- +physical foregrounds to help with convergence and compu- +tational time reduction. For example, the synchrotron spec- +tral index is known to be around −3.1, although experi- +ments such as S-PASS found a more negative value. Here +we use the estimated value obtained with Planck polariza- +tion data by the SMICA method, βs = −3.1 ± 0.06 (Planck +Collaboration 2020d) and use a broad Gaussian distribu- +tion N(−3.1, 0.3)3 as a prior on βs. When we include a +curvature in the synchrotron model we apply a Gaussian +prior N(0, 0.1) on cs. Moreover, we apply Gaussian priors +N(1.55, 0.1) and N(21, 3) on both βd and Td respectively. +Finally, flat priors are used in the characterization of the +amplitude parameters. +4 +DATA +The aim of this work is to obtain a better characterization +of the low-frequency foregrounds by including the newly re- +leased QUIJOTE-MFI maps in component separation anal- +yses. In this Section we summarize the basic details of these +maps as well as those from the other experiments used in +the analysis, i.e., the K and Ka bands from WMAP and +Planck’s third and fourth public releases (PR3 and PR4, re- +spectively). We also discuss some technical issues related to +the instruments such as the estimated noise, RFI, and the +colour corrections. +4.1 +Datasets +In this analysis we have used the data from the following +experiments: +• QUIJOTE. We have used the low frequency QUI- +JOTE MFI 11 and 13 GHz channels (MFI) (Rubi˜no-Mart´ın +et al. 2023) due to their better signal-to-noise ratio. Al- +though QUIJOTE has observed 70% of the sky there are re- +gions with poorer sensitivity due to the presence of artificial +3 N(x, σ) represents a normal distribution with mean x and vari- +ance σ2. +satellites and high atmospheric masses in some directions. +Thus, in this analysis we have considered the mask shown +in Fig. 1, as the observable sky. This mask (satband+NCP) +is described in Rubi˜no-Mart´ın et al. (2023). +• WMAP. We have used the low frequency Nine- +Year Wilkinson Microwave Anisotropy Probe (WMAP) K +(22.8 GHz) and Ka (33.1 GHz) bands (Bennett et al. 2013)4. +• Planck. We have used the full set of Planck polariza- +tion maps i.e., the low frequency instrument (LFI) 30, 44 and +70 GHz frequency maps and the high frequency instrument +(HFI) 100, 143, 217 and 353 GHz maps. We have obtained +results from both PR35 (Planck Collaboration 2020b,c) and +PR4 (Planck Collaboration 2020f) data releases. +Before component separation analyses, the frequency +maps are all convolved (taking appropriately into account +the beam window function of each particular frequency map) +with a common beam, a Gaussian beam of FWHM = 2◦, +and downgraded to the same resolution through spherical +harmonics, given by the HEALPix parameter Nside = 64. +The procedure followed is described below: +(i) We calculate the spherical harmonics coefficients +(tℓm, eℓm, bℓm) using the healpy routine map2alm. +(ii) To convolve all channels with the same beam we +multiply the (tℓm, eℓm, bℓm) by bℓ(2◦) pℓ(64)/(bi,ℓ pℓ(Nside)), +where bℓ(α) is a gaussian beam window function whose +FWHM is α, bi,ℓ is the i-th channel beam window function +and, pℓ(Nside) is the pixel window function at the resolution +Nside. +(iii) We obtain the downgraded maps at Nside = 64 apply- +ing the healpy routine alm2map to the new (tℓm, eℓm, bℓm). +Several combinations of the previous data sets have +been tested. Each configuration’s name is given by the +“sum” of the sets of maps included in the analysis. For +example, the configuration composed of PR4 channels in +combination with WMAP’s K and Ka bands is referred as +K/Ka+PR4, or MFI-QUIJOTE low frequency channels in +combination with PR4 and WMAP channels is specified as +MFI+K/Ka+PR4. +4.2 +Instrumental Effects +Real data present different instrumental effects that need to +be accounted for. For example, an important contribution to +the observed signal is the noise produced by the detectors of +each experiment. A proper characterization of the noise lev- +els is key for component separation analyses. In this work, we +have calculated the covariance matrix among the frequency +channels per pixel, required by the parametric component +separation method, using realistic noise simulations specific +to each instrument. Each experiment’s noise simulations are +obtained as follows: +• QUIJOTE We have used the correlated noise simula- +tions described in Rubi˜no-Mart´ın et al. (2023). They account +4 The other bands were not included since they have a much +lower synchrotron signal-to-noise ratio and do not contribute to +the determination of the synchrotron characteristics. +5 We used the Planck maps corrected from bandpass leakage. +MNRAS 000, 000–000 (0000) + +6 +E. de la Hoz +for the 1/f noise present in the maps, and the correlated +noise component between 11 and 13 GHz. +• WMAP We have generated a set of white noise simula- +tions using the RMS noise per pixel provided by the WMAP +collaboration (Hinshaw et al. 2003). The RMS noise σ is cal- +culated as σ = σ0/√Nobs.6 +• Planck. For PR3 we have used the FFP10 simula- +tions generated by the Planck Collaboration (Planck Col- +laboration 2020a). In the case of the PR4, we have em- +ployed the noise simulations described in Planck Collabo- +ration (2020f).7 +While the frequency channels of different experiments +are uncorrelated, there might be correlations between chan- +nels of a given instrument. This is the case for the 11 and +13 GHz low-frequency MFI channels. On the other hand, +we have assumed no correlations between frequency chan- +nels for WMAP and Planck. Thus, for a given pixel p, the +Planck and WMAP frequency covariance matrices are diag- +onal while QUIJOTE’s has non-zero off-diagonal terms. For +a given configuration, the covariance matrix is obtained as a +block matrix, where each block corresponds to the frequency +covariance matrix of each instrument included in that con- +figuration. +To obtain the experiments’ frequency covariance matri- +ces, first we pre-process the noise simulations in the same +manner as the data maps. Then, for Planck and WMAP, +the diagonal terms are calculated as the variance of the +noise simulations at the corresponding pixel for each fre- +quency. Each pixel covariance matrix between QUIJOTE +11 and 13 GHz is calculated as the sample covariance ma- +trix using the values of the 11 and 13 GHz noise simulations +at that specific pixel. +One test to verify that our covariance matrices are well +estimated is the following. We obtained a distribution of χ2 +n,i +values as: +χ2 +n,i = nT +i C−1 +i +ni , +(8) +where ni is a noise simulated map8 at the frequency i and +Ci is the noise covariance matrix described above. The χ2 +n,i +distributions should have the expected form with Npix de- +grees of freedom (d.o.f.). This is consistent with the values +obtained for Planck and WMAP. In the case of QUIJOTE, +the distribution deviates slightly from the expected Npix +d.o.f. χ2-distribution since they are not end-to-end noise +simulations and hence not as accurate (see Rubi˜no-Mart´ın +et al. 2023, for details). However, as subsequent analyses +will show, we find that when the astrophysical emission is +included, the obtained χ2 is correct as expected, i.e., in the +regions where the model properly explains the data (outside +the Galactic plane). Thus, QUIJOTE’s noise simulations are +accurate enough to perform scientific analyses. +We explored the possibility of including correlations +6 σ0 and Nobs are given in https://lambda.gsfc.nasa.gov/ +product/wmap/dr5/skymap_info.html. +7 Simulations +available +at +NERSC +under +/global/cfs/cdirs/cmb/data/planck2020. +8 The noise simulations used in this test are different from the +noise simulations used to calculate the noise covariance matrices. +among neighbouring pixels within a 1 degree radius9. The +smoothing process of the maps induces noise correlations +among different pixels and, although this does not affect +our pixel-by-pixel analyses, it can affect analyses where we +assume a uniform parameter value within one region. There- +fore, for each pixel, we calculated the covariance matrix +among its neighbouring pixels from noise simulations. Then +we generated a sparse covariance matrix where the only non- +zero values in each row were the diagonal element and the +correlation with the neighbouring pixels. In this case the +distribution does not follow a Npix degrees-of-freedom χ2 +distribution as one would expect. The recovered values were +smaller than expected, more notably for Planck maps. This +is a consequence of not having enough noise simulations, +which prevents us from obtaining a good characterization +of the noise correlations. Therefore, we use the covariance +matrices that do not take into account possible noise corre- +lations among neighbouring pixels in the following. +As explained in Rubi˜no-Mart´ın et al. (2023), in order +to correct residual RFI signals emerging after co-adding all +data in the map-making process of the QUIJOTE-MFI data, +the polarization maps are corrected using a function of the +declination (FDEC). This correction is equivalent to apply- +ing a filter to QUIJOTE data, which removes the zero mode +in lines of constant declination. In Appendix C we studied +whether this correction affects the recovery of foregrounds +spectral parameters such as βs. We found that if only QUI- +JOTE maps are filtered with FDEC the recovered βs map +is biased in regions such as the North Polar Spur. When all +data maps are filtered in the same way this bias disappears. +Thus, for this analysis we have filtered all signal maps with +their corresponding FDEC function. +Another important instrumental effect arises from de- +tectors having a finite bandwidth. This issue has to be +taken into account when dealing with foreground compo- +nents whose amplitude varies within that frequency band. +This effect can be corrected by adding a multiplicative fac- +tor, called colour correction, to the signal that depends on +the spectral behaviour. We have used the fastcc Python +code (Peel et al. 2022, G´enova-Santos et al. 2023) to ob- +tain the colour corrections of each experiment considered +here. Therefore, our model for the sky signal presented in +Section 2 is corrected as follows: +Xν = Xν,cmb + +Xν,s +Cs(α, ν) + +Xν,d +Cd(βd, Td, ν), +(9) +where X is either Q or U, Cs(α, ν) is synchrotron colour +correction whose spectral behaviour is modelled as a power +law with α = βs + 2. The spectral behaviour of dust colour +correction Cd(βd, Td, ν) is assumed to be a modified black +body and it is determined by its βd and Td parameters. +The colour correction values are updated in each MCMC +iteration. +5 +RESULTS +In this Section we present the component separation prod- +ucts obtained using the recently released QUIJOTE low- +9 The pixels contained within this radius are the ones with the +largest correlations induced by the smoothing process. +MNRAS 000, 000–000 (0000) + +QUIJOTE-MFI Diffuse polarized foregrounds +7 +s +K/Ka+PR4 +-3.4 +-2.7 +s +0.01 +0.4 +2 +red +0 +10 +MFI+PR4 +-3.4 +-2.7 +0.01 +0.4 +0 +10 +MFI+K/Ka+PR4 +-3.4 +-2.7 +0.01 +0.4 +0 +10 +MFI+K/Ka+PR3 +-3.4 +-2.7 +0.01 +0.4 +0 +10 +Figure 2. Synchrotron spectral index (top row) and uncertainty maps (middle row) obtained after component separation with four +different datasets. The synchrotron emission is modelled with a power law. Bottom row: reduced χ2 map obtained for each dataset. +MFI data along with the already available Planck and +WMAP data. We have focused primarily on the synchrotron +spectral parameters since those are the parameters where +a greater improvement is found, see Section 5.1 and Sec- +tion 5.2. Moreover, we show the recovered amplitudes of the +CMB, synchrotron and thermal dust and, compare them +with those obtained by Commander using PR4 data in Sec- +tion 5.3. In Section 5.4 we present the spectral parameters +of the thermal dust. Finally, we evaluate the robustness of +these results in Section 5.5. +5.1 +Synchrotron Spectral Index +The major improvement obtained from including the low- +frequency QUIJOTE-MFI channels is having the sufficient +sensitivity to study the synchrotron spectral index with +great accuracy. Here we have conducted a deep study on sev- +eral aspects with regard to this parameter. First, we have +compared the recovered βs maps using different combina- +tions of the available datasets (Section 5.1.1). Section 5.1.2 +studies the spatial variability of βs. Finally, we compare our +results to the available βs models that are often exploited in +simulations used in CMB science forecasts in Section 5.1.3. +5.1.1 +Datasets +We have obtained different βs maps from component sep- +aration analyses using the four following datasets: WMAP +K and Ka bands with PR4 (K/Ka+PR4); QUIJOTE-MFI +11 and 13 GHz channels with PR4 (MFI+PR4); QUIJOTE- +MFI 11 and 13 GHz channels, WMAP K and Ka bands +and PR4 (MFI+K/Ka+PR4) and QUIJOTE-MFI 11 and +13 GHz channels, WMAP K and Ka bands and PR3 +(MFI+K/Ka+PR3). The results are shown in Fig. 2. It is +clear from the comparison of the synchrotron spectral index +uncertainty maps obtained in the K/Ka+PR4 case (first col- +umn) with respect to the MFI+K/Ka+PR4 case (third col- +umn), that the inclusion of QUIJOTE channels significantly +improves the estimation of βs. Moreover, we observe that, +outside the Galactic plane, the estimation of βs is very close +to the mean value of the prior set on this parameter, in this +case −3.1. In other words, the information contained in that +fraction of the data, i.e., the likelihood, is very poor and the +estimation is driven by the prior. +This improvement does not come from the inclusion of +more channels, but from channels where the synchrotron +contribution is larger. This is evident from the comparison +of the results from K/Ka+PR4 with respect to MFI+PR4, +where the number of frequency channels is the same but the +results are significantly better for the latter. +Finally, we have compared also the results obtained +with MFI+K/KA+PR3 and MFI+K/Ka+PR4 (fourth and +third column respectively). In this case the recovered un- +certainty maps are virtually the same but there are some +distinct differences between the βs maps that should be as- +cribed to changes in Planck maps. +One of the advantages of using a parametric compo- +nent separation method is that we can evaluate the good- +ness of the fit with certain estimators. In this work we use +the reduced χ2 estimator, whose value at a given pixel p is +calculated as: +χ2 +red,p = +1 +Ndof +� +i∈{Q,U} +(dp,i − Sp,i)C−1 +p,i(dp,i − Sp,i) , +(10) +where the sum is over all Q and U frequency channels, and +Ndof is the number of d.o.f.. The bottom row of Fig. 2 shows +the χ2 +red maps obtained for each dataset combination. These +maps show that our default model, i.e., a power law and a +MNRAS 000, 000–000 (0000) + +8 +E. de la Hoz +0 +2 +4 +6 +8 +10 +2 +red (K/Ka+PR4) +0 +2 +4 +6 +8 +10 +2 +red (MFI+K/Ka+PR4) +0 +2 +4 +6 +8 +10 +2 +red (MFI+PR4) +0 +2 +4 +6 +8 +10 +2 +red (MFI+K/Ka+PR3) +Figure 3. Reduced χ2, χ2 +red, obtained using the MFI+K/Ka+PR4 dataset vs. the χ2 +red obtained using K/Ka+PR4 (left), MFI+PR4 +(center) and MFI+K/Ka+PR3 (right). The color scale is related to the density of points, redder (bluer) corresponds to denser (sparser) +regions. The orange rectangle shows the χ2 +red within a 95% confidence region. The slope calculated with the points within this 95% +confidence region is m = 0.686 ± 0.004 (left column), m = 0.732 ± 0.003 (center column) and m = 0.731 ± 0.003 (right column), shown +with a green dashed line. The orange dashed line shows the one-to-one line. The synchrotron emission is modelled with a power law. +modified black body to model the synchrotron and thermal +dust emission respectively, provides a good fit (low values of +χ2 +red) outside the Galactic plane. Within the Galactic plane, +this model is not able to capture all the physical complexity +and the χ2 +red values are quite large. However, we note that in +this analysis we have considered statistical uncertainties but +not calibration errors, which in QUIJOTE are of 5%. Apart +from the higher complexity of the Galactic plane emission, +the higher χ2 +red in this region could also be due, in part, to +having neglected calibration errors. +We have also used the χ2 +red estimator to select the +dataset that is used as the default for further tests be- +tween the MFI+K/Ka+PR3 and the MFI+K/Ka+PR4 +datasets, i.e., the only combinations that include all the +channels considered. In Fig. 3 the χ2 +red obtained using the +MFI+K/Ka+PR4 dataset is plotted against the χ2 +red ob- +tained with K/Ka+PR4, MFI+PR4 and MFI+K/Ka+PR3. +The 95% confidence regions are delimited by orange lines. +These lines indicate the χ2 values, from the reduced χ2- +distribution with Ndof d.o.f.10, that satisfy that the normal- +ized area covered to their left is equal to 0.95. We have also +fitted the points within this confidence regions to a straight +line to determine which dataset has more pixels with smaller +χ2 +red. If the slope is larger than unity, the dataset on the hor- +izontal axis has more pixels with smaller χ2. If the slope is +smaller than unity, the dataset on the vertical axis is the +one which satisfies that condition. +Although it is not clear from the left plot of Fig. 3 +which dataset is better, the slope m = 0.686 ± 0.004 indi- +cates that the MFI+K/Ka+PR4 dataset provides a better +fit. Moreover, the K/Ka+PR4 dataset has larger uncertain- +ties which can mask model inconsistencies. On the other +hand, from the middle plot of Fig. 3, we observe that the +inclusion of the K and Ka WMAP bands to the MFI+PR4 +dataset improves the goodness of the fit. Finally, compar- +ing MFI+K/Ka+PR4 with MFI+K/Ka+PR3, we see that +PR3 provides a better fit in the Galactic plane, while PR4 +10 The χ2-distribution with Ndof divided by Ndof. +fits better outside the Galactic plane (Fig. 2). Since the fit +in the Galactic plane is bad in both cases we have chosen +the MFI+K/Ka+PR4 as our default dataset as it retrieves +better fits within the 95% confidence regions (pixels outside +the Galactic plane, Fig. 4). +5.1.2 +Spatial Variability +We have also studied the spatial variability of the syn- +chrotron spectral index in several high signal-to-noise re- +gions of the sky, see Fig. 5. These connected regions satisfy +the condition that βs is estimated with a signal-to-noise ra- +tio larger than 15. In particular, R1 is associated with the +North Polar Spur (NPS), and R2 encompasses the Galac- +tic plane. R3, R4 and R5 are other sky regions where the +polarized synchrotron intensity has a large signal-to-noise +ratio. +Fig. 6 shows the estimated synchrotron spectral index +against the uncertainty on the estimation of all the pixels +within a given region. We have limited this study to those +pixels with a χ2 +red within the 95% confidence region. The +area delimited by the dotted lines contains the values that +are consistent within 3σ with the weighted mean in each +region. The top left panel indicates that βs has a large spatial +variability across the whole available QUIJOTE-MFI sky +(QS). Therefore, a constant value of βs is not a good model +of the synchrotron emission. On the contrary, the R1, R3, +R4 and R5 pixels values are well within those lines, i.e., +a uniform βs value could be a good model for all pixels +within each region. Finally, R2 (top right panel) shows a +significant spatial variability which is consistent with the +large heterogeneity observed in the βs map. +The study of regions with uniform βs values helps with +improving the detectability of primordial B-modes. Allow- +ing spatial variations of the spectral parameters at the pixel +level results in a very robust parametrization of the signal +sky. However, this robustness comes at the expense of an in- +crease in the statistical uncertainty of the parameters as less +information is provided in the fit (Errard & Stompor 2019). +MNRAS 000, 000–000 (0000) + +QUIJOTE-MFI Diffuse polarized foregrounds +9 +s +-3.4 +-2.7 +s +0.01 +0.4 +2 +red +0 +10 +Figure 4. Synchrotron spectral index (top), its uncertainty (mid- +dle) and reduced χ2 (bottom) maps obtained after component +separation with the default dataset MFI+K/Ka+PR4. The syn- +chrotron emission is modelled with a power law. +Thus, several approaches have been proposed in the litera- +ture to define sky regions with uniform spectral parameters. +For example, in Errard & Stompor (2019), these regions are +chosen as super-pixels at a lower HEALPix maps resolution, +whereas in Grumitt et al. (2020), the regions are obtained +using clustering algorithms such as the mean-shift cluster- +ing algorithm. Recently, Puglisi et al. (2022) has presented +a new methodology based on spectral clustering to define +R1 +R2 +R3 +R4 +R5 +Figure 5. R1, R3, R4 and R5 are sky regions where βs is assumed +uniform in Section 5.1.2 and R2, which encompasses the Galactic +plane seen by QUIJOTE, is a very heterogeneous region. These +regions satisfy that βs is recovered with a signal-to-noise larger +than 15. +Table 1. Synchrotron spectral index estimation βR +s and its uncer- +tainty σ(βR +s ) obtained assuming uniform value across the regions +R1, R3, R4 and R5 shown in Fig. 5. +Region +fsky (%) +βR +s +σ(βR +s ) +R1 +4.84 +-3.028 +0.002 +R3 +0.96 +-2.945 +0.008 +R4 +0.56 +-3.319 +0.011 +R5 +0.21 +-3.228 +0.019 +geometrical affine regions with similar spectral parameters. +It is worth noting that if the assumption of uniform spectral +parameters within those regions does not hold, the mod- +elling errors introduced might bias cosmological parameters +measurements obtained from the output CMB map after +component separation, as well as foreground model param- +eters. +We have calculated the value of βs in some of these +regions assuming a constant value within each region. We +have performed the fit in the following manner: +• First we fix βs to a given value and fit the rest of the +model parameters in each pixel of the region. +• Then, the rest of the parameters are fixed to the estima- +tion from the previous fit, and we fit βs assuming a unique +value in the whole region under study. +• βs is fixed to the new obtained value and the process is +repeated until it reaches convergence. +We have chosen the median of the βs values (obtained pixel- +wise) within that region as the initial guess of βs. The results +are shown in Table 1. Notice that the uncertainty on the +recovered βs has dramatically decreased. This is simply a +result of having N R +pix (the number of pixels contained within +the region R) times more information to fit the parameter. +The βs values recovered in each region (R1, R3, R4 and +R5) are not consistent among them. These results further +showcase the spatial variability of the synchrotron’s spectral +parameter. +MNRAS 000, 000–000 (0000) + +10 +E. de la Hoz +0.0 +0.5 +1.0 +1.5 +4.0 +3.5 +3.0 +2.5 +s +QS +0.0 +0.1 +0.2 +R1 +0.0 +0.5 +1.0 +1.5 +R2 +0.0 +0.2 +0.4 +s +4.0 +3.5 +3.0 +2.5 +s +R3 +0.0 +0.1 +0.2 +0.3 +s +R4 +0.0 +0.1 +0.2 +s +R5 +Figure 6. Synchrotron spectral index estimate against its uncertainty within different sky regions: QUIJOTE-MFI sky (QS) (Fig. 1); +R1, R2, R3, R4 and R5 are shown in Fig. 5. The solid, dashed, and dotted lines enclose the values of βs within 1σ, 2σ and 3σ of the +weighted mean respectively. The study is limited to those pixels whose χ2 +red lies within the 95% confidence region. +3.6 +3.4 +3.2 +3.0 +2.8 +2.6 +s +Model 4 M-D et al. (QS95) +MFI+K/Ka+PR4 (QS95) +MFI+K/Ka+PR4 (HS2N95) +Figure 7. Distribution of the synchrotron spectral index from +“Model 4” of Miville-Deschˆenes et al. (2008) and from our esti- +mation using the MFI+K/Ka+PR4 dataset. Vertical dashed lines +indicate the mean value for each distribution. +5.1.3 +Comparison with current βs models +In this Section we compare our βs map with the cur- +rently most used βs template11, the “Model 4” Miville- +11 Used for example in the Planck Sky Model (Ashdown et al. +2012), or in the Python Sky Model (PySM) a Python library to +simulate foregrounds (Thorne et al. 2017). +Deschenes et al. template, which was constructed with +Haslam and WMAP observations in temperature (Miville- +Deschˆenes et al. 2008). Fig. 7 shows the distribution of the +spectral index value for this model (blue) and for our anal- +ysis (orange), considering only those QUIJOTE-MFI pixels +that lie within the 95% confidence region of the χ2 (QS95). +In the QS95 region, the mean and the standard deviation +from the “Model 4” of Miville-Deschˆenes et al. (2008) tem- +plate are −3.00 ± 0.05 while those from our estimate are +−3.08±0.13. It is interesting to note that the variability ob- +served in our analysis is significantly larger. A direct compar- +ison of the dispersion of both maps (using the same mask) +indicates an increment of the spatial variability in our study +around a factor of 2.6, i.e., σ(βMFI+K/Ka+PR4 +s +)/σ(βModel 4 +s +) ∼ +2.6. +One may wonder if this result can be affected by the +considered prior, since the estimated spectral indices for low +signal-to-noise pixels are significantly constrained by it (see +Section 5.5.2). In order to test this point, we have repeated +the previous analysis considering only those pixels satisfy- +ing that the recovered βs values have a signal-to-noise larger +than 15 (i.e., where the synchrotron signal-to-noise is high +and thus the results are not driven by the prior) and lie +within the 95% confidence region of the χ2 (HS2N95). In +this case, we find that the mean value and dispersion of the +distribution of βs are −3.12±0.15 for our analysis (see green +histogram in Fig. 7) versus −3.00 ± 0.05 for “Model 4” in +the same region, confirming our finding. Although our esti- +mations can be affected by the presence of noise, the results +show that the variability of the synchrotron spectral index +assumed in current templates is underestimated. A similar +MNRAS 000, 000–000 (0000) + +QUIJOTE-MFI Diffuse polarized foregrounds +11 +increment in the variability was also noted by analysing the +S-PASS data in the Southern Hemisphere (Krachmalnicoff +et al. 2018). +Recently Weiland et al. (2022) published a composite +map of βs using publicly available data covering approxi- +mately 44% of the sky. In the region covered in our study, +they obtained βs estimates in the Galactic Plane and the +North Polar Spur using information from WMAP K and Ka +band, and estimates at latitudes larger than 40◦ using K, Ka +and DRAO 1.41 GHz map (Wolleben et al. 2006). From a vi- +sual inspection our results are compatible within the North +Polar Spur. We find that our derived spectral indices are +steeper at the Galactic plane. Weiland et al. (2022) found +discrepancies between the βs values obtained in the Fan Re- +gion when they performed the analysis using WMAP K and +Ka band versus WMAP K band and Planck LFI 30 GHz +channel. In the latter case, the recovered βs were signifi- +cantly steeper. We repeated our analysis excluding the PR4 +30 GHz channel and did not observe a discrepancy concern- +ing the βs recovered from the default analysis in Fan Region. +This results from the fact that the βs recovery is mainly +driven by QUIJOTE-MFI data. At high latitudes we can- +not make a reasonable comparison since our βs estimates +are driven by the prior. They also show that DRAO data +have some unexplained systematics and can be affected by +Faraday Rotation depolarization. +Other studies, such as those presented in Vidal et al. +(2015); Fuskeland et al. (2014, 2021); Martire et al. (2022), +also find variability of the spectral index analyzing differ- +ent regions of the sky. However it is difficult to compare +the same regions in our map, since they compute a global +spectral index for large areas, while we work pixel by pixel. +For example, near the center of the Galactic plane we see +a fair amount of structure that cannot be accounted for in +the T–T scatter plots analyses carried out in some of the +cited papers, that use several pixels to obtain a single βs +value. In that sense, the methodology followed here is more +complete given that we perform a full component separation +in each pixel, retrieving information at smaller scales for a +large fraction of the sky. +Rubi˜no-Mart´ın et al. (2023) obtain an estimate of the +synchrotron spectral index map directly from the compari- +son of the QUIJOTE-MFI 11 GHz map with the WMAP K +band map. The results obtained there are fully consistent +with the ones from this work. +5.2 +Synchrotron Curvature +We have also explored a synchrotron model with curva- +ture, i.e., the model presented in equation (3), using the +MFI+K/Ka+PR4 dataset. Fig. 8 shows the estimation and +uncertainty maps of the curvature parameter as well as the +χ2 +red map and the cs signal-to-noise map. +We observe from the signal-to-noise map that curvature +is detected at more than 3σ in the Galactic plane, in regions +where the fit is not good as it can be seen from the χ2 +red map. +Even though the inclusion of a curvature parameter is not +able to explain the complexity of this region, this parameter +can account for some effects along the Galactic plane, e.g., +Faraday rotation. +Outside the Galactic plane the estimated cs values are +close to zero and their uncertainties are around 0.1, which +Table 2. Estimated values of the curvature and its uncertainty +obtained assuming the curvature is uniform within the region. +Region +fsky (%) +cR +s +σcR +s +���cR +s +��� /σcR +s +RC1 +45.48 +-0.0797 +0.0012 +63.75 +RC2 +5.93 +-0.2768 +0.0017 +161.57 +Haze +0.94 +0.041 +0.010 +4.23 +North bubble +0.63 +-0.083 +0.007 +11.43 +are the expected value and the spread of the prior set on cs. +Moreover, the recovered βs map in this case is very similar +to the one obtained when the synchrotron is model with a +power law. This means that we do not have enough sensitiv- +ity to detect a spatially varying curvature. Hopefully, joint +analyses with future releases of the Northern Celestial Hemi- +sphere data like the new MFI2 instrument and C-BASS at +5 GHz (Jones et al. 2018) might elucidate more details on +changes of the power law spectrum. +In Fig. 9 we compare the goodness of fit using a power +law versus a power law with curvature as the synchrotron +model. We see that there are more points located below the +bisector. Besides, the slope 0.9227±0.0005 calculated at the +95% confidence region, shows that, given the current data, +the power law model is slightly preferred over the power law +plus curvature model. +Furthermore, we have considered modelling the syn- +chrotron emission with a power law with uniform curvature. +We have assumed a constant cs in four regions: RC1, RC2, +and the Haze and North bubble (Fig. 10). The recovered +curvature values are shown in Table 2. RC1 encompasses all +the pixels whose χ2 +red is within 95% confidence region. RC2 +is composed of the RC1 pixels that also satisfy that the syn- +chrotron polarized intensity signal-to-noise ratio at 30 GHz +is larger than 5. We detect curvature in all regions. The de- +tection is more evident in RC1 and RC2, mostly due to the +higher sensitivity (lower σC) in these regions. However, it is +important to highlight that there is no physical reasoning +behind the definition of RC1 and RC2, and the assumption +of uniform curvature in all synchrotron high signal-to-noise +regions is arbitrary12. In the Haze and North bubble, we +find a curvature value different from zero at more than 3σ. +These regions are studied in greater detail in Guidi et al. +(2023). +We have studied how βs changes when we impose the +constraint of having a uniform cs value within each region. +The results are displayed in Fig. 11. For RC2, we observe +that βs steepens considerably. The weighted mean value of +βs in RC2 is ⟨βs⟩ = −3.022 ± 0.011 in the pixel-wise anal- +ysis and, ⟨βs⟩ = −3.375 ± 0.002 when cs is imposed to be +uniform in RC2. For RC1, this effect is not as considerable. +The weighted mean values are ⟨βs⟩ = −3.079 ± 0.002 and +⟨βs⟩ = −3.1651 ± 0.0014 when cs varies pixel-wise and is +uniform respectively. The steepening of βs leads to values of +the exponent βs +cs log(ν/νs) within [-3.04,-3.10] at 11 GHz +which are compatible with the average value of βs when we +fit to a power law model. From these results, we infer that +the βs and cs parameters are not independent. More sensi- +12 Any curvature will be more easily detected in high signal-to- +noise regions than in low signal-to-noise regions. +MNRAS 000, 000–000 (0000) + +12 +E. de la Hoz +cs +-0.4 +0.4 +cs +0.01 +0.2 +2 +red +0 +10 +|cs|/ +cs +0 +3 +Figure 8. Top row: Synchrotron curvature estimate (left) and uncertainty (right) maps obtained after component separation using the +default dataset (MFI+K/Ka+PR4). The synchrotron emission is modelled using a power law with spatially varying curvature (pixel-wise). +Bottom row: reduced χ2 map (left) and cs signal-to-noise map (right). +0 +2 +4 +6 +8 +10 +2 +red, plc +0 +2 +4 +6 +8 +10 +2 +red, pl +Figure 9. Reduced χ2 calculated using a power law as a model +of the synchrotron emission (χ2 +red,pl) vs. χ2 +red when the model +is a power law with spatially varying curvature (χ2 +red,plc). The +color scale is related to the density of points, redder (bluer) cor- +responds to denser (sparser) regions. The red rectangle shows the +χ2 +red within a 95% confidence region. The slope at the 95% confi- +dence region is m = 0.9227 ± 0.0005, shown with a green dashed +line. The orange dashed line shows the one-to-one line. +tive data at the QUIJOTE frequencies and at lower and/or +higher frequencies are required to break the degeneracy. +In order to test which model provides a better goodness +of fit we calculate the reduced χ2 of a given region R as +(a) RC1 (coloured) and RC2 (orange) regions. +Haze +North Bubble +(b) Haze and North bubble regions. +Figure 10. Regions where cs has been assumed to be uniform. +MNRAS 000, 000–000 (0000) + +QUIJOTE-MFI Diffuse polarized foregrounds +13 +3.75 +3.50 +3.25 +3.00 +2.75 +2.50 +s (uniform cs) +3.75 +3.50 +3.25 +3.00 +2.75 +2.50 +s (spatially varying cs) +RC1 +RC2 +Figure 11. Comparison between the pixel βs values obtained +when fitting the synchrotron emission with a spatially vary- +ing curvature model (y-axis) versus with a model with uni- +form curvature (x-axis) in the regions RC1 and RC2 using the +MFI+K/Ka+PR4 dataset. +Table 3. Reduced χ2 obtained using either a power law or a +power law with curvature model in different regions, R. We have +considered two curvature models: one where cs varies spatially +(spatial) and other where cs is assumed constant in R. +Model +Curvature +Region +χ2 +red,R +power law +– +RC1 +0.892 +power law + curvature +spatial +RC1 +0.965 +power law + curvature +uniform +RC1 +0.936 +power law +– +RC2 +1.010 +power law + curvature +spatial +RC2 +1.088 +power law + curvature +uniform +RC2 +1.081 +power law +– +Haze +0.845 +power law + curvature +spatial +Haze +0.936 +power law + curvature +uniform +Haze +0.885 +power law +– +North bubble +0.961 +power law + curvature +spatial +North bubble +1.041 +power law + curvature +uniform +North bubble +0.986 +follows: +χ2 +red,R = +1 +Ndof +NR +pix +� +p=1 +� +i∈{Q,U} +(dp,i − Sp,i)C−1 +p,i(dp,i − Sp,i) , +(11) +where we sum over all pixels N R +pix within R. The d.o.f. are +given as Ndof = N R +pix(2N − Nθ) when all model parameters +are allowed to vary pixel-wise, and Ndof = N R +pix(2N −(Nθ − +1)) − 1 when cs is assumed uniform in the analysis, where +Nθ is the number of model parameters. We calculated the +value of this estimator in three cases: i) when the model +parameters are allowed to vary spatially using a power law +model for the synchrotron component, ii) when the model +parameters vary from pixel-to-pixel using a power law with +curvature model, iii) when we fit the data assuming uniform +curvature using a power law with curvature model. The re- +sults are given in Table 3. +The χ2 +red results show that the models we used, i.e., +power law and power law with curvature, are compatible +with the data. However there is not enough statistical sig- +nificance to discern which model suits better the data. Es- +pecially, considering that we have not been able to take into +account possible correlations between pixels and that the +power law with curvature model is degenerate13. +5.3 +Recovered Amplitudes and Comparison with +Planck results +We have compared our baseline results, i.e., using the +MFI+K/Ka+PR4 dataset and a power law as the syn- +chrotron model, to those obtained from the Commander +pipeline (Eriksen et al. 2008) applied to PR4 data14. We +have only considered this pipeline among those used by +Planck, since it is the reference method with regard to the +recovery of foreground components. In Figs. 12 to 14, we +show a comparison of the CMB, the synchrotron emission +at 30 GHz, and the thermal dust emission at 353 GHz be- +tween Commander and our results. In order to perform a +direct comparison we have filtered Commander results with +FDEC. The left column shows the Q and U Commander +amplitudes, the center column our amplitudes and the right +column the corresponding uncertainties. A visual inspection +shows that both estimates are very similar, especially the +synchrotron and thermal dust emissions which are the dom- +inant contributions in polarization. +5.3.1 +CMB +Regarding CMB, the left column of Fig 15 shows the pixel- +to-pixel comparison for the recovered CMB map from our +analysis and from Commander both in Q and U. We have +applied a combination of the QUIJOTE observed sky and +the common polarization confidence mask provided by the +Planck Collaboration15 (Planck Collaboration 2020d). +We observe from the maps that there is a discrepancy. +We found that the application of the FDEC filter, before +the component separation process, leads to a decrease of +the amplitude in the power spectra of our recovered CMB +map. This power reduction appears only when Planck and +WMAP are filtered with FDEC, since the CMB information +is extracted mainly from those channels. Instead of applying +the FDEC filter, one could apply a filter that suppresses the +large scales. This would be equivalent to applying a linear +function to the CMB and there would not be a reduction +of power. However, since we want to study all scales we +decided to apply the FDEC filter. Since the aim of this work +is the study of the foregrounds, we keep the results obtained +with all the data filtered with FDEC to recover the βs map +without any bias. One can in principle recover the unbiased +CMB following one of the approaches described below: +13 We considered applying other statistics such as the Bayesian +evidence to do model selection. However, since the QUIJOTE- +MFI noise simulations are not end-to-end and the Bayesian evi- +dence is very computationally expensive we did not perform any +model selection analysis. This is left for future work. +14 Data available at NERSC under /cmb/daa/planck20. +15 Available at https://pla.esac.esa.int/#maps. +MNRAS 000, 000–000 (0000) + +14 +E. de la Hoz +(CMBQ) (MFI+K/Ka+PR4) +0.01 +0.75 +(CMBU) (MFI+K/Ka+PR4) +0.01 +0.75 +CMBQ (MFI+K/Ka+PR4) +-2 +2 +CMBU (MFI+K/Ka+PR4) +-2 +2 +CMBQ (Commander) +-2 +2 +CMBU (Commander) +-2 +2 +Figure 12. Left column: Commander Q (top) and U (bottom) CMB maps at Nside = 64, smoothed with a Gaussian beam to a final +resolution of FWHM = 2◦. Centre column: CMB Q and U maps using the MFI+K/Ka+PR4 dataset. Right column: uncertainty of the +CMB maps. Maps are in thermodynamic temperature (µK). We apply the common polarization confidence mask provided by Planck. +(aQ +s ) (MFI+K/Ka+PR4) +0.01 +2 +(aU +s ) (MFI+K/Ka+PR4) +0.01 +2 +aQ +s (MFI+K/Ka+PR4) +-80 +80 +aU +s (MFI+K/Ka+PR4) +-80 +80 +aQ +s (Commander) +-80 +80 +aU +s (Commander) +-80 +80 +Figure 13. Left column: Commander Q (top) and U (bottom) synchrotron amplitude maps at 30 GHz at Nside = 64, smoothed with +a Gaussian beam to a final resolution of FWHM = 2◦. Centre column: our estimate of the synchrotron amplitude at 30 GHz, using the +MFI+K/Ka+PR4 dataset. Right column: uncertainty of the estimated synchrotron amplitude. Maps are in antenna temperature (µK). +MNRAS 000, 000–000 (0000) + +QUIJOTE-MFI Diffuse polarized foregrounds +15 +(aQ +d ) (MFI+K/Ka+PR4) +0.01 +2 +(aU +d) (MFI+K/Ka+PR4) +0.01 +2 +aQ +d (MFI+K/Ka+PR4) +-80 +80 +aU +d (MFI+K/Ka+PR4) +-80 +80 +aQ +d (Commander) +-80 +80 +aU +d (Commander) +-80 +80 +Figure 14. Left column: Commander Q (top) and U (bottom) thermal dust amplitude maps at 353 GHz at Nside = 64, smoothed with +a Gaussian beam to a final resolution of FWHM = 2◦. Centre column: our estimate of the thermal dust amplitude at 353 GHz, using +the MFI+K/Ka+PR4 dataset. Right column: uncertainty of the estimated thermal dust ampltitude. Maps are in antenna temperature +(µK). +2 +1 +0 +1 +2 +2 +1 +0 +1 +2 +MFI+K/Ka+PR4 [ KCMB] +Q +200 +0 +200 +400 +200 +0 +200 +400 +MFI+K/Ka+PR4 [ KRJ] +Q +0 +100 +50 +0 +50 +100 +150 +MFI+K/Ka+PR4 [ KRJ] +Q +2 +1 +0 +1 +2 +Commander [ KCMB] +2 +1 +0 +1 +2 +MFI+K/Ka+PR4 [ KCMB] +U +CMB +150 +100 +50 +0 +50 +100 +Commander [ KRJ] +150 +100 +50 +0 +50 +100 +MFI+K/Ka+PR4 [ KRJ] +U +Synchrotron +0 +100 +Commander [ KRJ] +50 +0 +50 +100 +150 +MFI+K/Ka+PR4 [ KRJ] +U +Thermal dust +Figure 15. Comparison of CMB (left), synchrotron at 30 GHz (centre) and thermal dust at 353 GHz (right) amplitudes recovered using +the MFI+K/Ka+PR4 dataset and the ones obtained by Commander using PR4 data. The correlation factors are ρQ = 0.543 and +ρU = 0.817 (CMB), ρQ = 0.992 and ρU = 0.973 (synchrotron) and ρQ = 1.000 and ρU = 0.997 (thermal dust). +MNRAS 000, 000–000 (0000) + +16 +E. de la Hoz +aQ +s +-5 +5 +aU +s +-5 +5 +Figure 16. Difference between the synchrotron amplitude aQ +s +(aU +s ) obtained with the MFI+K/Ka+PR4 and the Commander +estimate, top row (bottom row). Maps are in antenna temperature +(µK). +• Perform the component separation analysis without fil- +tering the data with FDEC and including the FDEC correc- +tion in QUIJOTE-MFI data as part of the model; or +• Given the unbiased βs map16 and Planck data, one can +construct a template with the modes that QUIJOTE-MFI +data is missing after being filtered with FDEC. Then per- +form the analysis with the reconstructed QUIJOTE-MFI +maps. +Since the estimation of the CMB is out of the scope of this +paper, we leave this analysis for future works. +5.3.2 +Synchrotron +Fig. 16 shows the difference between the synchrotron am- +plitude maps obtained using the MFI+K/Ka+PR4 and +the Commander reconstruction using the PR4 data. The +largest differences observed are located in the Galactic plane +where the model fails to reproduce the sky signal. We also +observe large scale structures in the difference map. These +structures can originate from the fact that we have obtained +a more accurate estimation of the scaling law as our fit is per- +formed using additional frequencies. However, overall, the +correlation between both methods is very good. +16 Obtained in the component separation analysis using the data +filtered with FDEC. +This can also be seen in the centre column of Fig. 15, +where a pixel-to-pixel comparison is given, showing that +both methods present a synchrotron amplitude at 30 GHz +highly correlated for Q and U except in some pixels where +the synchrotron emission is very large. Those pixels are lo- +cated primarily in the Galactic plane. These discrepancies +are likely to arise from differences in the amplitude of the +polarised intensity instead of from differences in the polar- +ization angles. In Fig. 15 we observe that both the slopes, +in the Q and U plots, are higher than unity. If the discrep- +ancies were originated from differences in the polarization +angle, one slope would be higher than unity and the other +lower. +5.3.3 +Dust +Regarding thermal dust emission, this foreground strongly +dominates the 353 GHz Planck frequency map and, there- +fore, the recovered amplitude is very much determined by +this channel. This was also the case in the Commander +analysis done by the Planck Collaboration. Thus, our recov- +ered Q and U components of the thermal dust are strongly +correlated with those obtained using Commander, see the +right column of Fig. 15. +5.4 +Dust Spectral Parameters +Although the frequencies of QUIJOTE-MFI do not overlap +with the spectral range where the thermal dust is more dom- +inant, we have studied whether the inclusion of this data set +in the analysis can help with the thermal dust characteriza- +tion due to an improvement on the determination of the rest +of the polarized foreground parameters. Fig. 17 shows the +thermal dust spectral index βd recovered with the default +data set, modelling the synchrotron emission as a power law, +in two cases: +• Td is included as a model parameter. +• Td is fixed to Commander’s estimation of the thermal +dust temperature from the component separation analysis in +intensity (Planck Collaboration 2016) like Commander did +in their polarization analysis. Fixing Td helps breaking its +degeneracy with βd in the Rayleigh-Jeans part of the ther- +mal dust spectrum, which is the one observed with Planck +in polarization. +In both maps we find that the recovered βd values are close +to the expected value of the prior, i.e., 1.55, except close to +the Galactic plane where the thermal dust signal is larger17. +The results differ significantly along the Galactic plane, see +Fig. 18. This difference originates since our recovered Td map +does not resemble the used Td template as shown in Fig. 19. +We remark that although in the first case Td is estimated +from the polarization analysis, the Td recovered values lie +close to the expected value of the prior (22 K) except along +the Galactic plane where the fit is not good. Moreover, it +is very difficult to fit Td from polarization data only, as the +17 Notice that the uncertainty does not improve in the regions +where the βd values are close to the mean value of the prior when +we fix one parameter. The uncertainty in those pixels is the spread +of the prior. +MNRAS 000, 000–000 (0000) + +QUIJOTE-MFI Diffuse polarized foregrounds +17 +d +1.3 +1.7 +d|Td +1.3 +1.7 +d +0.001 +0.15 +d|Td +0.001 +0.15 +Figure 17. Left column: estimate (top) and uncertainty (bottom) of thermal dust spectral index obtained when Td is included as a +model parameter. Right column: estimate (top) and uncertainty (bottom) of thermal dust spectral index obtained when the Td template +obtained by Commander in the intensity analysis is used to fix Td in the component separation process. +s +-3 +3 +d +-3 +3 +Figure 18. βd (top row) and βs (bottom row) relative difference +map between the maps obtained when we include Td as a model +parameter and when we fix it. +highest frequency is 353 GHz, and thus we are not able to +trace the thermal dust peak. +In Fig. 18 we show the relative difference between spec- +tral index map of the thermal dust and synchrotron obtained +when Td is included as a model parameter and when it is +fixed. The relative difference is calculated as follows: +� +∆β1,2 = +β1 − β2 +� +σ2 +β1 + σ2 +β2 − 2σβ1,β2 +, +(12) +where σ2 +β1 (σ2 +β2) is the variance of the β1 (β2) map, and +σβ1,β2 is the covariance between the β1 and β2 maps that +are being compared. As expected from Fig. 17 the differences +close to the Galactic plane are significantly large in the case +of βd. On the other hand, we find that, the βs maps recovered +in both cases are compatible and the differences resemble +Gaussian noise except along the Galactic plane where the +model fails. +We also studied the relative difference between the βd +map obtained with the MFI+K/Ka+PR4 and K/Ka+PR4 +data sets in Fig. 20. The top panel shows the relative dif- +ference when Td is included as a model parameter and the +bottom panel when Td is fixed. We observed that both maps +are compatible except in regions where the fit is not good. +Moreover, when we compare the uncertainty maps we find +that there is not a significant improvement when we in- +clude QUIJOTE-MFI channels. Thus, we conclude that the +improvement in the characterization of low-frequency fore- +grounds does not help necessarily with the estimation of +thermal dust spectral parameters. +MNRAS 000, 000–000 (0000) + +18 +E. de la Hoz +Td +15 +25 +Td +-7 +7 +Figure 19. Top row: thermal dust temperature map recovered +in the default case. Bottom row: difference map between the top +row map and the Td template used in the analysis. Maps are in +Kelvin. +5.5 +Goodness of fit +In this Section we study in depth the quality of the results +obtained using the default dataset. In Section 5.5.1 we an- +alyze the χ2 distribution of the results as well as the Q +and U residuals of each channel. Section 5.5.2 investigates +the robustness of our results regarding the estimation of the +synchrotron spectral index with respect to the prior applied +to this parameter. +5.5.1 +χ2 distribution and residuals +We have studied the pixel χ2 distribution obtained from the +fit using MFI+K/Ka+PR4 (see Fig. 21): +χ2 +p = Ndof · χ2 +red . +(13) +Moreover, we have also calculated the residuals per channel +involved in the analysis: +rp,ν = (dp,ν − Spν) +σp,ν +. +(14) +In the perfect scenario, residuals maps are consistent with +instrumental noise alone. Therefore, they are a valuable tool +to look for either systematic effects or mismatches in the +foreground modelling. +First of all, we recall that the number of d.o.f. for this +analysis is 13 (11 channels × 2 (Q and U) minus 9 free pa- +rameters). We find ⟨χ2 +p⟩ = 14.3 and σ = 8.9 slightly larger +than what is expected for the theoretical number of d.o.f.. +Fig. 21 shows that the χ2 +p values follow a χ2-like distribu- +tion, whose peak lies close to Ndof = 13. However, there +d +-3 +3 +d +-3 +3 +Figure 20. βd relative difference map between the map ob- +tained using the MFI+K/Ka+PR4 and the one obtaiened with +K/Ka+PR4 datasets when we include Td as a model parameter +(top row) and when we fix it (bottom row). +0 +10 +20 +30 +40 +50 +60 +70 +2 +p +Figure 21. χ2 +p distribution obtained using the default dataset. +The orange curve shows the theoretical χ2 probability density +function with Ndof = 13. The area to the left of the gray dashed +line shows values within the 95% confidence region. +is an excess of pixels at large values of χ2 with respect to +the χ2 +Ndof -distribution. That excess appears since there are +pixels where the model is not able to track the true sky +emission, mainly in the Galactic plane. Thus, those pixels +are highly inconsistent with this χ2 distribution. +Fig. 22 shows the Q and U residuals maps of every +frequency channel from the MFI+K/Ka+PR4 dataset. We +MNRAS 000, 000–000 (0000) + +QUIJOTE-MFI Diffuse polarized foregrounds +19 +MFI 11 GHz (Q) +MFI 11 GHz (U) +MFI 13 GHz (Q) +-200 +200 +MFI 13 GHz (U) +-200 +200 +WMAP K (Q) +WMAP K (U) +WMAP Ka (Q) +-20 +20 +WMAP Ka (U) +-20 +20 +PR4 30 GHz (Q) +PR4 30 GHz (U) +PR4 44 GHz (Q) +PR4 44 GHz (U) +PR4 70 GHz (Q) +-5 +5 +PR4 70 GHz (U) +-5 +5 +PR4 100 GHz (Q) +PR4 100 GHz (U) +PR4 143 GHz (Q) +PR4 143 GHz (U) +PR4 217 GHz (Q) +PR4 217 GHz (U) +PR4 353 GHz (Q) +-5 +5 +PR4 353 GHz (U) +-5 +5 +Figure 22. Q and U residual maps for each frequency channel at Nside = 64. Maps are displayed in thermodynamic temperature (µK). +find that Planck and WMAP residuals maps are reasonably +consistent with the expected noise except along the Galactic +plane. The residuals in this region are a consequence of an +incorrect modelling of the sky as we saw in the χ2 +red maps. +For the MFI channels we observe that the largest residu- +als are located in compact regions along the Galactic plane. +We observe in the 11 GHz U channel a redder region in the +NPS’s closest part to the Galactic centre. This region over- +laps with the area where we obtain a better goodness-of-fit if +Faraday Rotation effects are taken into account, see Fig. B2 +in Appendix B. Furthermore, artefacts that resemble the +FDEC morphology are present in MFI 13 GHz. +In light of these tests, we are confident of the results ob- +tained in those pixels, that are properly modelled by our as- +sumed parametric model. The pixels outside the confidence +region are located mainly in the Galactic plane, probably be- +cause our model fails to account for the complexity of this +region. It would be convenient to study these regions in more +detail with more complex models. However, the aim of this +work is to study the diffuse components and the study of +specific regions has been conducted in other works (Watson +et al. 2023; Ruiz-Granados et al. 2023; Guidi et al. 2023). +5.5.2 +Robustness with respect to the prior +As previously stated, the use of prior information is essential +in Bayesian analysis, helping with convergence and compu- +tational time reduction. Besides, when the data do not have +enough sensitivity, i.e., there is not enough information to +obtain a reliable estimation of the spectral index, the prior +tends to provide a value close to the mean value of the dis- +tribution. In other words, a conservative value is assigned to +the spectral index in those pixels. Thus, in order to detect +which pixels are prior-dependent we have also performed +component separation using two additional Gaussian priors +on βs, N(−3.1, 0.6) and N(−3.0, 0.3). The βs estimation +and uncertainty maps with these new priors are shown in +Fig. 23 together with those obtained with the prior used in +the default analysis (left columns). +Comparing +the +results +using +the +default +prior, +i.e., +N(−3.1, 0.3), +versus +a +less +restrictive +prior, +i.e., +N(−3.1, 0.6), we observe that the uncertainty on the re- +covered βs increases at the prior-dominated pixels. On the +other hand, in those regions where the synchrotron emission +is very intense the uncertainty remains the same. Likewise, +the estimated βs in the latter pixels are very similar whereas +the other pixels are visually different. The βs distribution of +the pixels outside the low-uncertainty regions are compati- +ble with the prior distribution. This is the reason why the +estimated values are different and the spread is larger when +the prior is relaxed. +When we use a prior with a different expected value, +i.e., N(−3.0, 0.3), but equal standard deviation we obtain +a similar uncertainty map. The estimated βs is almost the +same in the low-uncertainty regions, i.e., the high-intensity +synchrotron regions. However, a flatter spectrum (closer to +−3.0 instead of −3.1) is recovered outside those areas. This +is more evident from the bottom panel of Fig. 24 where +the difference between the βs map estimated with the de- +fault prior and the N(−3.0, 0.3) prior is shown. Outside the +regions where the synchrotron emission is the largest, the +difference is close to −0.1 which is the difference between +the expected value of the priors. In other words, when there +is not enough information from the data the recovered βs is +close to the expected value of the prior. This is an advan- +tage of using prior information, since it assigns a conserva- +tive value to the spectral index instead of unphysical values +or simply failing to perform the fit. +6 +CONCLUSIONS +In this work, we have presented the component separa- +tion products in polarization obtained from combining the +MNRAS 000, 000–000 (0000) + +20 +E. de la Hoz +s +( +3.1, 0.3) +-3.4 +-2.7 +s +0.01 +0.4 +( +3.1, 0.6) +-3.4 +-2.7 +0.01 +0.4 +( +3.0, 0.3) +-3.4 +-2.7 +0.01 +0.4 +Figure 23. Synchrotron spectral index estimate (top row) and uncertainty (bottom row) obtained using different Gaussian prior +distributions and the default dataset (MFI+K/Ka+PR4). The synchrotron emission is modelled as a power law. +QUIJOTE-MFI data at 11 and 13 GHz, with the WMAP +K and Ka bands and all Planck polarized channels. We +have seen that the inclusion of the QUIJOTE-MFI data is +crucial to improve the parameter estimation of the low fre- +quency foregrounds, in particular for the estimation of the +synchrotron spectral index. +We have obtained the first detailed βs map of the North- +ern Celestial Hemisphere at a scale of 2◦ assuming the syn- +chrotron emission is modelled as a power law. This model +represents well the data except in the Galactic plane where +the physics might be more complex. We find, using the pixels +whose χ2 +red lies within the 95% confidence region, an average +value of −3.08 and a dispersion of 0.13. The latter is broader +than the dispersion of commonly used βs templates. More- +over, we have found that the spectral index is not compati- +ble with a uniform value, i.e., there are statistical significant +differences of βs across the observable sky. +We have also modelled the synchrotron emission as a +power law with curvature. The pixel-based analysis of the +curvature shows that cs is only detected in some regions in +the Galactic plane where the fit is bad. When we assume +a model with uniform curvature in RC1 (the region that +includes all pixels whose χ2 +red is within the 95% confidence +region for the power law with curvature model) we found +a cs = −0.0797 ± 0.0012. We found that both models, i.e., +power law and power law with uniform curvature, provide +a good fit given the available data. However there is not +enough statistical significance to distinguish which model is +better. A more thorough study is left for future work. +We found that our recovered synchrotron and ther- +mal dust maps are highly correlated with the maps pre- +sented by the Planck collaboration using Commander, even +though we found some large scale difference between the syn- +chrotron emission maps which arise from better estimation +of the SED due to the addition of more frequency channels. +On the other hand, we recovered a CMB with less power +when we use the filtered K, Ka and PR4 with FDEC. Since +our analysis focuses on the characterization of foregrounds +we keep the results obtained with the filtered maps. How- +ever, as commented in Section 5.3.1 an unbiased CMB map +can be recovered following other approaches. +We have also performed different analyses to test the +validity of our results. First, we found that our results are +compatible with a χ2 distribution in those pixels where +the power law model fits well the data. Furthermore, we +have calculated the normalized residuals of the pixels with +an acceptable goodness of fit of all frequency channels and +they are all consistent within the 3σ level. Finally, we have +evaluated the robustness of the estimated βs varying the +prior imposed in this parameter. We found that the estima- +tions in the high signal-to-noise synchrotron areas are prior- +independent, while outside these regions the prior governs +the βs estimation. +ACKNOWLEDGMENTS +We thank the staff of the Teide Observatory for invalu- +able assistance in the commissioning and operation of +QUIJOTE. The QUIJOTE experiment is being developed +by the Instituto de Astrofisica de Canarias (IAC), the +Instituto de Fisica de Cantabria (IFCA), and the Univer- +sities of Cantabria, Manchester and Cambridge. Partial +financial support was provided by the Spanish Ministry of +Science and Innovation under the projects AYA2007-68058- +C03-01, AYA2007-68058-C03-02, AYA2010-21766-C03-01, +AYA2010-21766-C03-02, +AYA2014-60438-P, +ESP2015- +70646-C2-1-R, AYA2017-84185-P, ESP2017-83921-C2-1-R, +AYA2017-90675-REDC +(co-funded +with +EU +FEDER +funds), PGC2018-101814-B-I00, PID2019-110610RB-C21, +PID2020-120514GB-I00, +IACA13-3E-2336, +IACA15-BE- +3707, EQC2018-004918-P, the Severo Ochoa Programs +SEV-2015-0548 +and +CEX2019-000920-S, +the +Maria +de +Maeztu Program MDM-2017-0765, and by the Consolider- +MNRAS 000, 000–000 (0000) + +QUIJOTE-MFI Diffuse polarized foregrounds +21 +s +-0.3 +0.3 +s +-0.3 +0.3 +Figure 24. Difference map between the estimated βs using +the default prior, i.e., N(−3.1, 0.3), and the one obtained us- +ing an alternative prior, see Fig. 23. Top: N(−3.1, 0.6) Bottom: +N(−3.0, 0.3). +Ingenio +project +CSD2010-00064 +(EPI: +Exploring +the +Physics of Inflation). We acknowledge support from the +ACIISI, Consejeria de Economia, Conocimiento y Empleo +del Gobierno de Canarias and the European Regional +Development Fund (ERDF) under grant with reference +ProID2020010108. This project has received funding from +the European Union’s Horizon 2020 research and inno- +vation program under grant agreement number 687312 +(RADIOFOREGROUNDS). EdlH acknowledges financial +support from the Concepci´on Arenal Programme of the +Universidad de Cantabria. DT acknowledges the support +from the Chinese Academy of Sciences (CAS) President’s +International Fellowship Initiative (PIFI) with Grant N. +2020PM0042. FP acknowledges support from the Span- +ish State Research Agency (AEI) under grant number +PID2019-105552RB-C43. +The +authors +acknowledge +the +computer +resources, +technical +expertise +and +assistance +provided by the Spanish Supercomputing Network (RES) +node at Universidad de Cantabria. Some of the presented +results are based on observations obtained with Planck +(http://www.esa.int/Planck), an ESA science mission with +instruments and contributions directly funded by ESA +Member States, NASA, and Canada. We acknowledge +the use of the Legacy Archive for Microwave Background +Data Analysis (LAMBDA) and the Planck Legacy Archive +(PLA). Support for LAMBDA is provided by the NASA +Office of Space Science. Some of the results in this paper +have been derived using the HEALPix package (G´orski +et al. 2005), and the healpy (Zonca et al. 2019), numpy +(Harris et al. 2020), emcee (Foreman-Mackey et al. 2013), +and matplotlib (Hunter 2007) Python packages. +DATA AVAILABILITY +The parameter maps obtained from the component sep- +aration +analysis +in +the +default +case, +i.e., +with +the +MFI+K/Ka+PR4 dataset using a power law to model the +synchrotron emission, are included in the released data prod- +ucts associated to the QUIJOTE-MFI wide survey. +These data products as well as the maps can be freely +downloaded from the QUIJOTE web page18, as well as from +the RADIOFOREGROUNDS platform19. They include also +an Explanatory Supplement describing the data formats. +Any other derived data products described in this paper +are available upon request to the QUIJOTE collaboration. +REFERENCES +Abazajian K. N., et al., 2016, arXiv e-prints, p. arXiv:1610.02743 +Ade P., et al., 2019, J. 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T., et al., 2023, +MNRAS, accepted +Ruiz-Granados B., et al., 2023, MNRAS, in prep. +Rybicki G. B., Lightman A. P., 2008, Radiative Processes in As- +trophysics. John Wiley & Sons +Sunyaev R. A., Zeldovich Y. B., 1972, Comments on Astrophysics +and Space Physics, 4, 173 +Thorne B., Dunkley J., Alonso D., Næss S., 2017, MNRAS, 469, +2821 +Tristram M., et al., 2022, Phys. Rev. D, 105, 083524 +Vidal M., Dickinson C., Davies R. D., Leahy J. P., 2015, MNRAS, +452, 656 +Watson R. A., et al., 2023, MNRAS, in prep. +Weiland J. L., Addison G. E., Bennett C. L., Halpern M., Hinshaw +G., 2022, ApJ, 936, 24 +Wolleben M., Landecker T. L., Reich W., Wielebinski R., 2006, +A&A, 448, 411 +Zonca A., Singer L., Lenz D., Reinecke M., Rosset C., Hivon E., +Gorski K., 2019, Journal of Open Source Software, 4, 1298 +APPENDIX A: INDEPENDENT Q AND U +SYNCHROTRON SPECTRAL INDEX +In order to test the assumption of having the same βs in both +Q and U, we fit Q and U signals independently. Fig. A1 +shows the spectral index, the uncertainty of the spectral +index as well as the reduced χ2 maps obtained from the +three independent fits using the MFI+K/Ka+PR4 dataset. +We infer from the χ2 +red maps that the fit outside the Galactic +plane is better when Q and U are fitted together. When +we fit just U we observe that the goodness of fit improves +significantly in the Galactic plane. However, this effect is +due to the low signal-to-noise ratio in that area, not due to +a better modelling of the signal. +The βQ +s and βU +s maps are distinctly different. The βQU +s +map resembles more the βQ +s map. This is expected, since Q +has more signal than U in Galactic coordinates. That is also +the reason why the uncertainty on the recovered βs is smaller +when we fit just Q compared to U. However, in those regions +where σβU +s +is smaller than σβQ +s , i.e., regions where U has +more signal than Q, the βQU +s +values obtained are closer to +those of βU +s . This is clearly seen in Fig. A2 where the relative +difference between βQU +s +with respect to βQ +s (top row) and βU +s +(bottom row) is shown. The largest differences shown in the +top (bottom) panel are located in regions where the signal- +to-noise is larger in U (Q). On the other hand the relative +difference decreases significantly in the regions where the +uncertainty on βQ +s (top) or βU +s (bottom) is smaller. +APPENDIX B: FARADAY ROTATION +We have also studied the significance of the difference be- +tween the βQ +s +and βU +s +maps, see top row of Fig. B1. The +discrepancies larger than 3σ are concentrated in the Galac- +tic plane, close to the Galactic centre. This could be a tracer +of Faraday rotation. If Faraday rotation is non-negligible at +QUIJOTE frequencies there will be a difference between the +polarization angles at QUIJOTE frequencies and those at +WMAP/Planck frequencies. This yields a βQ +s map different +from βU +s due to the bias introduced by the change in angle. +That bias is reasonably cancelled out when combining both +Q and U to obtain a single index. +We have studied the possibility of correcting the Fara- +day rotation effect in the QUIJOTE MFI maps using the +model from Hutschenreuter et al. (2022). The rotation of +the polarization plane experienced due to the Faraday Ro- +tation effect can be described by: +∆φ = RMλ2 , +(B1) +where λ is the wavelength, and RM is the rotation measure. +We use the RM map estimated by Hutschenreuter et al. +(2022) to calculate the rotation angle maps at 11 and 13 GHz +MNRAS 000, 000–000 (0000) + +QUIJOTE-MFI Diffuse polarized foregrounds +23 +s +QU +-3.4 +-2.7 +s +0.01 +0.4 +2 +red +0 +10 +Q +-3.4 +-2.7 +0.01 +0.4 +0 +10 +U +-3.4 +-2.7 +0.01 +0.4 +0 +10 +Figure A1. Synchrotron spectral index estimate (top row) and uncertainty maps (second row) obtained after component separation +using the MFI+K/Ka+PR4 dataset. The left column shows the βs recovered when we assume that Q and U share the same spectral +index, while the centre and right columns depict the Q and U βs when they are assumed to be independent. Bottom row: reduced χ2 +map for each case study considered. The synchrotron emission is modelled as a power law. +QU, Q +-3 +3 +QU, U +-3 +3 +Figure A2. Relative difference map between the βs map ob- +tained when we assume the same βs in both Q and U, and βs +recovered from the fit with just Q (top) and just U (bottom). +The synchrotron emission is modelled as a power law. +Q, U +-3 +3 +Q(FR), U(FR) +-3 +3 +Figure B1. Relative difference map between the βs map from +the independent Q and U fit using the MFI+K/Ka+PR4 dataset +(top), and using the MFI(FR)+K/Ka+PR4 dataset (bottom). +The synchrotron emission is modelled as a power law. +MNRAS 000, 000–000 (0000) + +24 +E. de la Hoz +2 +red +2 +red, FR +-3 +3 +Figure B2. Difference map between the χ2 +red obtained with the +MFI+K/Ka+PR4 dataset with respect to the χ2 +red,FR obtained +with MFI(FR)+K/Ka+PR4 dataset. In both fits we have as- +sumed that Q and U share the same spectral indices. The syn- +chrotron emission is modelled as a power law. +QUIJOTE frequencies. Then, QUIJOTE Q and U maps at +a given frequency ν are de-rotated as follows: +� +QFR +UFR +� +ν += +� +cos(2∆φν) +− sin(2∆φν) +sin(2∆φν) +cos(2∆φν) +� +ν +� +Q +U +� +ν +, +(B2) +The variance of the de-rotated QFR and UFR is: +σ2 +QFR = cos2(2∆φν)σ2 +Q + sin2(2∆φν)σ2 +U +(B3) ++ 4 +� +sin(2∆φν)Q + cos(2∆φν)U +�2 σ2 +∆φ +σ2 +UFR = sin2(2∆φν)σ2 +Q + cos2(2∆φν)σ2 +U +(B4) ++ 4 +� +cos(2∆φν)Q − sin(2∆φν)U +�2 σ2 +∆φ +Therefore, we have repeated the same analysis but us- +ing the MFI(FR)+K/Ka+PR4 dataset, where MFI(FR) in- +dicates that the QUIJOTE 11 and 13 GHz maps have been +de-rotated using the angle obtained from the Hutschenreuter +et al. (2022) model, to correct any possible mismatch due to +the Faraday Rotation effect, see bottom row of Fig. B1. +We compare these maps (Fig. B1) with the difference +map between the reduced χ2 map (χ2 +red) obtained with +the MFI+K/Ka+PR4 dataset with respect to the reduced +χ2 (χ2 +red,FR) obtained with MFI(FR)+K/Ka+PR4 dataset +shown in Fig. B2. The sky regions where the absolute value +of the relative difference � +∆βQ(FR),U(FR) is smaller than +� +∆βQ,U are correlated to those regions where the χ2 +red,FR is +smaller than χ2 +red (reddish regions) and vice versa (bluish +regions). This result suggests that Faraday rotation might +be playing a role in some of the significant differences areas +observed between βQ +s and βU +s . +APPENDIX C: FUNCTION-OF-DECLINATION +CORRECTION SIMULATIONS +We studied using simulations if the application of a function- +of-declination (FDEC) filter to QUIJOTE-MFI maps biases +the βs map obtained from component separation. We gen- +erated sky simulation maps with the following components +at the QUIJOTE-MFI 11 and 13 GHz, K and Ka, and PR4 +frequencies: +all +s +-3 +3 +MFI +s +-3 +3 +Figure C1. Relative difference map between the βs template +used in the simulation and the βs map from the fit using the +simulated data with an FDEC filter applied to all maps (top), +and an FDEC filter applied only to QUIJOTE-MFI frequencies +(bottom). +• CMB. Generated as Gaussian random samples using +the power spectra obtained from CAMB (Lewis & Challi- +nor 2011) with the latest Planck cosmological parameters +(Planck Collaboration 2020e). +• Synchrotron. Generated using the s1 model of the +Python Sky Model (PySM) (Thorne et al. 2017). +• Thermal dust. Generated using the d1 model of the +PySM. +• Realistic noise simulations. For each experiment we use +the ones described in Section 4. +All components are either generated or downgraded to +Nside = 512. Then the components maps are added and we +apply the corresponding FDEC filter to each signal map. Fi- +nally all maps are downgraded to Nside = 64 and smoothed +with a Gaussian beam of FWHM = 2 deg following the pro- +cedure described in Section 4. +We perform the component separation analysis in two +scenarios: i) when only the QUIJOTE-MFI frequency signal +maps are filtered, and ii) when all maps are filtered. Fig. C1 +shows the relative difference (equation 12) between the βs +map recovered from the component separation analysis and +the βs template (equation 12 taking into account that the +uncertainty of the template map is set to zero, σβs = 0). We +find that when only QUIJOTE-MFI channels are filtered +(bottom panel) the relative differences are larger in regions +MNRAS 000, 000–000 (0000) + +QUIJOTE-MFI Diffuse polarized foregrounds +25 +such as the North Polar Spur or the R3 region than when all +maps are filtered. Moreover, in those regions the βs relative +differences are larger than 3σ with respect to the template In +the case when all maps are filtered (top panel), these biases +are reduced significantly. +1Instituto de F´ısica de Cantabria (IFCA), CSIC-Univ. +de Cantabria, Avda. los Castros s/n, E-39005 Santander, +Spain. +2Dpto. de F´ısica Moderna, Universidad de Cantabria, Avda. +de los Castros s/n, E-39005 Santander, Spain. +3Instituto de Astrof´ısica de Canarias, E-38205 La Laguna, +Tenerife, Spain. +4Departamento de Astrof´ısica, Universidad de La Laguna, +E-38206 La Laguna, Tenerife, Spain. +5Institut d’Astrophysique de Paris, UMR 7095, CNRS & +Sorbonne Universit´e, 98 bis boulevard Arago, 75014 Paris, +France. +6Astrophysics Group, Cavendish Laboratory, University of +Cambridge, J J Thomson Avenue, Cambridge CB3 0HE, +U.K. +7Kavli Institute for Cosmology, University of Cambridge, +Madingley Road, Cambridge CB3 0HA, U.K. +8Dpto. de Ingenieria de COMunicaciones (DICOM), Edif. +Ingenieria de Telecomunicacion, Pl. de la Ciencia s/n, E- +39005 Santander, Spain. +9Dpto. de Matem´aticas, estad´ıstica y computaci´on, Univ. de +Cantabria, Avda. de los Castros s/n, E-39005 Santander, +Spain. +10Aurora Technology for the European Space Agency (ESA), +European Space Astronomy Centre (ESAC), Camino Bajo +del Castillo +s/n, 28692 Villanueva de la Ca˜nada, Madrid, Spain. +11Universidad Europea de Madrid, 28670, Madrid, Spain. +12Jodrell Bank Centre for Astrophysics, Alan Turing Build- +ing, Department of Physics and Astronomy, School of Nat- +ural Sciences, +University of Manchester, Oxford Road, Manchester M13 +9PL, U.K. +13Consejo Superior de Investigaciones Cientificas, E-28006 +Madrid, Spain +14Dpto. de F´ısica. Facultad de Ciencias. Univ. de C´ordoba. +Campus de Rabanales, Edif. C2. Planta Baja. E-14071 +C´ordoba, Spain. +15Purple Mountain Observatory, CAS, No.10 Yuanhua +Road, Qixia District, Nanjing 210034, China. +16NAOC-UKZN Computational Astrophysics Center (NU- +CAC), University of Kwazulu-Natal, Durban 4000, South +Africa. +MNRAS 000, 000–000 (0000) + diff --git a/HNE4T4oBgHgl3EQfgg1T/content/tmp_files/load_file.txt b/HNE4T4oBgHgl3EQfgg1T/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d431489d5a42bf1d028af587747bd3f05607a7f9 --- /dev/null +++ b/HNE4T4oBgHgl3EQfgg1T/content/tmp_files/load_file.txt @@ -0,0 +1,1625 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf,len=1624 +page_content='MNRAS 000, 000–000 (0000) Preprint 13 January 2023 Compiled using MNRAS LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='0 QUIJOTE scientific results – VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Diffuse polarized foregrounds from component separation with QUIJOTE-MFI E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' de la Hoz,1,2⋆R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Barreiro,1 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Vielva,1 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Mart´ınez-Gonz´alez,1 J.' metadata={'source': 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M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Peel,3,4 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Piccirillo,12 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Poidevin,3,4 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Rebolo,3,4,13 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Ruiz-Granados,3,4,14 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Tramonte,15,16,3,4 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Vansyngel,3,4 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Watson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='12 Affiliations are listed at the end of the paper Accepted 2022 October 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Received 2022 October 14;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' in original form 2022 July 27 ABSTRACT We derive linearly polarized astrophysical component maps in the Northern Sky from the QUIJOTE-MFI data at 11 and 13 GHz in combination with the WMAP K and Ka bands (23 and 33 GHz) and all Planck polarized channels (30-353 GHz), using the para- metric component separation method B-SeCRET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' The addition of QUIJOTE-MFI data significantly improves the parameter estimation of the low-frequency foregrounds, especially the estimation of the synchrotron spectral index, βs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' We present the first detailed βs map of the Northern Celestial Hemisphere at a smoothing scale of 2◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' We find statistically significant spatial variability across the sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' We obtain an average value of −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='08 and a dispersion of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='13, considering only pixels with reliable goodness- of-fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' The power law model of the synchrotron emission provides a good fit to the data outside the Galactic plane but fails to track the complexity within this region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Moreover, when we assume a synchrotron model with uniform curvature, cs, we find a value of cs = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='0797 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='0012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' However, there is insufficient statistical significance to determine which model is favoured, either the power law or the power law with uniform curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Furthermore, we estimate the thermal dust spectral parameters in polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Our CMB, synchrotron, and thermal dust maps are highly correlated with the corresponding products of the PR4 Planck release, although some large-scale differences are observed in the synchrotron emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Finally, we find that the βs es- timation in the high signal-to-noise synchrotron emission areas is prior-independent while, outside these regions, the prior governs the βs estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Key words: cosmology: observations – methods: data analysis – polarization – cosmic microwave background 1 INTRODUCTION Currently, most of the efforts of the CMB community are devoted to the search for primordial B-modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' These pre- dicted B-modes at large scales can only be produced by tensor modes, and their detection would constitute com- ⋆ e-mail:delahoz@ifca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='unican.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='es pelling evidence of an inflationary phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' The intensity of this primordial signal is determined by the tensor-to-scalar ratio r, the relative amplitude between the tensor and scalar modes at a given pivot scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' The current best upper bound on the tensor-to-scalar ratio is: r < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='032 at 95% CL, set by the combination of Planck, BICEP2/KeckArray and baryon- acoustic-oscillation data (Tristram et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' The weakness of the primordial B-modes makes its de- © 0000 The Authors arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='05117v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='CO] 12 Jan 2023 2 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' de la Hoz tection a tremendous experimental challenge, requiring high- sensitivity experiments as well as an exquisite control of sys- tematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Indeed, a large effort is currently on-going with the aim to detect, or at least to constrain, r with a sen- sitivity σr(r = 0) ⩽ 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' This includes many planned ground-based experiments, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=', GroundBIRD (Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 2020), LSPE-Strip (Lamagna et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 2020), CMB-S4 (Abaza- jian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 2016), Simons Observatory (Ade et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 2019) and BICEP array (Hui et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 2018), as well as satellite missions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=', LiteBIRD (LiteBIRD Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 2022) and PICO (Hanany et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' The detectability of the primordial B-modes could be improved by removing the secondary B-mode component in- duced by weak gravitational lensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Several delensing pro- cedures have been proposed in the literature (Planck Collab- oration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Millea et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 2019) and have been applied to data from current CMB experiments (Planck Collabora- tion et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Carron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' BICEP/Keck Collabora- tion et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 2021), and in forecasts of future CMB experiments (Diego-Palazuelos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Namikawa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' It is necessary to disentangle the CMB polarization signal from those coming from other microwave emissions, such as Galactic synchrotron, thermal dust and extragalactic point sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Thus, the problem of component separation is a crucial step in order to detect the primordial B-mode of CMB polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' This process benefits from the char- acterization of foreground emissions using complementary frequency ranges that provide unique information about the contaminants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' The main diffuse polarized contaminants are the syn- chrotron emission (at low-frequencies) and the thermal dust emission (at high-frequencies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' The best characterization of these diffuse foregrounds has been done by Planck (Planck Collaboration 2020d) using a data set covering frequencies from 30 to 353 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' This frequency range limited strongly the estimation of the synchrotron spectral parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' In Planck Collaboration (2020d) it is shown that, with Planck data only, one cannot test the spatial variability of the syn- chrotron spectral index due to limited sensitivity and fre- quency coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' The data only allows a measurement of a global spectral index of βs = −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' The synchrotron spectral index has also been estimated using other datasets, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=', Fuskeland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' (2014);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Krachmalnicoff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Fuskeland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' The Q-U-I JOint Tenerife Experiment (QUIJOTE) (Rubi˜no-Mart´ın et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 2010) is a polarimetric ground-based CMB experiment whose main scientific goal is the charac- terization of the polarization of the cosmic microwave back- ground (CMB) and other Galactic and extragalactic phys- ical processes in the frequency range 10–40 GHz and at large angular scales (≳ 1◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' The experiment is located at the Teide Observatory (at ∼ 2400 m above sea level) in Tenerife.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' It is composed of two telescopes equipped with three instruments: the Multi-Frequency Instrument (MFI), the Thirty-GHz Instrument (TGI), and the Forty-GHz In- strument (FGI), operating at 10–20 GHz, 26–36 GHz and 39–49 GHz respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' The MFI instrument has been operating from Novem- ber 2012 to October 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' It conducted two different sur- veys: i) a shallow Galactic survey (called “wide survey”) covering all the visible sky from Tenerife at elevations larger than 30◦, and ii) a deep cosmological survey covering ap- proximately 3000 deg2 in three separated sky patches in the northern sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' In this work we use the QUIJOTE-MFI wide survey maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' This survey provides an average sensitivity in polarization of ∼ 35–40 µK per 1-degree beam in four bands centred around 11, 13, 17 and 19 GHz (Rubi˜no-Mart´ın et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Those frequencies are crucial to achieving a better characterization of the low-frequency foregrounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' In inten- sity, this additional information helps breaking degenera- cies between the synchrotron, free-free and anomalous mi- crowave emissions while, in polarization, the QUIJOTE-MFI channels are key to characterize the synchrotron spectral de- pendence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' In the present work we perform a component separa- tion analysis to obtain more information about the polarized sky using the QUIJOTE-MFI data1 (Rubi˜no-Mart´ın et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 2023) in combination with the publicly available Planck (Planck Collaboration 2020f,a) and Nine-Year WMAP (Ben- nett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 2013) data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' To perform component separa- tion analysis we use B-SeCRET (Bayesian-Separation of Components and Residual Estimation Tool), a parametric maximum-likelihood method described in de la Hoz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' The paper is organized as follows: in Section 2 we pro- vide details of the main components in the polarized mi- crowave sky and the corresponding parametric models used to characterize them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Section 3 describes briefly the B- SeCRET method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' The data used in the analysis are pre- sented in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Then, the main component separation results obtained are shown in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Finally, the main conclusions from the analysis are given in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' In Ap- pendix A we provide maps of the synchrotron spectral index obtained from independent fits in linear Stokes parameters Q and U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Appendix B compares the variations on the syn- chrotron spectral index due to rotations of the polarized angle with Faraday rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 2 THE MICROWAVE SKY MODEL The polarized microwave sky is composed primarily of pho- tons from the CMB, synchrotron and thermal dust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' As stated before, the synchrotron emission dominates at low- frequencies while the thermal dust is the principal compo- nent at higher frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' The contribution from other com- ponents, discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='5, is expected to be insignif- icant and not taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Apart from these astro- nomical signals, the measured sky signal maps have another contribution from the instrumental noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' The characteris- tics of this noise depend on the specifications of the exper- iment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Furthermore, contaminants such as the atmosphere and artificial signals from satellites also contribute to the microwave sky, see Rubi˜no-Mart´ın et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' (2023) for more de- tails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Thus, the measured polarized sky signal for a given ν channel can be expressed as the following sum: � Q U � ν = � Qcmb Ucmb � ν + � Qs Us � ν + � Qd Ud � ν + � Qn Un � ν , (1) 1 This is one of the papers which are part of the MFI wide survey data release.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' MNRAS 000, 000–000 (0000) QUIJOTE-MFI Diffuse polarized foregrounds 3 where Xcmb, Xs, and Xd are the CMB, synchrotron and thermal dust signals respectively, and Xn is the instrumen- tal noise (X ∈ {Q, U}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' In the subsequent subsections we describe the main physical components that encompass the sky signal as well as some effects that alter this signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' More- over, we present the parametric models that we use in the component separation analysis for each polarized astronom- ical component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='1 Synchrotron The synchrotron emission arises from relativistic particles (cosmic rays) passing through the Galactic magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Its emissivity depends both on the magnetic field strength and energy distribution of the relativistic particles (generally electrons).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' These quantities are not uniform in the Galactic disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' For instance, the free electrons are more predominant in compact regions as supernovae remnants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' On the other hand, the magnetic field is amplified in some compact re- gions and can have different strength and direction across the sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' The synchrotron spectral energy distribution (SED) is generally described as a power law (Rybicki & Lightman 2008): � Qs Us � ν = � AQ s AU s � � ν νs �βs , (2) where As is the amplitude in brightness temperature at the pivot frequency νs = 30 GHz and βs is the spectral index which is assumed to be equal for both Q and U Stokes pa- rameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Previous works dedicated to the estimation of the spec- tral index, found values around βs ≃ −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='1 (Planck Col- laboration 2020d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' However, the spectral index is expected to vary spatially due to its dependence on the energy dis- tribution of the cosmic rays N(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Studies such as Fuske- land et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' (2014);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Vidal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' (2015);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Krachmalnicoff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Martire et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Weiland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' (2022) indicate that different polarized regions present different spectral in- dices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Here, we conduct a more detailed analysis of the βs spatial variations in the Northern Hemisphere by perform- ing a pixel-by-pixel component separation analysis using the QUIJOTE MFI polarized maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' The S-PASS survey (Carretti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 2019), has provided the most sensitive reconstruction of the βs variations of the South Celestial Hemisphere (Krachmalnicoff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' They found large variability over the sky, and a mean value of −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='22 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Those results were further confirmed in the analysis of Fuskeland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' (2021) that estimated the spec- tral index taking into account the Faraday Rotation effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' They also studied the Galactic plane and found compatible results to those where only WMAP data were used, finding a flatter index in the Galactic plane than at high Galactic latitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' We have also considered an extension of equation (2) where we include a possible curvature in the synchrotron’s SED: � Qs Us � ν = � AQ s AU s � ν � ν νs �βs+cs log � ν νs � , (3) where cs is the parameter that represents the curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' This extension is worth studying since a curved spectrum can account for steepening or flattening of the SED due to different effects, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=', cosmic ray aging effect or multiple synchrotron components along the line of sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' This model could also account for the presence of polarized anomalous microwave emission (AME).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='2 Thermal Dust The thermal dust radiation comes from dust grains present in the interstellar medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Those grains absorb ultraviolet light and re-emit as a grey body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' In general, these dust grains are not perfectly spherical and typically have their minor axis aligned with the direction of the local magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' This effect yields polarized thermal dust emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' The SED of this radiation is often described as a modified black body with emissivity index βd and dust temperature Td: � Qd Ud � ν = � AQ d AU d � � ν νd �βd+1 eγνd − 1 eγν − 1 , (4) where Ad is the amplitude of the dust in brightness tem- perature evaluated at the pivot frequency νd = 143 GHz and γ = h kBTd 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' The amplitude is well characterized by the higher frequency channels where the other components are clearly sub-dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' The current temperature map of the dust grains (Td) is obtained from temperature analy- sis and has values mostly between 14 K and 26 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' The polarized dust emissivity evaluated with Planck data is βd = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='55 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='05 (Planck Collaboration 2020d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Several works support the idea that a single component dust model is too simplistic and more components might be required to fully characterize this emission (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=', McBride et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Ritacco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Nonetheless, since this pa- per is focused on the low frequency foregrounds, we keep the model used in Planck Collaboration (2020d) which seems to provide a good description at the Planck polarized frequen- cies (30 GHz < ν < 353 GHz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='3 CMB The CMB radiation has a thermal black body spectrum with a temperature of To = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='7255 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='0006 K (Fixsen 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' CMB photons are linearly polarized due to the Thomson scattering experienced with the hot electron gas at the last scattering surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Unlike in intensity, where the CMB can be the dominant contribution at intermediate frequencies (70-150 GHz) and high Galactic latitudes, in polarization, the foreground contribution cannot be overlooked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Thus, in order to detect the primordial B-mode, experiments with very high sensitivity, exquisite control of systematics and a careful removal of foregrounds are mandatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' The CMB signal at each pixel is given by its amplitude Acmb, which is the only free parameter for this component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Since the rest of the components are given in brightness temperature we convert the thermodynamic temperature of the CMB to the same units: � Qcmb Ucmb � ν = � AQ cmb AU cmb � x2ex (ex − 1)2 , (5) 2 h and kB are Planck and Boltzmann constants respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' MNRAS 000, 000–000 (0000) 4 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' de la Hoz where x = hν kBTo .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='4 Faraday Rotation Another issue intrinsic to the polarization signal is the Fara- day rotation effect, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=', the rotation of the plane of polariza- tion that occurs when light passes through the interstellar medium in the presence of a magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' The magnitude of this effect scales with the square of the wavelength, hence its repercussions are more significant at low-frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' To properly account for this effect we require a broad knowl- edge of the Galactic magnetic field as well as the interstellar medium, in order to recognise the regions where the effect is more significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Moreover, since the instrumental beam has a finite size, the measured signal is an average of the emis- sion from various directions within the beam with slightly different rotation angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' This results in a “beam depolar- ization” of the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Hutschenreuter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' (2022) show that the possible Faraday Rotation effects at the QUIJOTE-MFI frequencies (10-20/,GHz) are very small in most of the sky, and partic- ularly at high Galactic latitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Thus, in this work we have not considered any Faraday Rotation effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Nevertheless, in Appendix B we study variations on the synchrotron spectral index due to rotations of the polarized angle and compare it to Faraday Rotation models such as the one proposed in Hutschenreuter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='5 Other contributions It is well known that there are other foreground components whose emissions are important for intensity analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' In par- ticular, at low frequencies, one needs to consider two ad- ditional Galactic emission components: the bremsstrahlung radiation generated from electron–ion scattering in interstel- lar plasma (free-free), and AME, whose physical origin still is not fully clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' At high frequencies, in addition to ther- mal dust, we find an isotropic extragalactic emission called the cosmic infrared background (CIB), coming from differ- ent sources, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=', dusty star-forming galaxies, quasars, in- tergalactic stars, inter-cluster dust in the Local group, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' We also have other contributions such as CO line emission or Sunyaev-Zeldovich effect (SZ) from clusters of galaxies (Sunyaev & Zeldovich 1972) that should be taken into ac- count in intensity analyses (Planck Collaboration 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' In addition, emission from extragalactic point sources, both at radio and infrared frequencies is an important contaminant at small scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' In polarization the problem is simplified since several of these emissions (free-free, CIB, SZ, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=') are not expected to be polarized (at least significantly), therefore we do not consider them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' The polarization of the anomalous microwave emission is still under study because its nature is still uncertain (Dickinson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Several models have been proposed such as spinning dust particles (Ali-Ha¨ımoud 2013), mag- netic dipole emission (Draine & Lazarian 1999) or more recently the proposal of spinning nano-diamonds (Greaves et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' The predicted polarization fraction of the AME emission for most of these models is below 5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' From the data analysis point of view, no evidence of polarization has been found in compact region studies (the most stringent con- straints on the polarization fraction, Π, have been provided by G´enova-Santos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' (2017), Π < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='22% at 41 GHz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Due to this lack of evidence, we do not take into account the AME component in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' On the other hand, point sources present some degree of polarization, which is in general small (a few percent).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' However, at the resolutions considered in this work, they are subdominant with respect to Galactic foregrounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Thus, we do not include them in our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' We note however that in the data, a few polarized point sources are present that are not taken into account in the component separation analysis, see Herranz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' (2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 3 COMPONENT SEPARATION METHODOLOGY In this work, we apply the parametric component separation method B-SeCRET to extract the polarized astrophysical signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Parametric methods are very powerful since they provide physical information of each sky component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' How- ever, they require a profound theoretical understanding of the nature of the foregrounds and accurate knowledge of the experiment’s characteristics to avoid biases in the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Below, in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='1, we outline the component separa- tion technique applied in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Then, in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='2, we describe the prior information that is used in the Bayesian analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='1 Bayesian analyses The B-SeCRET methodology is a parametric pixel-based maximum-likelihood method, which relies on an Affine- Invariant Markov Chain Monte Carlo Ensemble sampler to draw samples from a posterior distribution (Foreman- Mackey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' This methodology has already been applied in previous studies, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=', de la Hoz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' (2020, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' B-SeCRET applies Bayesian inference to determine the best-fit model parameters given some prior information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' In Bayesian statistics, the probability of the model parameters θp given the signal data dp at the pixel p is proportional to the probability of the dp given θp times the probability of θp, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=', P(θp|dp) ∝ P(dp|θp)P(θp) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' (6) P(θp) is commonly known as the prior information, whereas P(dp|θp) is usually referred to as the likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Assuming Gaussian noise, the likelihood of the data can be expressed as P(dp|θp) = exp � −1 2(dp − Sp)T C−1(dp − Sp) � � (2π)N det(C) , (7) where C is the noise covariance matrix, N is the number of elements in the dp array, and Sp is the parametric model considered, which has been described in detail in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' To draw samples from the posterior probability we use the Python implementation emcee (Foreman-Mackey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 2013) of an affine-invariant ensemble sampler for MCMC (Goodman & Weare 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' In each pixel, the best-fit pa- rameters and their uncertainties are obtained as the median and the standard deviation of their respective marginalized posterior probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' MNRAS 000, 000–000 (0000) QUIJOTE-MFI Diffuse polarized foregrounds 5 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' QUIJOTE observed sky after removing the geosta- tionary satellite band and the region around the north celestial pole, which is affected by high atmospheric air-mass (fsky = 51%, Galactic coordinates centred on (0,0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='2 Priors In this work we benefit from prior information about astro- physical foregrounds to help with convergence and compu- tational time reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' For example, the synchrotron spec- tral index is known to be around −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='1, although experi- ments such as S-PASS found a more negative value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Here we use the estimated value obtained with Planck polariza- tion data by the SMICA method, βs = −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='06 (Planck Collaboration 2020d) and use a broad Gaussian distribu- tion N(−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='3)3 as a prior on βs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' When we include a curvature in the synchrotron model we apply a Gaussian prior N(0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='1) on cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Moreover, we apply Gaussian priors N(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='55, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='1) and N(21, 3) on both βd and Td respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Finally, flat priors are used in the characterization of the amplitude parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 4 DATA The aim of this work is to obtain a better characterization of the low-frequency foregrounds by including the newly re- leased QUIJOTE-MFI maps in component separation anal- yses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' In this Section we summarize the basic details of these maps as well as those from the other experiments used in the analysis, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=', the K and Ka bands from WMAP and Planck’s third and fourth public releases (PR3 and PR4, re- spectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' We also discuss some technical issues related to the instruments such as the estimated noise, RFI, and the colour corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='1 Datasets In this analysis we have used the data from the following experiments: QUIJOTE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' We have used the low frequency QUI- JOTE MFI 11 and 13 GHz channels (MFI) (Rubi˜no-Mart´ın et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 2023) due to their better signal-to-noise ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Al- though QUIJOTE has observed 70% of the sky there are re- gions with poorer sensitivity due to the presence of artificial 3 N(x, σ) represents a normal distribution with mean x and vari- ance σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' satellites and high atmospheric masses in some directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Thus, in this analysis we have considered the mask shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 1, as the observable sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' This mask (satband+NCP) is described in Rubi˜no-Mart´ın et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' (2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' WMAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' We have used the low frequency Nine- Year Wilkinson Microwave Anisotropy Probe (WMAP) K (22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='8 GHz) and Ka (33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='1 GHz) bands (Bennett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 2013)4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Planck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' We have used the full set of Planck polariza- tion maps i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=', the low frequency instrument (LFI) 30, 44 and 70 GHz frequency maps and the high frequency instrument (HFI) 100, 143, 217 and 353 GHz maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' We have obtained results from both PR35 (Planck Collaboration 2020b,c) and PR4 (Planck Collaboration 2020f) data releases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Before component separation analyses, the frequency maps are all convolved (taking appropriately into account the beam window function of each particular frequency map) with a common beam, a Gaussian beam of FWHM = 2◦, and downgraded to the same resolution through spherical harmonics, given by the HEALPix parameter Nside = 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' The procedure followed is described below: (i) We calculate the spherical harmonics coefficients (tℓm, eℓm, bℓm) using the healpy routine map2alm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' (ii) To convolve all channels with the same beam we multiply the (tℓm, eℓm, bℓm) by bℓ(2◦) pℓ(64)/(bi,ℓ pℓ(Nside)), where bℓ(α) is a gaussian beam window function whose FWHM is α, bi,ℓ is the i-th channel beam window function and, pℓ(Nside) is the pixel window function at the resolution Nside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' (iii) We obtain the downgraded maps at Nside = 64 apply- ing the healpy routine alm2map to the new (tℓm, eℓm, bℓm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Several combinations of the previous data sets have been tested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Each configuration’s name is given by the “sum” of the sets of maps included in the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' For example, the configuration composed of PR4 channels in combination with WMAP’s K and Ka bands is referred as K/Ka+PR4, or MFI-QUIJOTE low frequency channels in combination with PR4 and WMAP channels is specified as MFI+K/Ka+PR4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='2 Instrumental Effects Real data present different instrumental effects that need to be accounted for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' For example, an important contribution to the observed signal is the noise produced by the detectors of each experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' A proper characterization of the noise lev- els is key for component separation analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' In this work, we have calculated the covariance matrix among the frequency channels per pixel, required by the parametric component separation method, using realistic noise simulations specific to each instrument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Each experiment’s noise simulations are obtained as follows: QUIJOTE We have used the correlated noise simula- tions described in Rubi˜no-Mart´ın et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' (2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' They account 4 The other bands were not included since they have a much lower synchrotron signal-to-noise ratio and do not contribute to the determination of the synchrotron characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 5 We used the Planck maps corrected from bandpass leakage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' MNRAS 000, 000–000 (0000) 6 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' de la Hoz for the 1/f noise present in the maps, and the correlated noise component between 11 and 13 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' WMAP We have generated a set of white noise simula- tions using the RMS noise per pixel provided by the WMAP collaboration (Hinshaw et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' The RMS noise σ is cal- culated as σ = σ0/√Nobs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='6 Planck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' For PR3 we have used the FFP10 simula- tions generated by the Planck Collaboration (Planck Col- laboration 2020a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' In the case of the PR4, we have em- ployed the noise simulations described in Planck Collabo- ration (2020f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='7 While the frequency channels of different experiments are uncorrelated, there might be correlations between chan- nels of a given instrument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' This is the case for the 11 and 13 GHz low-frequency MFI channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' On the other hand, we have assumed no correlations between frequency chan- nels for WMAP and Planck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Thus, for a given pixel p, the Planck and WMAP frequency covariance matrices are diag- onal while QUIJOTE’s has non-zero off-diagonal terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' For a given configuration, the covariance matrix is obtained as a block matrix, where each block corresponds to the frequency covariance matrix of each instrument included in that con- figuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' To obtain the experiments’ frequency covariance matri- ces, first we pre-process the noise simulations in the same manner as the data maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Then, for Planck and WMAP, the diagonal terms are calculated as the variance of the noise simulations at the corresponding pixel for each fre- quency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Each pixel covariance matrix between QUIJOTE 11 and 13 GHz is calculated as the sample covariance ma- trix using the values of the 11 and 13 GHz noise simulations at that specific pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' One test to verify that our covariance matrices are well estimated is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' We obtained a distribution of χ2 n,i values as: χ2 n,i = nT i C−1 i ni , (8) where ni is a noise simulated map8 at the frequency i and Ci is the noise covariance matrix described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' The χ2 n,i distributions should have the expected form with Npix de- grees of freedom (d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' This is consistent with the values obtained for Planck and WMAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' In the case of QUIJOTE, the distribution deviates slightly from the expected Npix d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' χ2-distribution since they are not end-to-end noise simulations and hence not as accurate (see Rubi˜no-Mart´ın et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 2023, for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' However, as subsequent analyses will show, we find that when the astrophysical emission is included, the obtained χ2 is correct as expected, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=', in the regions where the model properly explains the data (outside the Galactic plane).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Thus, QUIJOTE’s noise simulations are accurate enough to perform scientific analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' We explored the possibility of including correlations 6 σ0 and Nobs are given in https://lambda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='gsfc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='nasa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='gov/ product/wmap/dr5/skymap_info.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='html.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 7 Simulations available at NERSC under /global/cfs/cdirs/cmb/data/planck2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 8 The noise simulations used in this test are different from the noise simulations used to calculate the noise covariance matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' among neighbouring pixels within a 1 degree radius9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' The smoothing process of the maps induces noise correlations among different pixels and, although this does not affect our pixel-by-pixel analyses, it can affect analyses where we assume a uniform parameter value within one region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' There- fore, for each pixel, we calculated the covariance matrix among its neighbouring pixels from noise simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Then we generated a sparse covariance matrix where the only non- zero values in each row were the diagonal element and the correlation with the neighbouring pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' In this case the distribution does not follow a Npix degrees-of-freedom χ2 distribution as one would expect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' The recovered values were smaller than expected, more notably for Planck maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' This is a consequence of not having enough noise simulations, which prevents us from obtaining a good characterization of the noise correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Therefore, we use the covariance matrices that do not take into account possible noise corre- lations among neighbouring pixels in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' As explained in Rubi˜no-Mart´ın et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' (2023), in order to correct residual RFI signals emerging after co-adding all data in the map-making process of the QUIJOTE-MFI data, the polarization maps are corrected using a function of the declination (FDEC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' This correction is equivalent to apply- ing a filter to QUIJOTE data, which removes the zero mode in lines of constant declination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' In Appendix C we studied whether this correction affects the recovery of foregrounds spectral parameters such as βs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' We found that if only QUI- JOTE maps are filtered with FDEC the recovered βs map is biased in regions such as the North Polar Spur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' When all data maps are filtered in the same way this bias disappears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Thus, for this analysis we have filtered all signal maps with their corresponding FDEC function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Another important instrumental effect arises from de- tectors having a finite bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' This issue has to be taken into account when dealing with foreground compo- nents whose amplitude varies within that frequency band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' This effect can be corrected by adding a multiplicative fac- tor, called colour correction, to the signal that depends on the spectral behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' We have used the fastcc Python code (Peel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 2022, G´enova-Santos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 2023) to ob- tain the colour corrections of each experiment considered here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Therefore, our model for the sky signal presented in Section 2 is corrected as follows: Xν = Xν,cmb + Xν,s Cs(α, ν) + Xν,d Cd(βd, Td, ν), (9) where X is either Q or U, Cs(α, ν) is synchrotron colour correction whose spectral behaviour is modelled as a power law with α = βs + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' The spectral behaviour of dust colour correction Cd(βd, Td, ν) is assumed to be a modified black body and it is determined by its βd and Td parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' The colour correction values are updated in each MCMC iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 5 RESULTS In this Section we present the component separation prod- ucts obtained using the recently released QUIJOTE low- 9 The pixels contained within this radius are the ones with the largest correlations induced by the smoothing process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' MNRAS 000, 000–000 (0000) QUIJOTE-MFI Diffuse polarized foregrounds 7 s K/Ka+PR4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='7 s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='4 2 red 0 10 MFI+PR4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='4 0 10 MFI+K/Ka+PR4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='4 0 10 MFI+K/Ka+PR3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='4 0 10 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Synchrotron spectral index (top row) and uncertainty maps (middle row) obtained after component separation with four different datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' The synchrotron emission is modelled with a power law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Bottom row: reduced χ2 map obtained for each dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' MFI data along with the already available Planck and WMAP data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' We have focused primarily on the synchrotron spectral parameters since those are the parameters where a greater improvement is found, see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='1 and Sec- tion 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Moreover, we show the recovered amplitudes of the CMB, synchrotron and thermal dust and, compare them with those obtained by Commander using PR4 data in Sec- tion 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' In Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='4 we present the spectral parameters of the thermal dust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Finally, we evaluate the robustness of these results in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='1 Synchrotron Spectral Index The major improvement obtained from including the low- frequency QUIJOTE-MFI channels is having the sufficient sensitivity to study the synchrotron spectral index with great accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Here we have conducted a deep study on sev- eral aspects with regard to this parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' First, we have compared the recovered βs maps using different combina- tions of the available datasets (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='2 studies the spatial variability of βs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Finally, we compare our results to the available βs models that are often exploited in simulations used in CMB science forecasts in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='1 Datasets We have obtained different βs maps from component sep- aration analyses using the four following datasets: WMAP K and Ka bands with PR4 (K/Ka+PR4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' QUIJOTE-MFI 11 and 13 GHz channels with PR4 (MFI+PR4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' QUIJOTE- MFI 11 and 13 GHz channels, WMAP K and Ka bands and PR4 (MFI+K/Ka+PR4) and QUIJOTE-MFI 11 and 13 GHz channels, WMAP K and Ka bands and PR3 (MFI+K/Ka+PR3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' The results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' It is clear from the comparison of the synchrotron spectral index uncertainty maps obtained in the K/Ka+PR4 case (first col- umn) with respect to the MFI+K/Ka+PR4 case (third col- umn), that the inclusion of QUIJOTE channels significantly improves the estimation of βs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Moreover, we observe that, outside the Galactic plane, the estimation of βs is very close to the mean value of the prior set on this parameter, in this case −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' In other words, the information contained in that fraction of the data, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=', the likelihood, is very poor and the estimation is driven by the prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' This improvement does not come from the inclusion of more channels, but from channels where the synchrotron contribution is larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' This is evident from the comparison of the results from K/Ka+PR4 with respect to MFI+PR4, where the number of frequency channels is the same but the results are significantly better for the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Finally, we have compared also the results obtained with MFI+K/KA+PR3 and MFI+K/Ka+PR4 (fourth and third column respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' In this case the recovered un- certainty maps are virtually the same but there are some distinct differences between the βs maps that should be as- cribed to changes in Planck maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' One of the advantages of using a parametric compo- nent separation method is that we can evaluate the good- ness of the fit with certain estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' In this work we use the reduced χ2 estimator, whose value at a given pixel p is calculated as: χ2 red,p = 1 Ndof � i∈{Q,U} (dp,i − Sp,i)C−1 p,i(dp,i − Sp,i) , (10) where the sum is over all Q and U frequency channels, and Ndof is the number of d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='. The bottom row of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 2 shows the χ2 red maps obtained for each dataset combination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' These maps show that our default model, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=', a power law and a MNRAS 000, 000–000 (0000) 8 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' de la Hoz 0 2 4 6 8 10 2 red (K/Ka+PR4) 0 2 4 6 8 10 2 red (MFI+K/Ka+PR4) 0 2 4 6 8 10 2 red (MFI+PR4) 0 2 4 6 8 10 2 red (MFI+K/Ka+PR3) Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Reduced χ2, χ2 red, obtained using the MFI+K/Ka+PR4 dataset vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' the χ2 red obtained using K/Ka+PR4 (left), MFI+PR4 (center) and MFI+K/Ka+PR3 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' The color scale is related to the density of points, redder (bluer) corresponds to denser (sparser) regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' The orange rectangle shows the χ2 red within a 95% confidence region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' The slope calculated with the points within this 95% confidence region is m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='686 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='004 (left column), m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='732 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='003 (center column) and m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='731 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='003 (right column), shown with a green dashed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' The orange dashed line shows the one-to-one line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' The synchrotron emission is modelled with a power law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' modified black body to model the synchrotron and thermal dust emission respectively, provides a good fit (low values of χ2 red) outside the Galactic plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Within the Galactic plane, this model is not able to capture all the physical complexity and the χ2 red values are quite large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' However, we note that in this analysis we have considered statistical uncertainties but not calibration errors, which in QUIJOTE are of 5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Apart from the higher complexity of the Galactic plane emission, the higher χ2 red in this region could also be due, in part, to having neglected calibration errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' We have also used the χ2 red estimator to select the dataset that is used as the default for further tests be- tween the MFI+K/Ka+PR3 and the MFI+K/Ka+PR4 datasets, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=', the only combinations that include all the channels considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 3 the χ2 red obtained using the MFI+K/Ka+PR4 dataset is plotted against the χ2 red ob- tained with K/Ka+PR4, MFI+PR4 and MFI+K/Ka+PR3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' The 95% confidence regions are delimited by orange lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' These lines indicate the χ2 values, from the reduced χ2- distribution with Ndof d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='10, that satisfy that the normal- ized area covered to their left is equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' We have also fitted the points within this confidence regions to a straight line to determine which dataset has more pixels with smaller χ2 red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' If the slope is larger than unity, the dataset on the hor- izontal axis has more pixels with smaller χ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' If the slope is smaller than unity, the dataset on the vertical axis is the one which satisfies that condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Although it is not clear from the left plot of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 3 which dataset is better, the slope m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='686 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='004 indi- cates that the MFI+K/Ka+PR4 dataset provides a better fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Moreover, the K/Ka+PR4 dataset has larger uncertain- ties which can mask model inconsistencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' On the other hand, from the middle plot of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 3, we observe that the inclusion of the K and Ka WMAP bands to the MFI+PR4 dataset improves the goodness of the fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Finally, compar- ing MFI+K/Ka+PR4 with MFI+K/Ka+PR3, we see that PR3 provides a better fit in the Galactic plane, while PR4 10 The χ2-distribution with Ndof divided by Ndof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' fits better outside the Galactic plane (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Since the fit in the Galactic plane is bad in both cases we have chosen the MFI+K/Ka+PR4 as our default dataset as it retrieves better fits within the 95% confidence regions (pixels outside the Galactic plane, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='2 Spatial Variability We have also studied the spatial variability of the syn- chrotron spectral index in several high signal-to-noise re- gions of the sky, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' These connected regions satisfy the condition that βs is estimated with a signal-to-noise ra- tio larger than 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' In particular, R1 is associated with the North Polar Spur (NPS), and R2 encompasses the Galac- tic plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' R3, R4 and R5 are other sky regions where the polarized synchrotron intensity has a large signal-to-noise ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 6 shows the estimated synchrotron spectral index against the uncertainty on the estimation of all the pixels within a given region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' We have limited this study to those pixels with a χ2 red within the 95% confidence region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' The area delimited by the dotted lines contains the values that are consistent within 3σ with the weighted mean in each region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' The top left panel indicates that βs has a large spatial variability across the whole available QUIJOTE-MFI sky (QS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Therefore, a constant value of βs is not a good model of the synchrotron emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' On the contrary, the R1, R3, R4 and R5 pixels values are well within those lines, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=', a uniform βs value could be a good model for all pixels within each region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Finally, R2 (top right panel) shows a significant spatial variability which is consistent with the large heterogeneity observed in the βs map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' The study of regions with uniform βs values helps with improving the detectability of primordial B-modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Allow- ing spatial variations of the spectral parameters at the pixel level results in a very robust parametrization of the signal sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' However, this robustness comes at the expense of an in- crease in the statistical uncertainty of the parameters as less information is provided in the fit (Errard & Stompor 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' MNRAS 000, 000–000 (0000) QUIJOTE-MFI Diffuse polarized foregrounds 9 s 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='7 s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='4 2 red 0 10 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Synchrotron spectral index (top), its uncertainty (mid- dle) and reduced χ2 (bottom) maps obtained after component separation with the default dataset MFI+K/Ka+PR4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' The syn- chrotron emission is modelled with a power law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Thus, several approaches have been proposed in the litera- ture to define sky regions with uniform spectral parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' For example, in Errard & Stompor (2019), these regions are chosen as super-pixels at a lower HEALPix maps resolution, whereas in Grumitt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' (2020), the regions are obtained using clustering algorithms such as the mean-shift cluster- ing algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Recently, Puglisi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' (2022) has presented a new methodology based on spectral clustering to define R1 R2 R3 R4 R5 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' R1, R3, R4 and R5 are sky regions where βs is assumed uniform in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='2 and R2, which encompasses the Galactic plane seen by QUIJOTE, is a very heterogeneous region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' These regions satisfy that βs is recovered with a signal-to-noise larger than 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Synchrotron spectral index estimation βR s and its uncer- tainty σ(βR s ) obtained assuming uniform value across the regions R1, R3, R4 and R5 shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Region fsky (%) βR s σ(βR s ) R1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='84 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='028 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='002 R3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='96 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='945 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='008 R4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='56 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='319 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='011 R5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='21 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='228 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='019 geometrical affine regions with similar spectral parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' It is worth noting that if the assumption of uniform spectral parameters within those regions does not hold, the mod- elling errors introduced might bias cosmological parameters measurements obtained from the output CMB map after component separation, as well as foreground model param- eters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' We have calculated the value of βs in some of these regions assuming a constant value within each region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' We have performed the fit in the following manner: First we fix βs to a given value and fit the rest of the model parameters in each pixel of the region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Then, the rest of the parameters are fixed to the estima- tion from the previous fit, and we fit βs assuming a unique value in the whole region under study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' βs is fixed to the new obtained value and the process is repeated until it reaches convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' We have chosen the median of the βs values (obtained pixel- wise) within that region as the initial guess of βs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' The results are shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Notice that the uncertainty on the recovered βs has dramatically decreased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' This is simply a result of having N R pix (the number of pixels contained within the region R) times more information to fit the parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' The βs values recovered in each region (R1, R3, R4 and R5) are not consistent among them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' These results further showcase the spatial variability of the synchrotron’s spectral parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' MNRAS 000, 000–000 (0000) 10 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' de la Hoz 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='0 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='2 R1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='5 R2 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='3 s R4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='2 s R5 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Synchrotron spectral index estimate against its uncertainty within different sky regions: QUIJOTE-MFI sky (QS) (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' R1, R2, R3, R4 and R5 are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' The solid, dashed, and dotted lines enclose the values of βs within 1σ, 2σ and 3σ of the weighted mean respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' The study is limited to those pixels whose χ2 red lies within the 95% confidence region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='6 s Model 4 M-D et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' (QS95) MFI+K/Ka+PR4 (QS95) MFI+K/Ka+PR4 (HS2N95) Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Distribution of the synchrotron spectral index from “Model 4” of Miville-Deschˆenes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' (2008) and from our esti- mation using the MFI+K/Ka+PR4 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Vertical dashed lines indicate the mean value for each distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='3 Comparison with current βs models In this Section we compare our βs map with the cur- rently most used βs template11, the “Model 4” Miville- 11 Used for example in the Planck Sky Model (Ashdown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 2012), or in the Python Sky Model (PySM) a Python library to simulate foregrounds (Thorne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Deschenes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' template, which was constructed with Haslam and WMAP observations in temperature (Miville- Deschˆenes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 7 shows the distribution of the spectral index value for this model (blue) and for our anal- ysis (orange), considering only those QUIJOTE-MFI pixels that lie within the 95% confidence region of the χ2 (QS95).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' In the QS95 region, the mean and the standard deviation from the “Model 4” of Miville-Deschˆenes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' (2008) tem- plate are −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='00 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='05 while those from our estimate are −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='08±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' It is interesting to note that the variability ob- served in our analysis is significantly larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' A direct compar- ison of the dispersion of both maps (using the same mask) indicates an increment of the spatial variability in our study around a factor of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='6, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=', σ(βMFI+K/Ka+PR4 s )/σ(βModel 4 s ) ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' One may wonder if this result can be affected by the considered prior, since the estimated spectral indices for low signal-to-noise pixels are significantly constrained by it (see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' In order to test this point, we have repeated the previous analysis considering only those pixels satisfy- ing that the recovered βs values have a signal-to-noise larger than 15 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=', where the synchrotron signal-to-noise is high and thus the results are not driven by the prior) and lie within the 95% confidence region of the χ2 (HS2N95).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' In this case, we find that the mean value and dispersion of the distribution of βs are −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='12±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='15 for our analysis (see green histogram in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 7) versus −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='00 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='05 for “Model 4” in the same region, confirming our finding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Although our esti- mations can be affected by the presence of noise, the results show that the variability of the synchrotron spectral index assumed in current templates is underestimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' A similar MNRAS 000, 000–000 (0000) QUIJOTE-MFI Diffuse polarized foregrounds 11 increment in the variability was also noted by analysing the S-PASS data in the Southern Hemisphere (Krachmalnicoff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Recently Weiland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' (2022) published a composite map of βs using publicly available data covering approxi- mately 44% of the sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' In the region covered in our study, they obtained βs estimates in the Galactic Plane and the North Polar Spur using information from WMAP K and Ka band, and estimates at latitudes larger than 40◦ using K, Ka and DRAO 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='41 GHz map (Wolleben et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' From a vi- sual inspection our results are compatible within the North Polar Spur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' We find that our derived spectral indices are steeper at the Galactic plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Weiland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' (2022) found discrepancies between the βs values obtained in the Fan Re- gion when they performed the analysis using WMAP K and Ka band versus WMAP K band and Planck LFI 30 GHz channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' In the latter case, the recovered βs were signifi- cantly steeper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' We repeated our analysis excluding the PR4 30 GHz channel and did not observe a discrepancy concern- ing the βs recovered from the default analysis in Fan Region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' This results from the fact that the βs recovery is mainly driven by QUIJOTE-MFI data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' At high latitudes we can- not make a reasonable comparison since our βs estimates are driven by the prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' They also show that DRAO data have some unexplained systematics and can be affected by Faraday Rotation depolarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Other studies, such as those presented in Vidal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' (2015);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Fuskeland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' (2014, 2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Martire et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' (2022), also find variability of the spectral index analyzing differ- ent regions of the sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' However it is difficult to compare the same regions in our map, since they compute a global spectral index for large areas, while we work pixel by pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' For example, near the center of the Galactic plane we see a fair amount of structure that cannot be accounted for in the T–T scatter plots analyses carried out in some of the cited papers, that use several pixels to obtain a single βs value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' In that sense, the methodology followed here is more complete given that we perform a full component separation in each pixel, retrieving information at smaller scales for a large fraction of the sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Rubi˜no-Mart´ın et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' (2023) obtain an estimate of the synchrotron spectral index map directly from the compari- son of the QUIJOTE-MFI 11 GHz map with the WMAP K band map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' The results obtained there are fully consistent with the ones from this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='2 Synchrotron Curvature We have also explored a synchrotron model with curva- ture, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=', the model presented in equation (3), using the MFI+K/Ka+PR4 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 8 shows the estimation and uncertainty maps of the curvature parameter as well as the χ2 red map and the cs signal-to-noise map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' We observe from the signal-to-noise map that curvature is detected at more than 3σ in the Galactic plane, in regions where the fit is not good as it can be seen from the χ2 red map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Even though the inclusion of a curvature parameter is not able to explain the complexity of this region, this parameter can account for some effects along the Galactic plane, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=', Faraday rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Outside the Galactic plane the estimated cs values are close to zero and their uncertainties are around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='1, which Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Estimated values of the curvature and its uncertainty obtained assuming the curvature is uniform within the region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Region fsky (%) cR s σcR s ���cR s ��� /σcR s RC1 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='0797 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='0012 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='75 RC2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='93 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='2768 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='0017 161.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='57 Haze 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='041 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='010 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='23 North bubble 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='63 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='083 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='007 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='43 are the expected value and the spread of the prior set on cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Moreover, the recovered βs map in this case is very similar to the one obtained when the synchrotron is model with a power law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' This means that we do not have enough sensitiv- ity to detect a spatially varying curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Hopefully, joint analyses with future releases of the Northern Celestial Hemi- sphere data like the new MFI2 instrument and C-BASS at 5 GHz (Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 2018) might elucidate more details on changes of the power law spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 9 we compare the goodness of fit using a power law versus a power law with curvature as the synchrotron model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' We see that there are more points located below the bisector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Besides, the slope 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='9227±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='0005 calculated at the 95% confidence region, shows that, given the current data, the power law model is slightly preferred over the power law plus curvature model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Furthermore, we have considered modelling the syn- chrotron emission with a power law with uniform curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' We have assumed a constant cs in four regions: RC1, RC2, and the Haze and North bubble (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' The recovered curvature values are shown in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' RC1 encompasses all the pixels whose χ2 red is within 95% confidence region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' RC2 is composed of the RC1 pixels that also satisfy that the syn- chrotron polarized intensity signal-to-noise ratio at 30 GHz is larger than 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' We detect curvature in all regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' The de- tection is more evident in RC1 and RC2, mostly due to the higher sensitivity (lower σC) in these regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' However, it is important to highlight that there is no physical reasoning behind the definition of RC1 and RC2, and the assumption of uniform curvature in all synchrotron high signal-to-noise regions is arbitrary12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' In the Haze and North bubble, we find a curvature value different from zero at more than 3σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' These regions are studied in greater detail in Guidi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' (2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' We have studied how βs changes when we impose the constraint of having a uniform cs value within each region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' The results are displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' For RC2, we observe that βs steepens considerably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' The weighted mean value of βs in RC2 is ⟨βs⟩ = −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='022 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='011 in the pixel-wise anal- ysis and, ⟨βs⟩ = −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='375 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='002 when cs is imposed to be uniform in RC2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' For RC1, this effect is not as considerable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' The weighted mean values are ⟨βs⟩ = −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='079 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='002 and ⟨βs⟩ = −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='1651 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='0014 when cs varies pixel-wise and is uniform respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' The steepening of βs leads to values of the exponent βs +cs log(ν/νs) within [-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='04,-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='10] at 11 GHz which are compatible with the average value of βs when we fit to a power law model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' From these results, we infer that the βs and cs parameters are not independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' More sensi- 12 Any curvature will be more easily detected in high signal-to- noise regions than in low signal-to-noise regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' MNRAS 000, 000–000 (0000) 12 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' de la Hoz cs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='4 cs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='2 2 red 0 10 |cs|/ cs 0 3 Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Top row: Synchrotron curvature estimate (left) and uncertainty (right) maps obtained after component separation using the default dataset (MFI+K/Ka+PR4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' The synchrotron emission is modelled using a power law with spatially varying curvature (pixel-wise).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Bottom row: reduced χ2 map (left) and cs signal-to-noise map (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 0 2 4 6 8 10 2 red, plc 0 2 4 6 8 10 2 red, pl Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Reduced χ2 calculated using a power law as a model of the synchrotron emission (χ2 red,pl) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' χ2 red when the model is a power law with spatially varying curvature (χ2 red,plc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' The color scale is related to the density of points, redder (bluer) cor- responds to denser (sparser) regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' The red rectangle shows the χ2 red within a 95% confidence region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' The slope at the 95% confi- dence region is m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='9227 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='0005, shown with a green dashed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' The orange dashed line shows the one-to-one line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' tive data at the QUIJOTE frequencies and at lower and/or higher frequencies are required to break the degeneracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' In order to test which model provides a better goodness of fit we calculate the reduced χ2 of a given region R as (a) RC1 (coloured) and RC2 (orange) regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Haze North Bubble (b) Haze and North bubble regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Regions where cs has been assumed to be uniform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' MNRAS 000, 000–000 (0000) QUIJOTE-MFI Diffuse polarized foregrounds 13 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='75 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='50 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='25 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='00 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='50 s (uniform cs) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='75 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='50 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='25 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='00 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='50 s (spatially varying cs) RC1 RC2 Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Comparison between the pixel βs values obtained when fitting the synchrotron emission with a spatially vary- ing curvature model (y-axis) versus with a model with uni- form curvature (x-axis) in the regions RC1 and RC2 using the MFI+K/Ka+PR4 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Reduced χ2 obtained using either a power law or a power law with curvature model in different regions, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' We have considered two curvature models: one where cs varies spatially (spatial) and other where cs is assumed constant in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Model Curvature Region χ2 red,R power law – RC1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='892 power law + curvature spatial RC1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='965 power law + curvature uniform RC1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='936 power law – RC2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='010 power law + curvature spatial RC2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='088 power law + curvature uniform RC2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='081 power law – Haze 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='845 power law + curvature spatial Haze 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='936 power law + curvature uniform Haze 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='885 power law – North bubble 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='961 power law + curvature spatial North bubble 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='041 power law + curvature uniform North bubble 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='986 follows: χ2 red,R = 1 Ndof NR pix � p=1 � i∈{Q,U} (dp,i − Sp,i)C−1 p,i(dp,i − Sp,i) , (11) where we sum over all pixels N R pix within R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' The d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' are given as Ndof = N R pix(2N − Nθ) when all model parameters are allowed to vary pixel-wise, and Ndof = N R pix(2N −(Nθ − 1)) − 1 when cs is assumed uniform in the analysis, where Nθ is the number of model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' We calculated the value of this estimator in three cases: i) when the model parameters are allowed to vary spatially using a power law model for the synchrotron component, ii) when the model parameters vary from pixel-to-pixel using a power law with curvature model, iii) when we fit the data assuming uniform curvature using a power law with curvature model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' The re- sults are given in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' The χ2 red results show that the models we used, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=', power law and power law with curvature, are compatible with the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' However there is not enough statistical sig- nificance to discern which model suits better the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Es- pecially, considering that we have not been able to take into account possible correlations between pixels and that the power law with curvature model is degenerate13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='3 Recovered Amplitudes and Comparison with Planck results We have compared our baseline results, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=', using the MFI+K/Ka+PR4 dataset and a power law as the syn- chrotron model, to those obtained from the Commander pipeline (Eriksen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 2008) applied to PR4 data14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' We have only considered this pipeline among those used by Planck, since it is the reference method with regard to the recovery of foreground components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' In Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 12 to 14, we show a comparison of the CMB, the synchrotron emission at 30 GHz, and the thermal dust emission at 353 GHz be- tween Commander and our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' In order to perform a direct comparison we have filtered Commander results with FDEC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' The left column shows the Q and U Commander amplitudes, the center column our amplitudes and the right column the corresponding uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' A visual inspection shows that both estimates are very similar, especially the synchrotron and thermal dust emissions which are the dom- inant contributions in polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='1 CMB Regarding CMB, the left column of Fig 15 shows the pixel- to-pixel comparison for the recovered CMB map from our analysis and from Commander both in Q and U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' We have applied a combination of the QUIJOTE observed sky and the common polarization confidence mask provided by the Planck Collaboration15 (Planck Collaboration 2020d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' We observe from the maps that there is a discrepancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' We found that the application of the FDEC filter, before the component separation process, leads to a decrease of the amplitude in the power spectra of our recovered CMB map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' This power reduction appears only when Planck and WMAP are filtered with FDEC, since the CMB information is extracted mainly from those channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Instead of applying the FDEC filter, one could apply a filter that suppresses the large scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' This would be equivalent to applying a linear function to the CMB and there would not be a reduction of power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' However, since we want to study all scales we decided to apply the FDEC filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Since the aim of this work is the study of the foregrounds, we keep the results obtained with all the data filtered with FDEC to recover the βs map without any bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' One can in principle recover the unbiased CMB following one of the approaches described below: 13 We considered applying other statistics such as the Bayesian evidence to do model selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' However, since the QUIJOTE- MFI noise simulations are not end-to-end and the Bayesian evi- dence is very computationally expensive we did not perform any model selection analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' This is left for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 14 Data available at NERSC under /cmb/daa/planck20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 15 Available at https://pla.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='esac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='esa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='int/#maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' MNRAS 000, 000–000 (0000) 14 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' de la Hoz (CMBQ) (MFI+K/Ka+PR4) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='75 (CMBU) (MFI+K/Ka+PR4) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='75 CMBQ (MFI+K/Ka+PR4) 2 2 CMBU (MFI+K/Ka+PR4) 2 2 CMBQ (Commander) 2 2 CMBU (Commander) 2 2 Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Left column: Commander Q (top) and U (bottom) CMB maps at Nside = 64, smoothed with a Gaussian beam to a final resolution of FWHM = 2◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Centre column: CMB Q and U maps using the MFI+K/Ka+PR4 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Right column: uncertainty of the CMB maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Maps are in thermodynamic temperature (µK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' We apply the common polarization confidence mask provided by Planck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' (aQ s ) (MFI+K/Ka+PR4) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='01 2 (aU s ) (MFI+K/Ka+PR4) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='01 2 aQ s (MFI+K/Ka+PR4) 80 80 aU s (MFI+K/Ka+PR4) 80 80 aQ s (Commander) 80 80 aU s (Commander) 80 80 Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Left column: Commander Q (top) and U (bottom) synchrotron amplitude maps at 30 GHz at Nside = 64, smoothed with a Gaussian beam to a final resolution of FWHM = 2◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Centre column: our estimate of the synchrotron amplitude at 30 GHz, using the MFI+K/Ka+PR4 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Right column: uncertainty of the estimated synchrotron amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Maps are in antenna temperature (µK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' MNRAS 000, 000–000 (0000) QUIJOTE-MFI Diffuse polarized foregrounds 15 (aQ d ) (MFI+K/Ka+PR4) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='01 2 (aU d) (MFI+K/Ka+PR4) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='01 2 aQ d (MFI+K/Ka+PR4) 80 80 aU d (MFI+K/Ka+PR4) 80 80 aQ d (Commander) 80 80 aU d (Commander) 80 80 Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Left column: Commander Q (top) and U (bottom) thermal dust amplitude maps at 353 GHz at Nside = 64, smoothed with a Gaussian beam to a final resolution of FWHM = 2◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Centre column: our estimate of the thermal dust amplitude at 353 GHz, using the MFI+K/Ka+PR4 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Right column: uncertainty of the estimated thermal dust ampltitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Maps are in antenna temperature (µK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 2 1 0 1 2 2 1 0 1 2 MFI+K/Ka+PR4 [ KCMB] Q 200 0 200 400 200 0 200 400 MFI+K/Ka+PR4 [ KRJ] Q 0 100 50 0 50 100 150 MFI+K/Ka+PR4 [ KRJ] Q 2 1 0 1 2 Commander [ KCMB] 2 1 0 1 2 MFI+K/Ka+PR4 [ KCMB] U CMB 150 100 50 0 50 100 Commander [ KRJ] 150 100 50 0 50 100 MFI+K/Ka+PR4 [ KRJ] U Synchrotron 0 100 Commander [ KRJ] 50 0 50 100 150 MFI+K/Ka+PR4 [ KRJ] U Thermal dust Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Comparison of CMB (left), synchrotron at 30 GHz (centre) and thermal dust at 353 GHz (right) amplitudes recovered using the MFI+K/Ka+PR4 dataset and the ones obtained by Commander using PR4 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' The correlation factors are ρQ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='543 and ρU = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='817 (CMB), ρQ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='992 and ρU = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='973 (synchrotron) and ρQ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='000 and ρU = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='997 (thermal dust).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' MNRAS 000, 000–000 (0000) 16 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' de la Hoz aQ s 5 5 aU s 5 5 Figure 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Difference between the synchrotron amplitude aQ s (aU s ) obtained with the MFI+K/Ka+PR4 and the Commander estimate, top row (bottom row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Maps are in antenna temperature (µK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Perform the component separation analysis without fil- tering the data with FDEC and including the FDEC correc- tion in QUIJOTE-MFI data as part of the model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' or Given the unbiased βs map16 and Planck data, one can construct a template with the modes that QUIJOTE-MFI data is missing after being filtered with FDEC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Then per- form the analysis with the reconstructed QUIJOTE-MFI maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Since the estimation of the CMB is out of the scope of this paper, we leave this analysis for future works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='2 Synchrotron Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 16 shows the difference between the synchrotron am- plitude maps obtained using the MFI+K/Ka+PR4 and the Commander reconstruction using the PR4 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' The largest differences observed are located in the Galactic plane where the model fails to reproduce the sky signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' We also observe large scale structures in the difference map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' These structures can originate from the fact that we have obtained a more accurate estimation of the scaling law as our fit is per- formed using additional frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' However, overall, the correlation between both methods is very good.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 16 Obtained in the component separation analysis using the data filtered with FDEC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' This can also be seen in the centre column of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 15, where a pixel-to-pixel comparison is given, showing that both methods present a synchrotron amplitude at 30 GHz highly correlated for Q and U except in some pixels where the synchrotron emission is very large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Those pixels are lo- cated primarily in the Galactic plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' These discrepancies are likely to arise from differences in the amplitude of the polarised intensity instead of from differences in the polar- ization angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 15 we observe that both the slopes, in the Q and U plots, are higher than unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' If the discrep- ancies were originated from differences in the polarization angle, one slope would be higher than unity and the other lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='3 Dust Regarding thermal dust emission, this foreground strongly dominates the 353 GHz Planck frequency map and, there- fore, the recovered amplitude is very much determined by this channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' This was also the case in the Commander analysis done by the Planck Collaboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Thus, our recov- ered Q and U components of the thermal dust are strongly correlated with those obtained using Commander, see the right column of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='4 Dust Spectral Parameters Although the frequencies of QUIJOTE-MFI do not overlap with the spectral range where the thermal dust is more dom- inant, we have studied whether the inclusion of this data set in the analysis can help with the thermal dust characteriza- tion due to an improvement on the determination of the rest of the polarized foreground parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 17 shows the thermal dust spectral index βd recovered with the default data set, modelling the synchrotron emission as a power law, in two cases: Td is included as a model parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Td is fixed to Commander’s estimation of the thermal dust temperature from the component separation analysis in intensity (Planck Collaboration 2016) like Commander did in their polarization analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Fixing Td helps breaking its degeneracy with βd in the Rayleigh-Jeans part of the ther- mal dust spectrum, which is the one observed with Planck in polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' In both maps we find that the recovered βd values are close to the expected value of the prior, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=', 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='55, except close to the Galactic plane where the thermal dust signal is larger17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' The results differ significantly along the Galactic plane, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' This difference originates since our recovered Td map does not resemble the used Td template as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' We remark that although in the first case Td is estimated from the polarization analysis, the Td recovered values lie close to the expected value of the prior (22 K) except along the Galactic plane where the fit is not good.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Moreover, it is very difficult to fit Td from polarization data only, as the 17 Notice that the uncertainty does not improve in the regions where the βd values are close to the mean value of the prior when we fix one parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' The uncertainty in those pixels is the spread of the prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' MNRAS 000, 000–000 (0000) QUIJOTE-MFI Diffuse polarized foregrounds 17 d 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='7 d|Td 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='7 d 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='15 d|Td 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='15 Figure 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Left column: estimate (top) and uncertainty (bottom) of thermal dust spectral index obtained when Td is included as a model parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Right column: estimate (top) and uncertainty (bottom) of thermal dust spectral index obtained when the Td template obtained by Commander in the intensity analysis is used to fix Td in the component separation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' s 3 3 d 3 3 Figure 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' βd (top row) and βs (bottom row) relative difference map between the maps obtained when we include Td as a model parameter and when we fix it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' highest frequency is 353 GHz, and thus we are not able to trace the thermal dust peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 18 we show the relative difference between spec- tral index map of the thermal dust and synchrotron obtained when Td is included as a model parameter and when it is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' The relative difference is calculated as follows: � ∆β1,2 = β1 − β2 � σ2 β1 + σ2 β2 − 2σβ1,β2 , (12) where σ2 β1 (σ2 β2) is the variance of the β1 (β2) map, and σβ1,β2 is the covariance between the β1 and β2 maps that are being compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' As expected from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 17 the differences close to the Galactic plane are significantly large in the case of βd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' On the other hand, we find that, the βs maps recovered in both cases are compatible and the differences resemble Gaussian noise except along the Galactic plane where the model fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' We also studied the relative difference between the βd map obtained with the MFI+K/Ka+PR4 and K/Ka+PR4 data sets in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' The top panel shows the relative dif- ference when Td is included as a model parameter and the bottom panel when Td is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' We observed that both maps are compatible except in regions where the fit is not good.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Moreover, when we compare the uncertainty maps we find that there is not a significant improvement when we in- clude QUIJOTE-MFI channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Thus, we conclude that the improvement in the characterization of low-frequency fore- grounds does not help necessarily with the estimation of thermal dust spectral parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' MNRAS 000, 000–000 (0000) 18 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' de la Hoz Td 15 25 Td 7 7 Figure 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Top row: thermal dust temperature map recovered in the default case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Bottom row: difference map between the top row map and the Td template used in the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Maps are in Kelvin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='5 Goodness of fit In this Section we study in depth the quality of the results obtained using the default dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' In Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='1 we an- alyze the χ2 distribution of the results as well as the Q and U residuals of each channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='2 investigates the robustness of our results regarding the estimation of the synchrotron spectral index with respect to the prior applied to this parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='1 χ2 distribution and residuals We have studied the pixel χ2 distribution obtained from the fit using MFI+K/Ka+PR4 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 21): χ2 p = Ndof · χ2 red .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' (13) Moreover, we have also calculated the residuals per channel involved in the analysis: rp,ν = (dp,ν − Spν) σp,ν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' (14) In the perfect scenario, residuals maps are consistent with instrumental noise alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Therefore, they are a valuable tool to look for either systematic effects or mismatches in the foreground modelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' First of all, we recall that the number of d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' for this analysis is 13 (11 channels × 2 (Q and U) minus 9 free pa- rameters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' We find ⟨χ2 p⟩ = 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='3 and σ = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='9 slightly larger than what is expected for the theoretical number of d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='. Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 21 shows that the χ2 p values follow a χ2-like distribu- tion, whose peak lies close to Ndof = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' However, there d 3 3 d 3 3 Figure 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' βd relative difference map between the map ob- tained using the MFI+K/Ka+PR4 and the one obtaiened with K/Ka+PR4 datasets when we include Td as a model parameter (top row) and when we fix it (bottom row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 0 10 20 30 40 50 60 70 2 p Figure 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' χ2 p distribution obtained using the default dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' The orange curve shows the theoretical χ2 probability density function with Ndof = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' The area to the left of the gray dashed line shows values within the 95% confidence region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' is an excess of pixels at large values of χ2 with respect to the χ2 Ndof -distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' That excess appears since there are pixels where the model is not able to track the true sky emission, mainly in the Galactic plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Thus, those pixels are highly inconsistent with this χ2 distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 22 shows the Q and U residuals maps of every frequency channel from the MFI+K/Ka+PR4 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' We MNRAS 000, 000–000 (0000) QUIJOTE-MFI Diffuse polarized foregrounds 19 MFI 11 GHz (Q) MFI 11 GHz (U) MFI 13 GHz (Q) 200 200 MFI 13 GHz (U) 200 200 WMAP K (Q) WMAP K (U) WMAP Ka (Q) 20 20 WMAP Ka (U) 20 20 PR4 30 GHz (Q) PR4 30 GHz (U) PR4 44 GHz (Q) PR4 44 GHz (U) PR4 70 GHz (Q) 5 5 PR4 70 GHz (U) 5 5 PR4 100 GHz (Q) PR4 100 GHz (U) PR4 143 GHz (Q) PR4 143 GHz (U) PR4 217 GHz (Q) PR4 217 GHz (U) PR4 353 GHz (Q) 5 5 PR4 353 GHz (U) 5 5 Figure 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Q and U residual maps for each frequency channel at Nside = 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Maps are displayed in thermodynamic temperature (µK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' find that Planck and WMAP residuals maps are reasonably consistent with the expected noise except along the Galactic plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' The residuals in this region are a consequence of an incorrect modelling of the sky as we saw in the χ2 red maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' For the MFI channels we observe that the largest residu- als are located in compact regions along the Galactic plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' We observe in the 11 GHz U channel a redder region in the NPS’s closest part to the Galactic centre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' This region over- laps with the area where we obtain a better goodness-of-fit if Faraday Rotation effects are taken into account, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' B2 in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Furthermore, artefacts that resemble the FDEC morphology are present in MFI 13 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' In light of these tests, we are confident of the results ob- tained in those pixels, that are properly modelled by our as- sumed parametric model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' The pixels outside the confidence region are located mainly in the Galactic plane, probably be- cause our model fails to account for the complexity of this region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' It would be convenient to study these regions in more detail with more complex models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' However, the aim of this work is to study the diffuse components and the study of specific regions has been conducted in other works (Watson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Ruiz-Granados et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Guidi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='2 Robustness with respect to the prior As previously stated, the use of prior information is essential in Bayesian analysis, helping with convergence and compu- tational time reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Besides, when the data do not have enough sensitivity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=', there is not enough information to obtain a reliable estimation of the spectral index, the prior tends to provide a value close to the mean value of the dis- tribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' In other words, a conservative value is assigned to the spectral index in those pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Thus, in order to detect which pixels are prior-dependent we have also performed component separation using two additional Gaussian priors on βs, N(−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='6) and N(−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' The βs estimation and uncertainty maps with these new priors are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 23 together with those obtained with the prior used in the default analysis (left columns).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Comparing the results using the default prior, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=', N(−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='3), versus a less restrictive prior, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=', N(−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='6), we observe that the uncertainty on the re- covered βs increases at the prior-dominated pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' On the other hand, in those regions where the synchrotron emission is very intense the uncertainty remains the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Likewise, the estimated βs in the latter pixels are very similar whereas the other pixels are visually different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' The βs distribution of the pixels outside the low-uncertainty regions are compati- ble with the prior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' This is the reason why the estimated values are different and the spread is larger when the prior is relaxed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' When we use a prior with a different expected value, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=', N(−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='3), but equal standard deviation we obtain a similar uncertainty map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' The estimated βs is almost the same in the low-uncertainty regions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=', the high-intensity synchrotron regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' However, a flatter spectrum (closer to −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='0 instead of −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='1) is recovered outside those areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' This is more evident from the bottom panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 24 where the difference between the βs map estimated with the de- fault prior and the N(−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='3) prior is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Outside the regions where the synchrotron emission is the largest, the difference is close to −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='1 which is the difference between the expected value of the priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' In other words, when there is not enough information from the data the recovered βs is close to the expected value of the prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' This is an advan- tage of using prior information, since it assigns a conserva- tive value to the spectral index instead of unphysical values or simply failing to perform the fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 6 CONCLUSIONS In this work, we have presented the component separa- tion products in polarization obtained from combining the MNRAS 000, 000–000 (0000) 20 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' de la Hoz s ( 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='3) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='7 s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='4 ( 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='6) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='4 ( 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='3) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='4 Figure 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Synchrotron spectral index estimate (top row) and uncertainty (bottom row) obtained using different Gaussian prior distributions and the default dataset (MFI+K/Ka+PR4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' The synchrotron emission is modelled as a power law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' QUIJOTE-MFI data at 11 and 13 GHz, with the WMAP K and Ka bands and all Planck polarized channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' We have seen that the inclusion of the QUIJOTE-MFI data is crucial to improve the parameter estimation of the low fre- quency foregrounds, in particular for the estimation of the synchrotron spectral index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' We have obtained the first detailed βs map of the North- ern Celestial Hemisphere at a scale of 2◦ assuming the syn- chrotron emission is modelled as a power law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' This model represents well the data except in the Galactic plane where the physics might be more complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' We find, using the pixels whose χ2 red lies within the 95% confidence region, an average value of −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='08 and a dispersion of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' The latter is broader than the dispersion of commonly used βs templates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' More- over, we have found that the spectral index is not compati- ble with a uniform value, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=', there are statistical significant differences of βs across the observable sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' We have also modelled the synchrotron emission as a power law with curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' The pixel-based analysis of the curvature shows that cs is only detected in some regions in the Galactic plane where the fit is bad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' When we assume a model with uniform curvature in RC1 (the region that includes all pixels whose χ2 red is within the 95% confidence region for the power law with curvature model) we found a cs = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='0797 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='0012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' We found that both models, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=', power law and power law with uniform curvature, provide a good fit given the available data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' However there is not enough statistical significance to distinguish which model is better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' A more thorough study is left for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' We found that our recovered synchrotron and ther- mal dust maps are highly correlated with the maps pre- sented by the Planck collaboration using Commander, even though we found some large scale difference between the syn- chrotron emission maps which arise from better estimation of the SED due to the addition of more frequency channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' On the other hand, we recovered a CMB with less power when we use the filtered K, Ka and PR4 with FDEC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Since our analysis focuses on the characterization of foregrounds we keep the results obtained with the filtered maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' How- ever, as commented in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='1 an unbiased CMB map can be recovered following other approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' We have also performed different analyses to test the validity of our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' First, we found that our results are compatible with a χ2 distribution in those pixels where the power law model fits well the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Furthermore, we have calculated the normalized residuals of the pixels with an acceptable goodness of fit of all frequency channels and they are all consistent within the 3σ level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Finally, we have evaluated the robustness of the estimated βs varying the prior imposed in this parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' We found that the estima- tions in the high signal-to-noise synchrotron areas are prior- independent, while outside these regions the prior governs the βs estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' ACKNOWLEDGMENTS We thank the staff of the Teide Observatory for invalu- able assistance in the commissioning and operation of QUIJOTE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' The QUIJOTE experiment is being developed by the Instituto de Astrofisica de Canarias (IAC), the Instituto de Fisica de Cantabria (IFCA), and the Univer- sities of Cantabria, Manchester and Cambridge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Partial financial support was provided by the Spanish Ministry of Science and Innovation under the projects AYA2007-68058- C03-01,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' EQC2018-004918-P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' the Severo Ochoa Programs SEV-2015-0548 and CEX2019-000920-S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' the Maria de Maeztu Program MDM-2017-0765,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' and by the Consolider- MNRAS 000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 000–000 (0000) QUIJOTE-MFI Diffuse polarized foregrounds 21 s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='3 s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='3 Figure 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Difference map between the estimated βs using the default prior, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=', N(−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='3), and the one obtained us- ing an alternative prior, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Top: N(−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='6) Bottom: N(−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Ingenio project CSD2010-00064 (EPI: Exploring the Physics of Inflation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' We acknowledge support from the ACIISI, Consejeria de Economia, Conocimiento y Empleo del Gobierno de Canarias and the European Regional Development Fund (ERDF) under grant with reference ProID2020010108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' This project has received funding from the European Union’s Horizon 2020 research and inno- vation program under grant agreement number 687312 (RADIOFOREGROUNDS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' EdlH acknowledges financial support from the Concepci´on Arenal Programme of the Universidad de Cantabria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' DT acknowledges the support from the Chinese Academy of Sciences (CAS) President’s International Fellowship Initiative (PIFI) with Grant N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 2020PM0042.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' FP acknowledges support from the Span- ish State Research Agency (AEI) under grant number PID2019-105552RB-C43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' The authors acknowledge the computer resources, technical expertise and assistance provided by the Spanish Supercomputing Network (RES) node at Universidad de Cantabria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Some of the presented results are based on observations obtained with Planck (http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='esa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='int/Planck), an ESA science mission with instruments and contributions directly funded by ESA Member States, NASA, and Canada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' We acknowledge the use of the Legacy Archive for Microwave Background Data Analysis (LAMBDA) and the Planck Legacy Archive (PLA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Support for LAMBDA is provided by the NASA Office of Space Science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Some of the results in this paper have been derived using the HEALPix package (G´orski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 2005), and the healpy (Zonca et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 2019), numpy (Harris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 2020), emcee (Foreman-Mackey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 2013), and matplotlib (Hunter 2007) Python packages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' DATA AVAILABILITY The parameter maps obtained from the component sep- aration analysis in the default case, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=', with the MFI+K/Ka+PR4 dataset using a power law to model the synchrotron emission, are included in the released data prod- ucts associated to the QUIJOTE-MFI wide survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' These data products as well as the maps can be freely downloaded from the QUIJOTE web page18, as well as from the RADIOFOREGROUNDS platform19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' They include also an Explanatory Supplement describing the data formats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Any other derived data products described in this paper are available upon request to the QUIJOTE collaboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' REFERENCES Abazajian K.' metadata={'source': 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+page_content=', Lenz D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=', Reinecke M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=', Rosset C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=', Hivon E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=', Gorski K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=', 2019, Journal of Open Source Software, 4, 1298 APPENDIX A: INDEPENDENT Q AND U SYNCHROTRON SPECTRAL INDEX In order to test the assumption of having the same βs in both Q and U, we fit Q and U signals independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' A1 shows the spectral index, the uncertainty of the spectral index as well as the reduced χ2 maps obtained from the three independent fits using the MFI+K/Ka+PR4 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' We infer from the χ2 red maps that the fit outside the Galactic plane is better when Q and U are fitted together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' When we fit just U we observe that the goodness of fit improves significantly in the Galactic plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' However, this effect is due to the low signal-to-noise ratio in that area, not due to a better modelling of the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' The βQ s and βU s maps are distinctly different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' The βQU s map resembles more the βQ s map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' This is expected, since Q has more signal than U in Galactic coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' That is also the reason why the uncertainty on the recovered βs is smaller when we fit just Q compared to U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' However, in those regions where σβU s is smaller than σβQ s , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=', regions where U has more signal than Q, the βQU s values obtained are closer to those of βU s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' This is clearly seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' A2 where the relative difference between βQU s with respect to βQ s (top row) and βU s (bottom row) is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' The largest differences shown in the top (bottom) panel are located in regions where the signal- to-noise is larger in U (Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' On the other hand the relative difference decreases significantly in the regions where the uncertainty on βQ s (top) or βU s (bottom) is smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' APPENDIX B: FARADAY ROTATION We have also studied the significance of the difference be- tween the βQ s and βU s maps, see top row of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' The discrepancies larger than 3σ are concentrated in the Galac- tic plane, close to the Galactic centre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' This could be a tracer of Faraday rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' If Faraday rotation is non-negligible at QUIJOTE frequencies there will be a difference between the polarization angles at QUIJOTE frequencies and those at WMAP/Planck frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' This yields a βQ s map different from βU s due to the bias introduced by the change in angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' That bias is reasonably cancelled out when combining both Q and U to obtain a single index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' We have studied the possibility of correcting the Fara- day rotation effect in the QUIJOTE MFI maps using the model from Hutschenreuter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' The rotation of the polarization plane experienced due to the Faraday Ro- tation effect can be described by: ∆φ = RMλ2 , (B1) where λ is the wavelength, and RM is the rotation measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' We use the RM map estimated by Hutschenreuter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' (2022) to calculate the rotation angle maps at 11 and 13 GHz MNRAS 000, 000–000 (0000) QUIJOTE-MFI Diffuse polarized foregrounds 23 s QU 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='7 s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='4 2 red 0 10 Q 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='4 0 10 U 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='4 0 10 Figure A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Synchrotron spectral index estimate (top row) and uncertainty maps (second row) obtained after component separation using the MFI+K/Ka+PR4 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' The left column shows the βs recovered when we assume that Q and U share the same spectral index, while the centre and right columns depict the Q and U βs when they are assumed to be independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Bottom row: reduced χ2 map for each case study considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' The synchrotron emission is modelled as a power law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' QU, Q 3 3 QU, U 3 3 Figure A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Relative difference map between the βs map ob- tained when we assume the same βs in both Q and U, and βs recovered from the fit with just Q (top) and just U (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' The synchrotron emission is modelled as a power law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Q, U 3 3 Q(FR), U(FR) 3 3 Figure B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Relative difference map between the βs map from the independent Q and U fit using the MFI+K/Ka+PR4 dataset (top), and using the MFI(FR)+K/Ka+PR4 dataset (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' The synchrotron emission is modelled as a power law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' MNRAS 000, 000–000 (0000) 24 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' de la Hoz 2 red 2 red, FR 3 3 Figure B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Difference map between the χ2 red obtained with the MFI+K/Ka+PR4 dataset with respect to the χ2 red,FR obtained with MFI(FR)+K/Ka+PR4 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' In both fits we have as- sumed that Q and U share the same spectral indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' The syn- chrotron emission is modelled as a power law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' QUIJOTE frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' QUIJOTE Q and U maps at a given frequency ν are de-rotated as follows: � QFR UFR � ν = � cos(2∆φν) − sin(2∆φν) sin(2∆φν) cos(2∆φν) � ν � Q U � ν ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' (B2) The variance of the de-rotated QFR and UFR is: σ2 QFR = cos2(2∆φν)σ2 Q + sin2(2∆φν)σ2 U (B3) + 4 � sin(2∆φν)Q + cos(2∆φν)U �2 σ2 ∆φ σ2 UFR = sin2(2∆φν)σ2 Q + cos2(2∆φν)σ2 U (B4) + 4 � cos(2∆φν)Q − sin(2∆φν)U �2 σ2 ∆φ Therefore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' we have repeated the same analysis but us- ing the MFI(FR)+K/Ka+PR4 dataset,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' where MFI(FR) in- dicates that the QUIJOTE 11 and 13 GHz maps have been de-rotated using the angle obtained from the Hutschenreuter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' (2022) model, to correct any possible mismatch due to the Faraday Rotation effect, see bottom row of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' We compare these maps (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' B1) with the difference map between the reduced χ2 map (χ2 red) obtained with the MFI+K/Ka+PR4 dataset with respect to the reduced χ2 (χ2 red,FR) obtained with MFI(FR)+K/Ka+PR4 dataset shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' The sky regions where the absolute value of the relative difference � ∆βQ(FR),U(FR) is smaller than � ∆βQ,U are correlated to those regions where the χ2 red,FR is smaller than χ2 red (reddish regions) and vice versa (bluish regions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' This result suggests that Faraday rotation might be playing a role in some of the significant differences areas observed between βQ s and βU s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' APPENDIX C: FUNCTION-OF-DECLINATION CORRECTION SIMULATIONS We studied using simulations if the application of a function- of-declination (FDEC) filter to QUIJOTE-MFI maps biases the βs map obtained from component separation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' We gen- erated sky simulation maps with the following components at the QUIJOTE-MFI 11 and 13 GHz, K and Ka, and PR4 frequencies: all s 3 3 MFI s 3 3 Figure C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Relative difference map between the βs template used in the simulation and the βs map from the fit using the simulated data with an FDEC filter applied to all maps (top), and an FDEC filter applied only to QUIJOTE-MFI frequencies (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' CMB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Generated as Gaussian random samples using the power spectra obtained from CAMB (Lewis & Challi- nor 2011) with the latest Planck cosmological parameters (Planck Collaboration 2020e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Synchrotron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Generated using the s1 model of the Python Sky Model (PySM) (Thorne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Thermal dust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Generated using the d1 model of the PySM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Realistic noise simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' For each experiment we use the ones described in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' All components are either generated or downgraded to Nside = 512.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Then the components maps are added and we apply the corresponding FDEC filter to each signal map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Fi- nally all maps are downgraded to Nside = 64 and smoothed with a Gaussian beam of FWHM = 2 deg following the pro- cedure described in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' We perform the component separation analysis in two scenarios: i) when only the QUIJOTE-MFI frequency signal maps are filtered, and ii) when all maps are filtered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' C1 shows the relative difference (equation 12) between the βs map recovered from the component separation analysis and the βs template (equation 12 taking into account that the uncertainty of the template map is set to zero, σβs = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' We find that when only QUIJOTE-MFI channels are filtered (bottom panel) the relative differences are larger in regions MNRAS 000, 000–000 (0000) QUIJOTE-MFI Diffuse polarized foregrounds 25 such as the North Polar Spur or the R3 region than when all maps are filtered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Moreover, in those regions the βs relative differences are larger than 3σ with respect to the template In the case when all maps are filtered (top panel), these biases are reduced significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 1Instituto de F´ısica de Cantabria (IFCA), CSIC-Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' de Cantabria, Avda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' los Castros s/n, E-39005 Santander, Spain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 2Dpto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' de F´ısica Moderna, Universidad de Cantabria, Avda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' de los Castros s/n, E-39005 Santander, Spain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 3Instituto de Astrof´ısica de Canarias, E-38205 La Laguna, Tenerife, Spain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 4Departamento de Astrof´ısica, Universidad de La Laguna, E-38206 La Laguna, Tenerife, Spain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 5Institut d’Astrophysique de Paris, UMR 7095, CNRS & Sorbonne Universit´e, 98 bis boulevard Arago, 75014 Paris, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 6Astrophysics Group, Cavendish Laboratory, University of Cambridge, J J Thomson Avenue, Cambridge CB3 0HE, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 7Kavli Institute for Cosmology, University of Cambridge, Madingley Road, Cambridge CB3 0HA, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 8Dpto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' de Ingenieria de COMunicaciones (DICOM), Edif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Ingenieria de Telecomunicacion, Pl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' de la Ciencia s/n, E- 39005 Santander, Spain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 9Dpto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' de Matem´aticas, estad´ıstica y computaci´on, Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' de Cantabria, Avda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' de los Castros s/n, E-39005 Santander, Spain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 10Aurora Technology for the European Space Agency (ESA), European Space Astronomy Centre (ESAC), Camino Bajo del Castillo s/n, 28692 Villanueva de la Ca˜nada, Madrid, Spain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 11Universidad Europea de Madrid, 28670, Madrid, Spain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 12Jodrell Bank Centre for Astrophysics, Alan Turing Build- ing, Department of Physics and Astronomy, School of Nat- ural Sciences, University of Manchester, Oxford Road, Manchester M13 9PL, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' 13Consejo Superior de Investigaciones Cientificas, E-28006 Madrid, Spain 14Dpto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' de F´ısica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE4T4oBgHgl3EQfgg1T/content/2301.05117v1.pdf'} +page_content=' Facultad de Ciencias.' metadata={'source': 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Dennis3, and Juliana B.R. Loureiro1,3 +1Programa de Engenharia Mecˆanica, Coordena¸c˜ao dos Programas de P´os-Gradua¸c˜ao em Engenharia, +Universidade Federal do Rio de Janeiro, C.P. 68503, CEP: 21941-972, Rio de Janeiro, RJ, Brazil +2Instituto de F´ısica, Universidade Federal do Rio de Janeiro, +Av. Athos da Silveira Ramos 149, CEP: 21941-909, Rio de Janeiro, RJ, Brazil +and +3Interdisciplinary Center for Fluid Dynamics, Universidade Federal do Rio de Janeiro, +R. Moniz Arag˜ao 360, CEP: 21941-594, Rio de Janeiro, Brazil +Turbulent pipe flows exhibit organizational states (OSs) that are labelled by discrete azimuthal +wavenumber modes and are reminiscent of the traveling wave solutions of low Reynolds number +regimes. The discretized time evolution of the OSs, obtained through stereoscopic particle image +velocimetry, is shown to be non-Markovian for data acquisition carried out at a structure-resolved +sampling rate. In particular, properly defined time-correlation functions for the OS transitions are +observed to decay as intriguing power laws, up to a large-eddy time horizon, beyond which they +decorrelate at much faster rates. We are able to establish, relying upon a probabilistic descrip- +tion of the creation and annihilation of streamwise streaks, a lower-level Markovian model for the +OS transitions, which reproduces their time-correlated behavior with meaningful accuracy. These +findings indicate that the OSs are distributed along the pipe as statistically correlated packets of +quasi-streamwise vortical structures. +Notwithstanding the large body of knowledge accumu- +lated since the landmark experiments of Reynolds [1], +turbulent pipes comprise flow patterns which have re- +mained surprisingly unsuspected until recent years. They +can be depicted as relatively organized sets of wall- +attached low-speed streaks coupled to pairs of counter- +rotating quasi-streamwise vortices [2–4]. These organi- +zational states (OSs) actually characterize the turbulent +velocity fluctuations at high Reynolds numbers and are +topologically similar to traveling waves – a class of ex- +act (but unstable) low-Reynolds number solutions of the +Navier-Stokes equations [5, 6]. +As for traveling waves, the OSs can be classified by the +number of low-speed streaks they contain. Observation +tells us, however, that this quantity changes in an ap- +parently random way along the turbulent pipe. For the +sake of illustration, Fig. 1 shows a transition between +OSs, visualized from a pair of cross-sectional snapshots +of the flow obtained through stereoscopic particle image +velocimetry (sPIV). +The existence of spatial transitions among the OS +modes suggests, within the perspective of dynamical sys- +tems, that the turbulent pipe flow could be described as +a chaotic attractor and its unstable periodic orbits in a +phase space of much reduced dimensionality [7–11]. In +connection with this circle of ideas, we are motivated to +study the OS transitions in the framework of stochastic +processes, focusing particular attention on their recurrent +dynamics. +To start, let u = u(r, θ) be, in polar coordinates, the +fluctuating streamwise component of the velocity field +∗Corresponding author: moriconi@if.ufrj.br +FIG. 1: Example of a transition between organized states, +as sampled out from our measurements, which are associated +to two and three low-speed streaks. Blue and red colors re- +fer, respectively, to negative and positive streamwise velocity +fluctuations around the mean (the systematic procedure to +ascertain a well-defined number of low-speed structures to a +given flow snapshot is discussed in the text). +defined over a fixed pipe’s cross-sectional plane. We may +introduce, accordingly, the instantaneous spectral power +density, +I(kn) = +���� +� 2π +0 +dθeiknθfuu(r0, θ) +���� +2 +, +(1) +where +fuu(r0, θ) = +� 2π +0 +dθ′u(r0, θ′)u(r0, θ′ + θ) , +(2) +kn = n ∈ Z+ is an azimuthal wavenumber, and r0 is a ref- +erence radial distance which falls within the log-region of +the pipe’s turbulent boundary layer. Empirical evidence +shows that I(kn) is in general peaked at some clearly +dominant wavenumber ¯k (to be identified to the number +of snapshotted low-speed streaks), which can be used to +label the probed velocity profile u(r, θ). As time evolves, +arXiv:2301.05344v1 [physics.flu-dyn] 13 Jan 2023 + +3 +22 +FIG. 2: Statistical results for the OS mode ¯k = 5. Left image: +positive (red) and negative (blue) level curves of Ruu, defined +by |Ruu(r − r0|¯k)| = 5% and 10% of (Ruu)max, with the +reference point r0 depicted as a black dot. Right image: a +closer look at the averaged streamwise velocity fluctuations +(red for positive, blue for negative), conditioned on u(r0) > 0. +The cross-sectional averaged velocity field reveals the vortical +structures that are usually coupled with velocity streaks. +I(kn) changes, and so does the wavenumber position of +its dominant peak. +Therefore, if u(r, θ) is recorded at +equally spaced time intervals ∆, the dynamical evolu- +tion of the pipe turbulent field can be mapped into the +stochastic process +S ≡ {¯k(t), ¯k(t + ∆), ¯k(t + 2∆), ... } . +(3) +In order to investigate the still very open statistical prop- +erties of S, we have performed a pipe flow experiment, at +Reynolds number Re = 24415, in the large pipe rig facil- +ity of the Interdisciplinary Nucleus for Fluid Dynamics +(NIDF) at the Federal University of Rio de Janeiro. The +pipe’s diameter and length are, respectively, D = 15 cm +and L = 12 m. By means of sPIV, with sampling rate +of 10 Hz (i.e., ∆ = 0.1 s), we have collected 104 cross- +sectional snapshots of the flow, each one containing the +three components of the turbulent velocity field over a +uniform grid of size 78 × 78. It turns out that all the ob- +served OS modes fall into the range 0 ≤ ¯k ≤ ¯kmax = 10. +Our experimental data has been validated with the +help of previous benchmark pipe flow experiments [12], +through the inspection of the performance of first and +second order single-point statistics for the streamwise +component of velocity field. +We have also attained a +further validation of the entire measured velocity field, +from the evaluation of particularly defined streamwise +velocity-velocity correlation functions conditioned on the +OS modes ¯k, more precisely, +Ruu(∆r|¯k) ≡ E[u(r0)u(r0 + ∆r)|¯k] , +(4) +which has its level curves depicted in Fig. 2, for the case +¯k = 5, in close correspondence with the results of Ref. [4]. +The first immediate question that can be raised about +the stochastic process S is whether it is Markovian or +|λh| +10−3 +10−2 +10−1 +100 +10−3 +10−2 +10−1 +100 +h = time lag between sPIV snapshots / Δ +0 +1 +2 +3 +0 +1 +2 +3 +FIG. 3: Eigenvalues of the probability transition matrices for +the original process (h = 1) and a decimated one (h = 2). +The dashed lines should intercept eigenvalue pairs if S were +a Markovian process. +not. Of course, while it is not possible to answer this +in full rigor, one may check if the Chapman-Kolmogorov +(CK) equation holds for the time series (3), a necessary +condition for S to be Markovian [13]. The CK equation +would imply that the eigenvalues of the transition prob- +ability matrix for OS modes separated by the time inter- +val h∆ can be represented, in some arbitrary ordering, as +the set of powers {λh +1, λh +2, ..., λh +¯kmax}. A straightforward +computation of the transition matrix eigenvalues for the +cases h = 1 and h = 2 indicates, however, that S is not +Markovian; see Fig. 3. +We expect that the decimated process for h large +enough is essentially Markovian, since in this situation +the OS modes become weakly correlated. The transition +to Markovian behavior can be alternatively addressed +from the analysis of correlation functions which we intro- +duce as it follows. Taking 0 ≤ m, m′ ≤ ¯kmax, let Vm(t) +and Mm′m(t) be, respectively, the components of vector +and matrix valued stochastic processes derived from S as +Vm(t) = +� +1, +if ¯k(t) = m +0, +otherwise +(5) +and +Mm′m(t) = +� +1, +if ¯k(t) = m and ¯k(t + ∆) = m′ +0, +otherwise . +(6) +Define, now, the correlation functions +˜F(t − t′) ≡ E[V(t) · V(t′)] − (E[V])2 , +(7) +˜G(t − t′) ≡ Tr +� +E[MT(t)M(t′)] − E[M]TE[M] +� +, (8) +and their normalized versions, +F(t − t′) ≡ +˜F(t − t′) +˜F(0) +, G(t − t′) ≡ +˜G(t − t′) +˜G(0) +. +(9) + +O3 +F(t-t') +10−3 +10−2 +10−1 +100 +10−3 +10−2 +10−1 +100 +|t-t'| (s) +10−1 +100 +10−1 +100 +δt +(a) +G(t-t') +10−2 +10−1 +100 +10−2 +10−1 +100 +|t-t'| (s) +10−1 +100 +10−1 +100 +δt +(b) +FIG. 4: The time-dependent correlation functions defined in +(9) are noticed to decay as power laws for |t − t′| ≤ δt ≈ 2 s. +The dotted lines in (a) and (b) have scaling exponent −1 for +both F(t − t′) and G(t − t′). +It is not difficult to see that F(t − t′) and G(t − t′) +describe, respectively, the correlations of returning OS +modes and transitions which are apart from each other +by the time interval |t − t′|. They are plotted in Fig. 4 +and are noticed to have interesting power law decays +(with the same approximate scaling exponent −1) up to +|t − t′| ≡ δt ≈ 20∆ = 2 s, which suggests some sort of +self-similarity across the spatial distribution of about ten +OS modes (their mean lifetime is 0.2 s ≈ δt/10). +For +time separations larger than δt, the correlation func- +tions become suddenly undersampled, meaning that they +crossover to a faster law of decay, probably exponential, +as it should be for the putative asymptotic Markovian +behavior of a large-time decimated S. +It is worth emphasizing that the non-Markovian nature +of S does not mean at all that it cannot be modeled as +a Markov process defined in terms of lower-level state +variables. In this connection, it is reasonable to assume +that there is a combinatoric degeneracy factor +Ω(¯kmax, m) = +�¯kmax +m +� +(10) +associated to a given OS mode ¯k = m. We simply mean +here that the m wall-attached low-speed streaks can be +spatially arranged for this particular mode in Ω(¯kmax, m) +different ways, since the pipe’s cross-sectional plane is +taken to hold at most ¯kmax low-speed streak channels. +The phase space of the “microscopic” state variables +for the underlying Markovian model of S is spanned, +therefore, by all the possible sets of ¯kmax streak bits, +X ≡ {s1, s2, ..., s¯kmax}, where +si = +� +1, +if the i-th streak channel is active +0, +otherwise . +(11) +We postulate, now, that the time evolution of the micro- +scopic states X is produced from the independent fluctu- +ations of streak bits, which have persistence probabilities +that depend on the total number of active streak chan- +nels, that is the OS label m. In this way, we define qm +and pm to be the persistence probabilities for any given +streak bit to keep its value 0 or 1, respectively, along +subsequent sPIV snapshots. There are, thus, four dif- +ferent types of streak bit flips, which appear in different +occurrence numbers for a given OS mode transition, as +summarized in Table I. +Transition Type # of Streak Channels Transition Prob. +0 → 0 +n1 +qm +0 → 1 +n2 +1 − qm +1 → 0 +n3 +1 − pm +1 → 1 +n4 +pm +TABLE I: Definition of the four possible transition types for +the streak channel states, together with the notations for their +occurrence numbers and individual transition probabilities. +Above, m = n3 + n4 labels the OS mode. +The parameters reported in Table I are related to +the OS mode transition m → m′, where m = n3 + n4 +and m′ = n2 + n4. The transition probability between +any specific pair of associated microstates is, as a conse- +quence, qn1 +m (1−qm)n2(1−pm)n3pn4 +m . Taking into account, +furthermore, the role of degeneracy factors, we may write +the transition probability between the OS modes m and +m′ as + +4 +Tm′m = +�¯kmax +m +�−1 ¯kmax +� +n1=0 +¯kmax +� +n2=0 +¯kmax +� +n3=0 +¯kmax +� +n4=0 +δ(n1 + n2 + n3 + n4, ¯kmax)δ(n3 + n4, m)δ(n2 + n4, m′) × +× +�¯kmax +n1 +��¯kmax − n1 +n2 +��¯kmax − n1 − n2 +n3 +� +qn1 +m (1 − qm)n2(1 − pm)n3pn4 +m . +(12) +Using, from now on, ¯kmax = 10, the Markovian model +just introduced may not appear very phenomenologically +attractive at first glance, since Tm′m is parametrized by +a large number of unknown parameters (q0, q1, ..., q9 and +p1, p2, ..., p10). +Note, however, that there are, in prin- +ciple, 90 independent entries in the empirical transition +matrix (the one derived from the sPIV measurements), +so the model is rather underdetermined (as we would ex- +pect for a phase-space reduced description of turbulent +fluctuations). +Instead of attempting to provide a detailed and com- +putationally costly model of the empirical transition ma- +trix, we address a much simpler approach, where we focus +on the asymptotic probability eigenvector of the modeled +transition matrix, +P = (P1, P2, ..., P10) , +(13) +which satifies to TP = P, that is, �10 +m=0 Tm′mPm = Pm′. +Here, Pm is the probability that the OS mode m be ob- +served in the statistically stationary regime. In an analo- +gous way, denoting by P∞ the empirical probability vec- +tor, determined from the sPIV measurements, we are in- +terested to find the set of probabilities qm and pm that +minimize the quadratic error +d({qm}, {pm}) ≡ ||P − P∞||2 . +(14) +While, as already commented, the original problem is +underdetermined, the optimization scheme related to +Eq. (14) is not: as a matter of fact, we would have to +model the 9 independent probability entries of (13) by +means of the 20 probability parameters qm and pm. To +reduce this large overdeterminacy, we rely on a few phe- +nomenological inputs: +(i) We assume that we can model the observed coher- +ence (time persistence) of low-speed streaks by a single +mode-independent and not small probability parameter +p, where p = p2 = p3 = ... = p10; +(ii) P0 turns out to be negligible, so we suppress transi- +tions from the OS mode m = 1 to m = 0, by imposing +that p1 = 1 (other transitions to the mode m = 0 from +modes m ̸= 1 are possible, but they are of O((1 − p)2). +Therefore, we end up with 11 parameters (q0, q1, ..., q9 +and p) to locate the minimum value of (14). The result +is a slightly overdetermined system, but if besides P∞, +the correlation functions F(t − t′) and G(t − t′) turn out +to be well reproduced with the same set of probability +parameters, as an extra bonus, then the model can be +taken as physically appealing. That is the heuristic setup +that we have in mind. +We have resorted to a straightforward Monte Carlo +procedure to obtain the set of qm’s that minimizes (14) +for various fixed values of p. We find, as shown in Fig. 5, +Min{q}[d({q},p)] +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +p (= p2 = p3 ... = p10) +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +FIG. 5: Minimization of the quadratic distance d({q}, p) for +various values of p. +Occurrence Probability (%) +0 +5 +10 +15 +20 +0 +5 +10 +15 +20 +OS Mode (k) +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 + +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +FIG. 6: The occurrence probability of OS modes obtained +from the experiment (dots) and from the stochastic model +(open circles: p = 0.86; crosses: p = 0.95), defined by the +transition matrix elements (12). + +5 +F(t-t') +10−3 +10−2 +10−1 +100 +10−3 +10−2 +10−1 +100 +|t-t'| (s) +10−1 +100 +10−1 +100 +(a) +G(t-t') +10−2 +10−1 +100 +10−2 +10−1 +100 +|t-t'| (s) +10−1 +100 +10−1 +100 +δt +δt +(b) +10−3 +10−2 +10−1 + +2 +4 +6 +8 + +FIG. 7: Empirical (dots) and modeled (crosses) correlation +functions F(t−t′) and G(t−t′). Crosses refer, in (a) and (b), +respectively, to modeling parameters p = 0.86 and p = 0.95. +The semi-log plot in the inset of (b) indicates the simple ex- +ponential form of G(t − t′) at large enough |t − t′|. +that the quadratic error quickly drops for p ≥ 0.85. The +modeled asymptotic probabilities for the occurrence of +OS modes are excellently compared, in Fig. 6, to the +empirical ones for the cases p = 0.86 and p = 0.95. These +are the values of p that lead to good accounts of F(t−t′) +and G(t−t′), as reported in Fig. 7. The related values of +the probabilities qm are listed in Table II. Even if a point +of subjective concern, the uncertainty of about 10% in +the definition of p should be taken as relatively small, +vis a vis the model’s accuracy in predicting the decaying +profiles of the OS correlation functions. +p +q0 +q1 +q2 +q3 +q4 +q5 +q6 +q7 +q8 +q9 +0.86 0.53 0.96 0.95 0.92 0.92 0.85 0.95 0.75 0.86 1.0 +0.95 0.22 0.98 0.98 0.97 0.97 0.96 0.97 0.93 0.94 0.49 +TABLE II: The list of probabilities qm’s which describe +the persistence of inactive streak channels, for the cases +p = 0.86 and p = 0.95. +Also evidenced in the inset Fig. 7 is the exponential +decay profile of the modeled G(t − t′) for time intervals +larger than δt. +At present, this point rests as a pre- +diction of the modeling scenario introduced in this work, +akin with the observed sudden undersampling of the time +series for larger decimations. We note that the crossover +to the faster exponential decay of correlation functions +takes place at δt ≈ 2D/U, where U is the bulk flow ve- +locity. Thus, the physical picture that emerges is that +the OSs are packed as chains of low-speed streaks and +vortical structures which are strongly correlated within +sizes that scale with the pipe’s diameter, although they +are merged along the entire turbulent flow. +To summarize, we have investigated the stochastic +properties of the non-Markovian OS mode transitions in a +turbulent pipe flow, recovering them as a surjective map- +ping of a lower-level Markov process. The essential idea +that underlies the model construction is that a given OS +mode may be associated to several spatial arrangements +of its low-speed streaks into a fixed number of “streak +channels” which azimuthally partition the pipe’s cross +section. +We find that the Markov model can account for the +scaling behavior of specifically introduced correlation +functions of OS mode transitions. Further work is in or- +der, not only to enlarge the size of sPIV ensembles, but +to address, in an analytical way, the very unexpected self- +similar dynamics of the OS mode transitions. We point +out that the dynamical scaling range of the recurrent OS +transitions reflects the existence of finite-sized OS packets +along the pipe flow, correlated at integral length scales +(i.e., the pipe’s diameter). +An interesting theoretical direction to pursue is related +to the use of instanton techniques [14] to evaluate the +transition probabilities between unstable flow configura- +tions as are the OS modes. In the turbulence or transi- +tional context, instantons are taken, respectively, as ex- +treme events or flow configurations that dominate the +probability measures in the weak coupling limit. They +have been successfully applied to a number of fluid dy- +namic problems, as in geophysical models, homogeneous +turbulence and the laminar-turbulent transition in shear +flows [15–17]. +We conclude by noting that the findings here presented +are likely to add relevant phenomenological information +to the discussion of fundamentally important issues in +pipe flow turbulence, as drag control and particle-laden +dynamics, once they are closely connected to the statis- +tical features of near-wall coherent structures [18–23]. +Acknowledgments +This work was partially supported by the Conselho +Nacional de Desenvolvimento Cient´ıfico e Tecnol´ogico +(CNPq) and by Funda¸c˜ao Coppetec/UFRJ (project num- +ber 20459). L.M. thanks E. Marensi for enlightening dis- +cussions about the phenomenology of traveling waves. + +6 +[1] O. Reynolds, Philos. Trans. R. Soc. 174, 935 (1883). +[2] B. Hof, C.W.H. van Doorne, J. Westerweel, F.T.M. +Nieuwstadt, H. Faisst, B. Eckhardt, H. Wedin, R.R. Ker- +swell, and F. Waleffe, Science 305, 1594 (2004). +[3] T.M. Schneider, B. Eckhardt, and J. Vollmer, Phys. Rev. +E 75, 066313 (2007). +[4] D.J.C. Dennis and F.M. Sogaro, Phys. Rev. Lett. 113, +234501 (2014). +[5] H. Faisst and B. Eckhardt, Phys. Rev. Lett. 91, 224502 +(2003). +[6] H. Wedin and R.R. Kerswell, J. Fluid Mech. 508, 333 +(2004). +[7] J. Gibson, J. Halcrow, and Cvitanovi´c, J. Fluid Mech. +611, 107 (2008). +[8] J. Moehlis, H. Faisst, and B. Eckhardt, SIAM J. Appl. +Dyn. Syst. 4, 352 (2005). +[9] N.B. Budanur, K.Y. Short, M. Farazmand, A.P. Willis, +and P. Cvitanovi´c, J. Fluid Mech. 883, 274 (2017). +[10] G. Yalnız, B. Hof, and N.B. Budanur Phys. Rev. Lett. +126, 244502 (2021). +[11] E. Marensi, G. Yalnız, B. Hof, and N.B. Budanur, J. +Fluid Mech. 954, A 10 (2023). +[12] J.M.J. Den Toonder and F.T.M. Nieuwstadt, Phys. Flu- +ids, 9, 3398 (1997). +[13] Erhan C¸irlan, Introduction to Stochastic Processes, Dover +(1975). +[14] T. Grafke, R. Grauer, and T. Sch¨afer, J. Phys. A: Math. +Theor. 48 333001 (2015). +[15] J. Laurie and F. Bouchet, New J. Phys. 17, 015009 +(2015). +[16] G.B. Apolin´ario, L. Moriconi, R.M. Pereira, and V.J. +Valad˜ao, Phys. Lett. A 449, 128360 (2022). +[17] S. Gom´e, L.S. Tuckerman, and D. Barkley, Phil. Trans. +R. Soc. A 380, 20210036 (2022). +[18] H. Choi, P. Moin, and J. Kim, J. Fluid Mech. 262, 75 +(1994). +[19] W. Schoppa and F. Hussain, Phys. Fluids 10, 1049 +(1998). +[20] I. Marusic, D. Chandran, A. Rouhi, M.K. Fu, D. Wine, +B. Holloway, D. Chung, and A.J. Smits, Nat. Commun. +12, 5805 (2021) . +[21] E. Gallorini, M. Quadrio, and D. Gatti, Phys. Rev. Fluids +7, 114602 (2022). +[22] G. Wang and D. Richter, J. Fluid Mech. 861, 901 (2019). +[23] L. Brandt and F. Coletti, Annu. Rev. Fluid Mech. 54, +159 (2022). + diff --git a/K9E4T4oBgHgl3EQf8A5c/content/tmp_files/load_file.txt b/K9E4T4oBgHgl3EQf8A5c/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..60683b8bb988bace8cfcdc403c5acf095ac2e7fe --- /dev/null +++ b/K9E4T4oBgHgl3EQf8A5c/content/tmp_files/load_file.txt @@ -0,0 +1,380 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf,len=379 +page_content='Stochastic Model of Organizational State Transitions in a Turbulent Pipe Flow Robert J¨ackel1,3, Bruno Magacho2,3, Bayode Owolabi2,3, Luca Moriconi2,3∗, David J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' Dennis3, and Juliana B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' Loureiro1,3 1Programa de Engenharia Mecˆanica, Coordena¸c˜ao dos Programas de P´os-Gradua¸c˜ao em Engenharia, Universidade Federal do Rio de Janeiro, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' 68503, CEP: 21941-972, Rio de Janeiro, RJ, Brazil 2Instituto de F´ısica, Universidade Federal do Rio de Janeiro, Av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' Athos da Silveira Ramos 149, CEP: 21941-909, Rio de Janeiro, RJ, Brazil and 3Interdisciplinary Center for Fluid Dynamics, Universidade Federal do Rio de Janeiro, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' Moniz Arag˜ao 360, CEP: 21941-594, Rio de Janeiro, Brazil Turbulent pipe flows exhibit organizational states (OSs) that are labelled by discrete azimuthal wavenumber modes and are reminiscent of the traveling wave solutions of low Reynolds number regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' The discretized time evolution of the OSs, obtained through stereoscopic particle image velocimetry, is shown to be non-Markovian for data acquisition carried out at a structure-resolved sampling rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' In particular, properly defined time-correlation functions for the OS transitions are observed to decay as intriguing power laws, up to a large-eddy time horizon, beyond which they decorrelate at much faster rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' We are able to establish, relying upon a probabilistic descrip- tion of the creation and annihilation of streamwise streaks, a lower-level Markovian model for the OS transitions, which reproduces their time-correlated behavior with meaningful accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' These findings indicate that the OSs are distributed along the pipe as statistically correlated packets of quasi-streamwise vortical structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' Notwithstanding the large body of knowledge accumu- lated since the landmark experiments of Reynolds [1], turbulent pipes comprise flow patterns which have re- mained surprisingly unsuspected until recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' They can be depicted as relatively organized sets of wall- attached low-speed streaks coupled to pairs of counter- rotating quasi-streamwise vortices [2–4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' These organi- zational states (OSs) actually characterize the turbulent velocity fluctuations at high Reynolds numbers and are topologically similar to traveling waves – a class of ex- act (but unstable) low-Reynolds number solutions of the Navier-Stokes equations [5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' As for traveling waves, the OSs can be classified by the number of low-speed streaks they contain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' Observation tells us, however, that this quantity changes in an ap- parently random way along the turbulent pipe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' For the sake of illustration, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' 1 shows a transition between OSs, visualized from a pair of cross-sectional snapshots of the flow obtained through stereoscopic particle image velocimetry (sPIV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' The existence of spatial transitions among the OS modes suggests, within the perspective of dynamical sys- tems, that the turbulent pipe flow could be described as a chaotic attractor and its unstable periodic orbits in a phase space of much reduced dimensionality [7–11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' In connection with this circle of ideas, we are motivated to study the OS transitions in the framework of stochastic processes, focusing particular attention on their recurrent dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' To start, let u = u(r, θ) be, in polar coordinates, the fluctuating streamwise component of the velocity field ∗Corresponding author: moriconi@if.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content='ufrj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content='br FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' 1: Example of a transition between organized states, as sampled out from our measurements, which are associated to two and three low-speed streaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' Blue and red colors re- fer, respectively, to negative and positive streamwise velocity fluctuations around the mean (the systematic procedure to ascertain a well-defined number of low-speed structures to a given flow snapshot is discussed in the text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' defined over a fixed pipe’s cross-sectional plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' We may introduce, accordingly, the instantaneous spectral power density, I(kn) = ���� � 2π 0 dθeiknθfuu(r0, θ) ���� 2 , (1) where fuu(r0, θ) = � 2π 0 dθ′u(r0, θ′)u(r0, θ′ + θ) , (2) kn = n ∈ Z+ is an azimuthal wavenumber, and r0 is a ref- erence radial distance which falls within the log-region of the pipe’s turbulent boundary layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' Empirical evidence shows that I(kn) is in general peaked at some clearly dominant wavenumber ¯k (to be identified to the number of snapshotted low-speed streaks), which can be used to label the probed velocity profile u(r, θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' As time evolves, arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content='05344v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content='flu-dyn] 13 Jan 2023 3 22 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' 2: Statistical results for the OS mode ¯k = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' Left image: positive (red) and negative (blue) level curves of Ruu, defined by |Ruu(r − r0|¯k)| = 5% and 10% of (Ruu)max, with the reference point r0 depicted as a black dot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' Right image: a closer look at the averaged streamwise velocity fluctuations (red for positive, blue for negative), conditioned on u(r0) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' The cross-sectional averaged velocity field reveals the vortical structures that are usually coupled with velocity streaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' I(kn) changes, and so does the wavenumber position of its dominant peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' Therefore, if u(r, θ) is recorded at equally spaced time intervals ∆, the dynamical evolu- tion of the pipe turbulent field can be mapped into the stochastic process S ≡ {¯k(t), ¯k(t + ∆), ¯k(t + 2∆), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' } .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' (3) In order to investigate the still very open statistical prop- erties of S, we have performed a pipe flow experiment, at Reynolds number Re = 24415, in the large pipe rig facil- ity of the Interdisciplinary Nucleus for Fluid Dynamics (NIDF) at the Federal University of Rio de Janeiro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' The pipe’s diameter and length are, respectively, D = 15 cm and L = 12 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' By means of sPIV, with sampling rate of 10 Hz (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=', ∆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content='1 s), we have collected 104 cross- sectional snapshots of the flow, each one containing the three components of the turbulent velocity field over a uniform grid of size 78 × 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' It turns out that all the ob- served OS modes fall into the range 0 ≤ ¯k ≤ ¯kmax = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' Our experimental data has been validated with the help of previous benchmark pipe flow experiments [12], through the inspection of the performance of first and second order single-point statistics for the streamwise component of velocity field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' We have also attained a further validation of the entire measured velocity field, from the evaluation of particularly defined streamwise velocity-velocity correlation functions conditioned on the OS modes ¯k, more precisely, Ruu(∆r|¯k) ≡ E[u(r0)u(r0 + ∆r)|¯k] , (4) which has its level curves depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' 2, for the case ¯k = 5, in close correspondence with the results of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' The first immediate question that can be raised about the stochastic process S is whether it is Markovian or |λh| 10−3 10−2 10−1 100 10−3 10−2 10−1 100 h = time lag between sPIV snapshots / Δ 0 1 2 3 0 1 2 3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' 3: Eigenvalues of the probability transition matrices for the original process (h = 1) and a decimated one (h = 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' The dashed lines should intercept eigenvalue pairs if S were a Markovian process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' Of course, while it is not possible to answer this in full rigor, one may check if the Chapman-Kolmogorov (CK) equation holds for the time series (3), a necessary condition for S to be Markovian [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' The CK equation would imply that the eigenvalues of the transition prob- ability matrix for OS modes separated by the time inter- val h∆ can be represented, in some arbitrary ordering, as the set of powers {λh 1, λh 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=', λh ¯kmax}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' A straightforward computation of the transition matrix eigenvalues for the cases h = 1 and h = 2 indicates, however, that S is not Markovian;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' We expect that the decimated process for h large enough is essentially Markovian, since in this situation the OS modes become weakly correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' The transition to Markovian behavior can be alternatively addressed from the analysis of correlation functions which we intro- duce as it follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' Taking 0 ≤ m, m′ ≤ ¯kmax, let Vm(t) and Mm′m(t) be, respectively, the components of vector and matrix valued stochastic processes derived from S as Vm(t) = � 1, if ¯k(t) = m 0, otherwise (5) and Mm′m(t) = � 1, if ¯k(t) = m and ¯k(t + ∆) = m′ 0, otherwise .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' (6) Define, now, the correlation functions ˜F(t − t′) ≡ E[V(t) · V(t′)] − (E[V])2 , (7) ˜G(t − t′) ≡ Tr � E[MT(t)M(t′)] − E[M]TE[M] � , (8) and their normalized versions, F(t − t′) ≡ ˜F(t − t′) ˜F(0) , G(t − t′) ≡ ˜G(t − t′) ˜G(0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=" (9) O3 F(t-t') 10−3 10−2 10−1 100 10−3 10−2 10−1 100 |t-t'| (s) 10−1 100 10−1 100 δt (a) G(t-t') 10−2 10−1 100 10−2 10−1 100 |t-t'| (s) 10−1 100 10−1 100 δt (b) FIG." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' 4: The time-dependent correlation functions defined in (9) are noticed to decay as power laws for |t − t′| ≤ δt ≈ 2 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' The dotted lines in (a) and (b) have scaling exponent −1 for both F(t − t′) and G(t − t′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' It is not difficult to see that F(t − t′) and G(t − t′) describe, respectively, the correlations of returning OS modes and transitions which are apart from each other by the time interval |t − t′|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' They are plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' 4 and are noticed to have interesting power law decays (with the same approximate scaling exponent −1) up to |t − t′| ≡ δt ≈ 20∆ = 2 s, which suggests some sort of self-similarity across the spatial distribution of about ten OS modes (their mean lifetime is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content='2 s ≈ δt/10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' For time separations larger than δt, the correlation func- tions become suddenly undersampled, meaning that they crossover to a faster law of decay, probably exponential, as it should be for the putative asymptotic Markovian behavior of a large-time decimated S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' It is worth emphasizing that the non-Markovian nature of S does not mean at all that it cannot be modeled as a Markov process defined in terms of lower-level state variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' In this connection, it is reasonable to assume that there is a combinatoric degeneracy factor Ω(¯kmax, m) = �¯kmax m � (10) associated to a given OS mode ¯k = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' We simply mean here that the m wall-attached low-speed streaks can be spatially arranged for this particular mode in Ω(¯kmax, m) different ways, since the pipe’s cross-sectional plane is taken to hold at most ¯kmax low-speed streak channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' The phase space of the “microscopic” state variables for the underlying Markovian model of S is spanned, therefore, by all the possible sets of ¯kmax streak bits, X ≡ {s1, s2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=', s¯kmax}, where si = � 1, if the i-th streak channel is active 0, otherwise .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' (11) We postulate, now, that the time evolution of the micro- scopic states X is produced from the independent fluctu- ations of streak bits, which have persistence probabilities that depend on the total number of active streak chan- nels, that is the OS label m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' In this way, we define qm and pm to be the persistence probabilities for any given streak bit to keep its value 0 or 1, respectively, along subsequent sPIV snapshots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' There are, thus, four dif- ferent types of streak bit flips, which appear in different occurrence numbers for a given OS mode transition, as summarized in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' Transition Type # of Streak Channels Transition Prob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' 0 → 0 n1 qm 0 → 1 n2 1 − qm 1 → 0 n3 1 − pm 1 → 1 n4 pm TABLE I: Definition of the four possible transition types for the streak channel states, together with the notations for their occurrence numbers and individual transition probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' Above, m = n3 + n4 labels the OS mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' The parameters reported in Table I are related to the OS mode transition m → m′, where m = n3 + n4 and m′ = n2 + n4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' The transition probability between any specific pair of associated microstates is, as a conse- quence, qn1 m (1−qm)n2(1−pm)n3pn4 m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' Taking into account, furthermore, the role of degeneracy factors, we may write the transition probability between the OS modes m and m′ as 4 Tm′m = �¯kmax m �−1 ¯kmax � n1=0 ¯kmax � n2=0 ¯kmax � n3=0 ¯kmax � n4=0 δ(n1 + n2 + n3 + n4, ¯kmax)δ(n3 + n4, m)δ(n2 + n4, m′) × × �¯kmax n1 ��¯kmax − n1 n2 ��¯kmax − n1 − n2 n3 � qn1 m (1 − qm)n2(1 − pm)n3pn4 m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' (12) Using, from now on, ¯kmax = 10, the Markovian model just introduced may not appear very phenomenologically attractive at first glance, since Tm′m is parametrized by a large number of unknown parameters (q0, q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=', q9 and p1, p2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=', p10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' Note, however, that there are, in prin- ciple, 90 independent entries in the empirical transition matrix (the one derived from the sPIV measurements), so the model is rather underdetermined (as we would ex- pect for a phase-space reduced description of turbulent fluctuations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' Instead of attempting to provide a detailed and com- putationally costly model of the empirical transition ma- trix, we address a much simpler approach, where we focus on the asymptotic probability eigenvector of the modeled transition matrix, P = (P1, P2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=', P10) , (13) which satifies to TP = P, that is, �10 m=0 Tm′mPm = Pm′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' Here, Pm is the probability that the OS mode m be ob- served in the statistically stationary regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' In an analo- gous way, denoting by P∞ the empirical probability vec- tor, determined from the sPIV measurements, we are in- terested to find the set of probabilities qm and pm that minimize the quadratic error d({qm}, {pm}) ≡ ||P − P∞||2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' (14) While, as already commented, the original problem is underdetermined, the optimization scheme related to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' (14) is not: as a matter of fact, we would have to model the 9 independent probability entries of (13) by means of the 20 probability parameters qm and pm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' To reduce this large overdeterminacy, we rely on a few phe- nomenological inputs: (i) We assume that we can model the observed coher- ence (time persistence) of low-speed streaks by a single mode-independent and not small probability parameter p, where p = p2 = p3 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' = p10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' (ii) P0 turns out to be negligible, so we suppress transi- tions from the OS mode m = 1 to m = 0, by imposing that p1 = 1 (other transitions to the mode m = 0 from modes m ̸= 1 are possible, but they are of O((1 − p)2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' Therefore, we end up with 11 parameters (q0, q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=', q9 and p) to locate the minimum value of (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' The result is a slightly overdetermined system, but if besides P∞, the correlation functions F(t − t′) and G(t − t′) turn out to be well reproduced with the same set of probability parameters, as an extra bonus, then the model can be taken as physically appealing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' That is the heuristic setup that we have in mind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' We have resorted to a straightforward Monte Carlo procedure to obtain the set of qm’s that minimizes (14) for various fixed values of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' We find, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' 5, Min{q}[d({q},p)] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content='5 p (= p2 = p3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' = p10) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content='0 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' 5: Minimization of the quadratic distance d({q}, p) for various values of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' Occurrence Probability (%) 0 5 10 15 20 0 5 10 15 20 OS Mode (k) 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' 6: The occurrence probability of OS modes obtained from the experiment (dots) and from the stochastic model (open circles: p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content='86;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' crosses: p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content='95), defined by the transition matrix elements (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=" 5 F(t-t') 10−3 10−2 10−1 100 10−3 10−2 10−1 100 |t-t'| (s) 10−1 100 10−1 100 (a) G(t-t') 10−2 10−1 100 10−2 10−1 100 |t-t'| (s) 10−1 100 10−1 100 δt δt (b) 10−3 10−2 10−1 2 4 6 8 FIG." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' 7: Empirical (dots) and modeled (crosses) correlation functions F(t−t′) and G(t−t′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' Crosses refer, in (a) and (b), respectively, to modeling parameters p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content='86 and p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content='95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' The semi-log plot in the inset of (b) indicates the simple ex- ponential form of G(t − t′) at large enough |t − t′|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' that the quadratic error quickly drops for p ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content='85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' The modeled asymptotic probabilities for the occurrence of OS modes are excellently compared, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' 6, to the empirical ones for the cases p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content='86 and p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content='95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' These are the values of p that lead to good accounts of F(t−t′) and G(t−t′), as reported in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' The related values of the probabilities qm are listed in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' Even if a point of subjective concern, the uncertainty of about 10% in the definition of p should be taken as relatively small, vis a vis the model’s accuracy in predicting the decaying profiles of the OS correlation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' p q0 q1 q2 q3 q4 q5 q6 q7 q8 q9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content='86 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content='53 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content='92 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content='92 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content='86 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content='93 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content='49 TABLE II: The list of probabilities qm’s which describe the persistence of inactive streak channels, for the cases p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content='86 and p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content='95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' Also evidenced in the inset Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' 7 is the exponential decay profile of the modeled G(t − t′) for time intervals larger than δt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' At present, this point rests as a pre- diction of the modeling scenario introduced in this work, akin with the observed sudden undersampling of the time series for larger decimations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' We note that the crossover to the faster exponential decay of correlation functions takes place at δt ≈ 2D/U, where U is the bulk flow ve- locity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' Thus, the physical picture that emerges is that the OSs are packed as chains of low-speed streaks and vortical structures which are strongly correlated within sizes that scale with the pipe’s diameter, although they are merged along the entire turbulent flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' To summarize, we have investigated the stochastic properties of the non-Markovian OS mode transitions in a turbulent pipe flow, recovering them as a surjective map- ping of a lower-level Markov process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' The essential idea that underlies the model construction is that a given OS mode may be associated to several spatial arrangements of its low-speed streaks into a fixed number of “streak channels” which azimuthally partition the pipe’s cross section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' We find that the Markov model can account for the scaling behavior of specifically introduced correlation functions of OS mode transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' Further work is in or- der, not only to enlarge the size of sPIV ensembles, but to address, in an analytical way, the very unexpected self- similar dynamics of the OS mode transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' We point out that the dynamical scaling range of the recurrent OS transitions reflects the existence of finite-sized OS packets along the pipe flow, correlated at integral length scales (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=', the pipe’s diameter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' An interesting theoretical direction to pursue is related to the use of instanton techniques [14] to evaluate the transition probabilities between unstable flow configura- tions as are the OS modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' In the turbulence or transi- tional context, instantons are taken, respectively, as ex- treme events or flow configurations that dominate the probability measures in the weak coupling limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' They have been successfully applied to a number of fluid dy- namic problems, as in geophysical models, homogeneous turbulence and the laminar-turbulent transition in shear flows [15–17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' We conclude by noting that the findings here presented are likely to add relevant phenomenological information to the discussion of fundamentally important issues in pipe flow turbulence, as drag control and particle-laden dynamics, once they are closely connected to the statis- tical features of near-wall coherent structures [18–23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' Acknowledgments This work was partially supported by the Conselho Nacional de Desenvolvimento Cient´ıfico e Tecnol´ogico (CNPq) and by Funda¸c˜ao Coppetec/UFRJ (project num- ber 20459).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' thanks E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' Marensi for enlightening dis- cussions about the phenomenology of traveling waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' 6 [1] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' 91, 224502 (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' [6] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' Wedin and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' Kerswell, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' Smits, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' 12, 5805 (2021) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' [21] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' Gallorini, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' Quadrio, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' Gatti, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' Fluids 7, 114602 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' [22] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' Wang and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' Richter, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' 861, 901 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' [23] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' Brandt and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' Coletti, Annu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} +page_content=' 54, 159 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E4T4oBgHgl3EQf8A5c/content/2301.05344v1.pdf'} diff --git a/KNAyT4oBgHgl3EQff_hn/content/tmp_files/2301.00350v1.pdf.txt b/KNAyT4oBgHgl3EQff_hn/content/tmp_files/2301.00350v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..67d517afc0d6fbf19439e28d1fd2d9d4840cbe18 --- /dev/null +++ b/KNAyT4oBgHgl3EQff_hn/content/tmp_files/2301.00350v1.pdf.txt @@ -0,0 +1,894 @@ +Rawat and Soni et. al. 2023 +1 + + Anisotropic Light-Matter Interactions in Single Crystal +Topological Insulator Bismuth Selenide +Divya Rawat, Aditya Singh, Niraj Kumar Singh and Ajay Soni* +School of Physical Sciences, Indian Institute of Technology Mandi, Mandi, 175005, HP India +*Author to whom correspondence should be addressed: ajay@iitmandi.ac.in + +Anisotropy of light-matter interactions in materials give remarkable information about the +phonons and their interactions with electrons. We report the angle-resolved polarized Raman +spectroscopy of single-crystal of Bi2Se3 to obtain the elements of Raman tensor for understanding +the strength of polarization along different crystallographic orientations. Intensity variation in the +polar plots corresponding to 𝐸𝑔 +1 ~ 37 cm-1, 𝐴1𝑔 +1 ~71 cm-1, 𝐸𝑔 +2 ~ 130 cm-1, and 𝐴1𝑔 +2 ~ 173 cm-1 +suggests the higher differential polarizability along cross-plane (bc-plane). The polar patterns and +the differences in elements of the Raman tensor provides the evidence of the fundamental electron- +phonon and anisotropic light matter interactions in Bi2Se3. + +Keywords: Bismuth Selenide, Anisotropic behaviour, Polarization Raman spectroscopy, Raman +tensor, Electron-phonon interactions + + + +Rawat and Soni et. al. 2023 +2 + +I. +INTRODUCTION +Light-matter interaction helps to understand the many body physics and fundamentals of +the electron and phonon coupling in materials.[1,2] Exploring the optical properties can provide +significant understanding of the (an)-isotropic interaction of light along with the electronic +susceptibility and permittivity (dielectric constant) of the materials. [3,4] Generally, the electric +field vector (𝐸⃗ ) of the incident and the scattered light are related through a complex matrix, known +as Raman tensor (Ʈ) associated with the polarizability (α) of materials along three crystallographic +orientations.[5] Recently, several layered materials such as MoS2 [6], WS2 , MoSe2 [5], PdTe2 [7] +have been studied using Raman spectroscopy by controlling the polarization vector of incident and +scattered light, to understand the dynamics of phonons along the different orientation of the crystal. +Layered chalcogenide materials have been known for their anisotropic carrier relaxation times, +which mainly arises due to their intriguing crystal structures and inherent anharmonicity.[8,9] +Additionally, the Raman studies on ternary chalcogenides, Bi2GeTe4, Sb2SnTe4 have shown that +electronic topological properties can also be coupled with phonons, which has been shown by the +anomalous thermal behaviour of the Raman modes associated with bonds involved heavy elements. +[8] Though several chalcogenide quantum materials have been explored extensively for their exotic +electronic phenomena such as Shubnikov-de Haas quantum oscillations, [10] weak +(anti)localization [11], thermoelectricity, superconductivity, charge-density waves and topological +quantum insulating properties, yet the coupling of their topological electrons with phonons is less +explored. [12-14] Bi2Se3 is one of the layered chalcogenides which has a fascinating layered crystal +structure of five atoms (quintuple layers) stacked with van der Waals (vdWs) gaps and a crystal +unit cell is composed of three quintuple layers. [15] Primarily, the topological studies on Bi2Se3 has +a focus on investigating surface and bulk electronic structures using magneto-transport and angle- +resolved photoemission spectroscopy studies, phonon dispersion, [16-19], but there are +imperceptible reports on the anisotropic response of the inelastic light scattering. Since the +topological quantum phenomena are associated with electrons, electron-phonon and electron- +photon interactions [3,20], thus the investigation of the anisotropy of the electron-phonon-photon +interaction, dynamics of phonon and evaluation of Raman-tensor are very important to explore. In +this regard, the polarized Raman spectroscopy can provide a significant information about the light +sensitive responses of single crystals along various orientations by controlling the polarization of +both the incident and scattered photons to acquire the evidences of electron-phonon interactions +and anisotropic behaviour. [21] In this work, we have discussed the angle resolved polarized Raman +spectroscopy (APRS) to corroborate the interaction between the polarized light (𝑘𝑖) and the + +Rawat and Soni et. al. 2023 +3 + +crystallographic orientation of the single crystal Bi2Se3. The isotropic and anisotropic behaviour of +phonons are studied with the rotation of crystal along two different configurations in ab-plane +(𝑘𝑖||c-axis) and bc-plane (𝑘𝑖||a-axis), respectively. The observed anisotropic behaviour and +polarizability of in-plane (𝐸𝑔) and out-of-plane (𝐴1𝑔) modes are quantified from the Raman tensor’s +elements. Our results open the opportunities to understand the role of anisotropic light-matter and +electron-phonon interactions by both classical as well as quantum treatment of the Raman tensors +obtained from the APRS analysis. The experimental details of synthesis and characterization of the +single crystal are mentioned in supplemental materials.[22] + + +FIG. 1. (a) Electron microscopy image of the fractured cross section of layered Bi2Se3, (b) +Powder X-ray diffraction pattern of single crystal showing the typical orientation along the +c-axis, (inset: photograph of the grown sample), (c) Schematic of the crystal structure +comprises of quintuple layers stacked with a weak Van der Waals gap, (d) Normalized +Raman spectra and (e) Schematic of the atomic displacements of the 𝐸𝑔, and 𝐴1𝑔 modes. + +The layered nature of the grown Bi2Se3 is shown in FESEM image (Fig. 1 (a)) and the XRD +pattern in Fig. 1 (b), which confirms the orientation of the grown sample along c-axis.[23] Rietveld +refinement of the XRD pattern of powdered Bi2Se3 provides the lattice parameters a =b ~ 4.13 Å, +c ~ 28.63 Å, and unit cell volume (V) ~ 425 Å3, (Fig. S1 of supplemental materials [22]). The +residual resistance ratio (RRR ~ 2.11) has been evaluated from the low temperature resistance +measurement (Fig. S2 of supplemental materials [22]), which shows a generate electron transport +in a high quality of single crystal. [22] Bi2Se3 crystallizes in a rhombohedral crystal structure with + +(0) +(o) +(d) +ntensity (arb.units) +2) +(b) +20 +160 +200 +Intensity (arb.units) +Ramanshift (cm*) +(e) +600 +( +20 (deg)Rawat and Soni et. al. 2023 +4 + +space group R3̅m (166), which is comprised of quintuple layers (SeI-Bi-SeII-Bi-SeI) separated by +weak vdW gap represented in Fig. 1(c). Here, SeI and SeII represents the different chemical +environment of Se atoms in the unit cell. [24,25] The primitive unit cell of Bi2Se3 has fifteen zone- +center vibrational modes, three acoustic and twelve optical, which can be represented by: Г = + 2𝐸𝑔 + 2𝐴1𝑔 + 2𝐸𝑢 + 2𝐴1𝑢, where 𝐴1𝑔 and doubly degenerate 𝐸𝑔 are Raman active modes, +whereas 2𝐴1𝑢, 2𝐸𝑢 are the infra-red active modes.[24] The normalized room temperature Raman +spectra, having modes at ~ 37 cm-1 (𝐸𝑔 +1), ~ 71 cm-1 (𝐴1𝑔 +1 ), ~ 130 cm-1 (𝐸𝑔 +2), and ~ 173 cm-1 (𝐴1𝑔 +2 ), is +shown in Fig. 1(d) and the corresponding schematics of atomic displacements are illustrated in Fig. +1(e). The modes 𝐴1𝑔 +1 (𝐴1𝑔 +2 ) and 𝐸𝑔 +1 (𝐸𝑔 +2) have a different polarizability as they involve the out-of- +plane and in-plane displacements in symmetric (anti-symmetric) stretching, respectively. Thus, +angle-resolved polarized spectra (APRS) is an important tool to provide the detailed information +on the interaction of the light along the different orientations of the crystal for estimation of +elements of Raman tensor. + +FIG. 2. Schematic representation of the two configurations used for APRS studies on Bi2Se3 +crystal, where polarized laser (ki) incidents along (a) c-axis (on ab-plane) and (b) normal to c- +axis (bc -plane). Here, ω and θ correspond to the angle between electric polarization vector (𝑒𝑖) +of incident light with a-axis (in ab-plane) and b-axis (in bc-plane), respectively. + + +(a) +(inab-plane) +(b) +(in bc-plane) +532nm Laser +532nmLaser +kill = (c-axis) +E(e) +Sn- +E(e) +D +x(a-axis) +y(b-axis) +xisRawat and Soni et. al. 2023 +5 + +Fig. 2 represents the two configurations used for the APRS measurements, where +crystallographic axes a, b, and c are taken as equivalent to x, y, and z axes of rotating stage. For the +first configuration (Fig. 2 (a)), the incident laser (ki) is parallel to the c-axis and electric polarization +vector (𝑒𝑖) is making an angle ω with the a-axis (in ab-plane). Hence, the scattering configuration +is defined as z(xx)𝑧̅, and the corresponding polarization vector of incident and scattered light are 𝑒𝑖⃗⃗ += 𝑒𝑠 +⃗⃗⃗ = (cos ω, sin ω, 0). For the second configuration (Fig. 2 (b)), the incident laser (ki) is parallel +to a-axis and electric polarization vector (𝑒𝑖) is making an angle θ with the b-axis (in bc-plane). +Correspondingly, the scattering configuration is defined as x(yy)𝑥̅ and the polarization vector of +incident and scattered light are 𝑒𝑖⃗⃗ = 𝑒𝑠 +⃗⃗⃗ = (0, cos θ, sin θ). Being isotropic in ab-plane, Bi2Se3 crystal +does not have any changes in intensity along a and b axes while the anisotropic light-matter +interactions along c axis and the details of Raman tensor is not reported in the literature. + +FIG. 3. Angle dependent polarized Raman spectra (a-b) and corresponding polarized Raman +colour plot with the rotation of the Bi2Se3 sample in parallel configuration of polarized +incident (ei) and scattered (es) light along ab as well as bc-plane. Colour scale on the right +side shows the intensity variation of Raman modes. + +Polarized Raman spectra with the rotation of crystal along both ab(/bc)-plane and +corresponding colour plot is shown in Fig. 3. The intensity of 𝐴1𝑔 +1 (𝐴1𝑔 +2 ) and 𝐸𝑔 +1 (𝐸𝑔 +2) modes are not +changing along ab-plane (Fig. 3 (a)), whereas a periodic alteration has been observed along bc- + +(a) +linensity (ab.anits) +ab-plane +0.0 +Intensity (arb.units) +150 +61 +006 +12 +600 +59.6 +300 +01.9 +30 +60 +06 +120 +150 +180 +210 +204060 +80100120146160180 +4.00 +Ramanshift(cm +o (deg) +(b) +bc-plane +Intensity (ab.nits) +Intensity (arb.units) +200 +2700 +2139 +1800 +1T +900 +30 +60 +90 +120150 +180210 +Ramanshift(cm) +0 (deg) +10012140106-189Rawat and Soni et. al. 2023 +6 + +plane (Fig. 3 (b)). The results indicate that there is an existence of anisotropy along the bc-plane as +compared to ab-plane, which can be examined clearly from polar plots. According to classical +treatment of Raman tensor, the inelastic process can be explained by the scattering from an extended +medium, where the variations of the polarization can be expressed as a derivative of the +susceptibility of the materials.[21] The contribution of such spatial symmetry to the Raman +scattering intensity (I) can be expressed as ⟨𝑒𝑖|Ʈ|𝑒𝑠⟩2, where Ʈ is the Raman tensor for a given +mode. [24] Thus, the elements of Raman tensor of 𝐴1𝑔 and double degenerate 𝐸𝑔 modes can be +represented as: +Ʈ (𝐴1𝑔) = [ +ƞ𝑒𝑖∅ƞ +0 +0 +0 +ƞ𝑒𝑖∅ƞ +0 +0 +0 +𝛽𝑒𝑖∅𝛽 +], +Ʈ (𝐸𝑔) = [ +𝛾𝑒𝑖∅𝛾 +0 +0 +0 +−𝛾𝑒𝑖∅𝛾 +𝛿𝑒𝑖∅𝛿 +0 +𝛿𝑒𝑖∅𝛿 +0 +] ; [ +0 +−𝛾𝑒𝑖∅𝛾 +−𝛿𝑒𝑖∅𝛿 +−𝛾𝑒𝑖∅𝛾 +0 +0 +−𝛿𝑒𝑖∅𝛿 +0 +0 +], +Here the values corresponding to ƞ, β, γ, and δ indicate the amplitudes whereas ∅ƞ, ∅𝛽, ∅𝛾, and ∅𝛿 +are the complex phases of the elements of Raman tensor. [21] Additionally, the magnitude of each +tensor element is related with the specific mode and the crystal symmetry of the material. The +calculated intensities for the estimation of the Ʈ (𝐸𝑔) has contributions from both the doubly +degenerate 𝐸𝑔 modes, thus added altogether. Using the Raman selection rule, |⟨𝑒𝑖|Ʈ∗|𝑒𝑠⟩|2, under +both ab(/bc)-plane, the scattering intensity of all modes have been calculated (Table I), which +clearly showed the distinct strength of interaction of polarized light with the crystal’s axes. +[5,6,26,27] +TABLE I. Mathematically derived intensity of modes using Raman selection rules. +S.no. Configuration Raman scattering intensity +1. ab-plane 𝑰𝑨𝟏𝒈 +|| +(ki||c-axis) = |ƞ|𝟐 + 𝑰𝑬𝒈 +|| (ki||c-axis) = |𝜸|𝟐 +2. bc-plane 𝑰𝑨𝟏𝒈 +|| +(ki||a-axis) = |ƞ|𝟐𝒔𝒊𝒏𝟒𝜽 + |𝜷|𝟐𝒄𝒐𝒔𝟒𝜽 + +𝟏 +𝟐 |ƞ||𝜷|𝒔𝒊𝒏𝟐(𝟐𝜽)𝒄𝒐𝒔𝝋ƞ𝜷 + 𝑰𝑬𝒈 +|| (ki||a-axis) = |𝜸|𝟐𝒄𝒐𝒔𝟒𝜽 + |𝜹|𝟐𝒔𝒊𝒏𝟐𝟐𝜽 − |𝜹||𝜸| 𝐬𝐢𝐧(𝟐𝜽) 𝒄𝒐𝒔𝟐𝜽  𝟐𝒄𝒐𝒔𝝋𝜸𝜹 + + +Rawat and Soni et. al. 2023 +7 + + +FIG. 4. Intensities of polar plots for 𝐴1𝑔 +1 , 𝐴1𝑔 +2 , 𝐸𝑔 +1, 𝐸𝑔 +2 modes in ab-plane (a-b), and in bc-plane +(c-f). Here, solid symbols and green line represent the experimental data fitting of the data using +equation in Table I, respectively. + +Further, the understanding of the isotropic behaviour along ab-plane of the intensity of 𝐴1𝑔 +and 𝐸𝑔 modes are depicted as circular shapes of the polar intensity plots (Fig. 4 (a-b)). On the other +hand, the shape of polar plots for 𝐴1𝑔 (Fig. 4 (c-d)) and 𝐸𝑔 (Fig. 4 (e-f)) modes along bc-plane are +different from ab-plane showing the anisotropy of the light matter interaction along crystallographic +orientation. The intensities of all modes are stronger along bc-plane in comparison to the ab-plane, +which advocates the higher differential polarizability along bc-plane. Similar observations on the +anisotropic light-matter interaction in bc-plane have been reported for Graphene, hBN, 2H- MoSe2, +MoS2.[5,6,28] Fascinatingly, the out of plane modes at ~ 71 cm-1 and ~ 173 cm-1, (Fig. 4 (c-d)), +have 𝐴1𝑔 symmetry but showing considerably different polar pattern at 90o and 270o rotations. The +anomalous polarization dependence of the Raman intensities appeared because of the difference in +Raman scattering cross-section through the second-order susceptibility or the electron–phonon +interactions.[21] +To understand the discrepancy, the microscopic quantum description of Raman tensor has +been adopted, which involved the electron-phonon interaction in addition to the electron-photon. +[29] Here, the total Raman intensity is described by the product of both the electron-photon and + +in ab-plane +inbc-plane +(a) +90 +Ai +fcj +120 +120 +06 +60 +(e) +120 +90 +1200 +60 +AT +3600 +AI +300 +50 +(Sun +800 +150 +30 +2400 +150 +30 +200 +150 +30 +400 +1200 +100 +Intensity(arb. +0480 +10 +0180 +40 +0/180 +400 +1200 +100 +008 +210 +330 +2400 +210 +330 +200 +210 +330 +1200 +240 +300 +3600 +240 +300 +240 +270 +270 +270 +300 +(b) +90 +E +(p) +() +120 +60 +06 +120 +60 +2400 +120 +90 +360F +a +E +1800/ +(sun +240 +150 +1800 +1200 +150 +30 +1200 +150 +F +30 +120 +600 +Intensity(arb. +600 +0180 +0480 +0180 +120 +600 +600 +240 +210 +330 +1200 +210 +1200 +1800 +210 +330 +360 +240 +300 +1800 +270 +240 +270 +300 +2400 L +240 +270 +300Rawat and Soni et. al. 2023 +8 + +electron-phonon interactions. Hence, the Raman tensor (Ʈ𝑖𝑗 +𝑘 ) associated with all modes can be given +by: +Ʈ𝑖𝑗 +𝑘 = 1 +𝑉 ∑ +∑ +⟨𝛹𝑣(𝑞 )|𝑒 𝑠. ∇⃗⃗ |𝛹𝑐′(𝑞 )⟩ ⟨𝛹𝑐′(𝑞 )|𝐻𝑒𝑝 +𝑘 |𝛹𝑐(𝑞 )⟩⟨𝛹𝑐(𝑞 )|𝑒 𝑖. 𝛻⃗ |𝛹𝑣(𝑞 )⟩ +(𝐸𝐿 − 𝐸𝑐𝑣(𝑞 ) − 𝑖𝛤𝑐)(𝐸𝐿 − ћ𝜔𝑝ℎ +𝑘 − 𝐸𝑐′𝑣(𝑞 ) − 𝑖𝛤𝑐′) +𝑞′ +𝑣,𝑐,𝑐′ + +Here, the numerator consists of the product of three matrix elements, (i) the electron-phonon (e-ph) +matrix elements (⟨𝛹𝑐′(𝑞 )|𝐻𝑒𝑝 +𝑘 |𝛹𝑐(𝑞 )⟩) and two electron-photon matrix elements for incident and +scattered light (ii) (⟨𝛹𝑐(𝑞 )|𝑒 𝑖. 𝛻⃗ |𝛹𝑣(𝑞 )⟩, (iii) ⟨𝛹𝑣(𝑞 )|𝑒 𝑠. ∇⃗⃗ |𝛹𝑐′(𝑞 )⟩), where 𝑒 𝑖 and 𝑒 𝑠 are the +polarization vectors of incident and scattered light, respectively.[29] The summation is over the +electronics branches in conduction (𝑐, 𝑐’) and valance (𝑣) bands along with all wave vectors with +first Brillouin zone. 𝛤𝑐 and 𝛤𝑐′ are the broadening factor associated with the lifetime of photo-excited +states. The inclusion of e-ph matrix element gives the major differences among both the out of plane +𝐴1𝑔 modes. Thus, different patterns of polar plots for 𝐴1𝑔 +1 , and 𝐴1𝑔 +2 modes indicate electron-phonon +interactions in Bi2Se3, similar to the observations in other anisotropic layered chalcogenides like +WS2, ReS2, GaTe, PdSe2 and black phosphorus.[21,29-31] In contrast to the 𝐴1𝑔 modes, the polar +plots of 𝐸𝑔 modes show four-lobbed polar pattern (Fig. 4 (e-f)) with the rotation of the crystal, +which indicates the maximum strength of anisotropic nature in bc-plane. To understand the +behaviour of polar plots related to 𝐸𝑔 modes, the spectra have been captured by controlling the +polarization of incident light (ei). This configuration is done by rotating half wave plate from 0o to +360o while keeping sample stage and analyzer fixed (Fig. S3 of supplemental materials. [22]) Here, +the intensity of both 𝐴1𝑔 modes (Fig. S3 (a-b) of supplemental materials [22]) showed analogous +polar pattern with polarization angle, whereas 𝐸𝑔 modes (Fig. S3 (c) of supplemental materials [22]) +exhibited a low dependency on the rotation of the half wave plate. This discrepancy of the 𝐸𝑔 modes +between the rotation of crystallographic axis and incident laser suggest the anisotropic behaviour +along bc-plane. [5,6] Anisotropic light-matter interaction has been understood by estimating the +amplitude and phase difference of Raman tensor’s element, which mainly contain the information +of differential polarizability along different orientation. To estimate the Raman tensor elements of +all modes, we have fitted the experimental data (in Fig 4) using the intensity’s expressions given in +Table I and the obtained details are presented in Table II. + +Rawat and Soni et. al. 2023 +9 + +TABLE II. Estimated Raman tensor elements obtained from the fitting of experimental data (Fig 4). + Modes Raman tensor + ab-plane bc-plane + + 𝑨𝟏𝒈 +𝟏 [ +𝟑𝟎 +𝟎 +𝟎 +𝟎 +𝟑𝟎 +𝟎 +𝟎 +𝟎 +𝜷 +] [ +𝟑𝟓 +𝟎 +𝟎 +𝟎 +𝟑𝟓 +𝟎 +𝟎 +𝟎 +𝟓𝟕𝒆𝒊𝟎.𝟑𝟕𝝅 +] + + 𝑨𝟏𝒈 +𝟐 [ +𝟏𝟕 +𝟎 +𝟎 +𝟎 +𝟏𝟕 +𝟎 +𝟎 +𝟎 +𝜷 +] [ +𝟐𝟏 +𝟎 +𝟎 +𝟎 +𝟐𝟏 +𝟎 +𝟎 +𝟎 +𝟒𝟏𝒆𝒊𝟎.𝟐𝟒𝝅 +] + + 𝑬𝒈𝟏 [ +𝟖 +−𝟖 +𝛅 +−𝟖 +−𝟖 + 𝛅 +𝛅 +𝛅 + 𝟎 +] [ +𝟖 +−𝟖 +−𝟏𝟑𝒆𝒊𝟎.𝟑𝟗𝝅 +−𝟖 +−𝟖 +𝟏𝟑𝒆𝒊𝟎.𝟑𝟗𝝅 +−𝟏𝟑𝒆𝒊𝟎.𝟑𝟗𝝅 +𝟏𝟑𝒆𝒊𝟎.𝟑𝟗𝝅 +𝟎 +] + + 𝑬𝒈𝟐 [ +𝟏𝟔 +−𝟏𝟔 +𝛅 +−𝟏𝟔 +−𝟏𝟔 + 𝛅 +𝛅 +𝛅 + 𝟎 +] [ +𝟏𝟒 +−𝟏𝟒 +−𝟑𝟖𝒆𝒊𝟎.𝟑𝟐𝝅 +−𝟏𝟒 +−𝟏𝟒 +𝟑𝟖𝒆𝒊𝟎.𝟑𝟐𝝅 +−𝟑𝟖𝒆𝒊𝟎.𝟑𝟐𝝅 +𝟑𝟖𝒆𝒊𝟎.𝟑𝟐𝝅 +𝟎 +] + + +In ab-plane, all modes show isotropic behaviour (Fig 4a and 4b), hence for Ʈ (𝐴1𝑔) and Ʈ (𝐸𝑔), the +component of Raman tensor, ƞ (𝐴1𝑔 +1 ~ 30 and 𝐴1𝑔 +2 ~ 17) and γ (𝐸𝑔 +1~ 8 and 𝐸𝑔 +2 ~ 16), have been +evaluated from the fitting of polar plots. As the propagation vector ki of incident light is along the +c-axis, there is no polarization along c-axis, thus, 𝛽 for out of plane 𝐴1𝑔 mode is not evaluated while +𝛽 is zero for in-plane 𝐸𝑔 modes. Here, the phase factor (∅ƞ) is zero due to isotropic responses in ab- +plane. On the other hand, in bc-plane (Fig. 4c and 4d), the component of Raman tensor, ƞ (𝐴1𝑔 +1 ~ 35 +and 𝐴1𝑔 +2 ~ 21) and 𝛽 (𝐴1𝑔 +1 ~ 57 and 𝐴1𝑔 +2 ~ 41) have been evaluated and the phase factor between ƞ +and 𝛽 (∅ƞ𝛽) is ~ 67.3o (0.37𝜋) for (𝐴1𝑔 +1 ) and ~ 44o (0.24𝜋) for (𝐴1𝑔 +2 ), which is arising due to the +anisotropic responses. Additionally, the elements of Raman tensor for in-plane modes are 𝛾 (𝐸𝑔 +1 ~ +8 and 𝐸𝑔 +2 ~ 14) and 𝛿 (𝐸𝑔 +1 ~ 13 and 𝐸𝑔 +2 ~ 38) and the phase factor between 𝛾 and 𝛿 (∅𝛾𝛿) is ~ 71o +(0.39𝜋) for (𝐸𝑔 +1) and ~ 58.4o (0.32𝜋) for (𝐸𝑔 +2). Overall, for out of plane 𝐴1𝑔 modes, 𝛽 > ƞ, (57 > +35 for 𝐴1𝑔 +1 and 41 > 21 for 𝐴1𝑔 +2 ), which indicates that differential polarizability is significantly +higher and anisotropic along c-axis (schematic Fig 1e). By comparing the tensor matrices of out of +plane modes, it is clearly evident that symmetric stretching (𝐴1𝑔 +1 ) induces larger dipole moment +(higher polarizability) than anti- symmetric stretching (𝐴1𝑔 +2 ) and the situation is completely +otherwise for in-plane modes 𝐸𝑔 +1 and 𝐸𝑔 +2 as confirmed by the smaller magnitude of Raman tensor +elements in Table II. For both the ab- and bc-plane, the comparison of relative magnitude of Raman +tensor elements for of 𝐴1𝑔 +1 (|ƞ𝑏𝑐−𝑝𝑙𝑎𝑛𝑒 ƞ𝑎𝑏−𝑝𝑙𝑎𝑛𝑒 +⁄ +|~ 1.16) and 𝐸𝑔 +2 (|𝛾𝑏𝑐−𝑝𝑙𝑎𝑛𝑒 𝛾𝑎𝑏−𝑝𝑙𝑎𝑛𝑒 +⁄ +|~ 1.14), + +Rawat and Soni et. al. 2023 +10 + +which authenticate the estimated elements of the Raman tensor. [6] Comparing the APRS estimated +Raman tensor elements with studies on MoSe2, MoS2, WSe2, PdTe2, it is clear that the laser +polarization dependence Raman spectra demonstrates the anisotropic light-matter interactions in +Bi2Se3. +In Summary, the Raman tensor for all modes of single crystal Bi2Se3 corresponds to 𝐸𝑔 +1 ~ +37 cm-1, 𝐴1𝑔 +1 ~70 cm-1, 𝐸𝑔2 ~ 129 cm-1, and 𝐴1𝑔 +2 ~ 172 cm-1 have been systematically studied by APRS +measurements along both ab(/bc)-plane under parallel polarization (𝑒𝑖 ∥ 𝑒𝑠) scattering +configuration. We have estimated the amplitude and phase difference of the tensor elements by +fitting the experimental results with the intensity expression obtained by applying Raman selection +rule. The different shapes of polar plot of the similar vibrational symmetry (𝐴1𝑔) represents the +different interaction of electrons with phonons, which provide the evidence of electron-phonon +coupling. Among two different orientations (ab(/bc)-plane) of single crystal, strong polarization +dependence has been observed along bc-plane for both 𝐴1𝑔 and 𝐸𝑔 modes, which is showing the +anisotropic light matter interaction in Bi2Se3. +Acknowledgement +We would like to thank IIT Mandi for the instruments and research facilities. A.S would like to +acknowledge DST-SERB for funding (Grant No. CRG/2018/002197). +References + +[1] +C. Grazianetti, C. Martella, and E. Cinquanta, Optical Materials: X 12, 100088 (2021). +[2] +Y. Xu et al., Advanced Optical Materials 6, 1800444 (2018). +[3] +Y. Xia et al., Nature Physics 5, 398 (2009). +[4] +J. Singh, Optical properties of condensed matter and applications (John Wiley & Sons, +2006), Vol. 6. +[5] +M. Jin et al., The Journal of Physical Chemistry Letters 11, 4311 (2020). +[6] +Y. Ding et al., Optics Letters 45 (2020). +[7] +L. Pi et al., Advanced Functional Materials 29, 1904932 (2019). +[8] +N. K. Singh et al., Physical Review B 105, 045134 (2022). +[9] +J. P. Heremans, Nature Physics 11, 990 (2015). +[10] +Z. Ren et al., Physical Review B 82, 241306 (2010). +[11] +N. K. Singh, A. Kashyap, and A. Soni, Applied Physics Letters 119, 223903 (2021). +[12] +J. E. Moore, Nature 464, 194 (2010). + +Rawat and Soni et. al. 2023 +11 + +[13] +J. Pandey and A. Soni, Physical Review Research 2, 033118 (2020). +[14] +S. Acharya, J. Pandey, and A. Soni, Applied Physics Letters 109, 133904 (2016). +[15] +W. Richter and C. R. Becker, physica status solidi b 84, 619 (1977). +[16] +Y. Kim et al., Applied Physics Letters 100, 071907 (2012). +[17] +A. C. Ferrari and D. M. Basko, Nature nanotechnology 8, 235 (2013). +[18] +S. R. Park et al., Physical Review Letters 108, 046805 (2012). +[19] +S. Sharma et al., Physical Review B 105, 115120 (2022). +[20] +M. Z. Hasan and C. L. Kane, Reviews of Modern Physics 82, 3045 (2010). +[21] +J. Kim, J.-U. Lee, and H. Cheong, Journal of Physics: Condensed Matter 32, 343001 +(2020). +[22] +See Supplementary Material..... for Synthesis, characterization and details of Raman +tensor. +[23] +K. Mazumder and P. M. Shirage, Journal of Alloys and Compounds 888, 161492 (2021). +[24] +J. Zhang et al., Nano Letters 11, 2407 (2011). +[25] +A. Soni et al., Nano Letters 12, 1203 (2012). +[26] +Y. Ding et al., The Journal of Physical Chemistry Letters 11, 10094 (2020). +[27] +J. R. Ferraro, K. Nakamoto, and C. W. Brown, in Introductory Raman Spectroscopy, +edited by J. R. Ferraro, K. Nakamoto, and C. W. Brown (Academic Press, San Diego, 2003), pp. +1. +[28] +J. Ribeiro-Soares et al., Physical Review B 90, 115438 (2014). +[29] +G. C. Resende et al., 2D Materials 8, 025002 (2020). +[30] +Y. Ding et al., Science China Materials 63, 1848 (2020). +[31] +S. Huang et al., ACS Nano 10, 8964 (2016). + + + + +1 + +Supplemental Material +Anisotropic Light-Matter Interactions in Single Crystal +Topological Insulator Bismuth Selenide +Divya Rawat, Aditya Singh, Niraj Kumar Singh and Ajay Soni* +School of Physical Sciences, Indian Institute of Technology Mandi, Mandi, 175005, HP India +*Author to whom correspondence should be addressed: ajay@iitmandi.ac.in + +In this supplemental file, we are providing the details of the synthesis, characterization techniques +and selected data complementing the main text. + +(a) Synthesis and characterization details. +Single crystal of Bi2Se3 was synthesized using dual zone vertical Bridgman furnace, by taking a +stoichiometric amounts of bismuth ingot and selenium shots (both 99.999% pure) in a quartz +ampoule, which was then vacuum sealed at 10-5 mbar. The ampoule was kept in a box furnace at +1123 K for 15 hr for homogenization followed by hanging it in Bridgman furnace. The temperature +of the hot zone and cold zone were kept at 1003 K and 953 K, respectively. The translation rate of +the motor for the vertical motion of quartz tube from hot zone to cold zone was fixed at 2 mm/hr. +X-ray diffraction (XRD) was carried out using rotating anode Rigaku SmartLab diffractometer +equipped with CuKα radiation (λ = 1.5406 Å) and in Bragg-Brentano geometry. Rietveld +refinement of the Powder-XRD pattern was done to determine the crystal structure, lattice +parameter, and phase purity. Resistance measurement was performed in the temperature range of +2 to 300 K using Quantum Design make physical properties measurement system (PPMS). Raman +spectroscopy measurements were carried out using a Horiba LabRAM HR Evolution Raman +spectrometer having 532 nm laser excitation, 1800 grooves/mm with the help of a Peltier cooled +(CCD) detector. Ultra-low frequency filters were used to access low-frequency spectra, very close +to laser line. To control the polarization state, a (λ/2) half-waveplate and an analyzer were used +before objective lens and spectrometer to select the desired polarization component of the incident + +2 + +and scattered light, respectively. To study the light-matter interaction on the crystallographic axis +of Bi2Se3, the sample was kept on the stage rotating from ~ 0o to ~ 360o with a step of ~ 20o. The +linearly polarized laser was directed on the sample and the scattered radiation was collected to the +detector in backscattering geometry. +(b) Rietveld refinement analysis. + + +FIG. S1:- Rietveld refined XRD pattern of single crystal Bi2Se3. Black closed circle represents +the experimental data point, Solid red line represents the refined data. + +The as-synthesized Bi2Se3 crystal was ground into fine powder for XRD analysis. The phase purity +of Bi2Se3 sample has been confirmed by Rietveld refinements of the powder XRD pattern. [1]The +Fig. S1 shows the Rietveld refined XRD data. Goodness of fitting was showed by the extracting +parameter, χ2 ~ 2.9. + + + +Observed +Simulated +Intensity (arb.units.) +Difference +Braggposition +10 +20 +30 +40 +50 +60 +70 +80 +90 +20 (deg)3 + +c) Resistance data of single crystal Bi2Se3: + + +FIG.S2:- Four-probe resistance measurement with the variation of the temperature. + +The electronic transport of the Bi2Se3 has been examined by the four probe resistance (R) and the +temperature dependence is consistent with the behavior of degenerate semiconductors. The +longitudinal resistance (R) is fitted using a phenomenological model: R = R0 + λe−θ /T + $T2, where +the λ and $ appear for phonon scattering and electron-electron scattering, respectively.[2,3] The +fitting parameter are evaluated and λ ~ 12× 10−3and $ ~ 1.74 × 10−7 𝐾−2, where smaller value of +$ suggests negligible electron-electron scattering in Bi2Se3. The residual resistance ratio (RRR ~ +2.11) shows a high quality of the single crystal. +(d) APRS spectra of Bi2Se3 in bc-plane with the rotation of polarization vector of incident +light while keeping the sample fixed. + +0.035 +Experimental data pount +9 +Fittingdata +0.020 +0 +50 +100 +150 +200 +250 +Temperature (K)4 + + +FIG. S3:- (a) APRS spectra and Polar plot of (b) 𝐴1𝑔 +1 (c) 𝐴1𝑔 +2 (d) 𝐸𝑔 +2 of Bi2Se3 single crystal +with the rotation of half-wave plate by keeping the sample fixed in bc-plane. Solid symbols +represent the experimental data point. +APRS measurements has been performed in in parallel configuration (𝑒𝑖 ∥ 𝑒𝑠), where polarization +vector of incident light has varied by rotating the half-wave plate, while keeping the stage of +sample and analyzer fixed. Here, the intensity of both 𝐴1𝑔 modes showed expected two-lobed +analogous polar pattern polar pattern with polarization angle. 𝐸𝑔 mode showed a low dependency +on the rotation of the half wave plate, showed isotropic interaction on the rotation of polarization +vector of incident light. [4] +References +[1] +N. K. Singh, A. Kashyap, and A. Soni, Applied Physics Letters 119, 223903 (2021). +[2] +T. Kino, T. Endo, and S. Kawata, Journal of the Physical Society of Japan 36, 698 (1974). +[3] +N. K. Singh et al., Physical Review B 105, 045134 (2022). +[4] +J. Kim, J.-U. Lee, and H. Cheong, Journal of Physics: Condensed Matter 32, 343001 (2020). + +(b) +120 +90 +(a) +Ata +00 +150 +180 +210 +330 +240 +270 +300 +(arb.units) +E +120 +90 +E: +(c) +00 +00 +150 +40° +Intensity +1803 +800 +210 +900 +240 +300 +1000 +120 +90 +(d) +150 +10 +1400 +180 +1800 +210 +330 +30 +60 +90 +120 +150 +180 +240 +270 +300 +Ramanshift(cm1) \ No newline at end of file diff --git a/KNAyT4oBgHgl3EQff_hn/content/tmp_files/load_file.txt b/KNAyT4oBgHgl3EQff_hn/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2d7568993fb3820b449347a61ad4dbee9d1b3c83 --- /dev/null +++ b/KNAyT4oBgHgl3EQff_hn/content/tmp_files/load_file.txt @@ -0,0 +1,375 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf,len=374 +page_content='Rawat and Soni et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' 2023 1 Anisotropic Light-Matter Interactions in Single Crystal Topological Insulator Bismuth Selenide Divya Rawat, Aditya Singh, Niraj Kumar Singh and Ajay Soni* School of Physical Sciences, Indian Institute of Technology Mandi, Mandi, 175005, HP India Author to whom correspondence should be addressed: ajay@iitmandi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content='in Anisotropy of light-matter interactions in materials give remarkable information about the phonons and their interactions with electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' We report the angle-resolved polarized Raman spectroscopy of single-crystal of Bi2Se3 to obtain the elements of Raman tensor for understanding the strength of polarization along different crystallographic orientations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' Intensity variation in the polar plots corresponding to 𝐸𝑔 1 ~ 37 cm-1, 𝐴1𝑔 1 ~71 cm-1, 𝐸𝑔 2 ~ 130 cm-1, and 𝐴1𝑔 2 ~ 173 cm-1 suggests the higher differential polarizability along cross-plane (bc-plane).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' The polar patterns and the differences in elements of the Raman tensor provides the evidence of the fundamental electron- phonon and anisotropic light matter interactions in Bi2Se3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' Keywords: Bismuth Selenide, Anisotropic behaviour, Polarization Raman spectroscopy, Raman tensor, Electron-phonon interactions Rawat and Soni et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' 2023 2 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' INTRODUCTION Light-matter interaction helps to understand the many body physics and fundamentals of the electron and phonon coupling in materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' [1,2] Exploring the optical properties can provide significant understanding of the (an)-isotropic interaction of light along with the electronic susceptibility and permittivity (dielectric constant) of the materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' [3,4] Generally, the electric field vector (𝐸⃗ ) of the incident and the scattered light are related through a complex matrix, known as Raman tensor (Ʈ) associated with the polarizability (α) of materials along three crystallographic orientations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' [5] Recently, several layered materials such as MoS2 [6], WS2 , MoSe2 [5], PdTe2 [7] have been studied using Raman spectroscopy by controlling the polarization vector of incident and scattered light, to understand the dynamics of phonons along the different orientation of the crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' Layered chalcogenide materials have been known for their anisotropic carrier relaxation times, which mainly arises due to their intriguing crystal structures and inherent anharmonicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' [8,9] Additionally, the Raman studies on ternary chalcogenides, Bi2GeTe4, Sb2SnTe4 have shown that electronic topological properties can also be coupled with phonons, which has been shown by the anomalous thermal behaviour of the Raman modes associated with bonds involved heavy elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' [8] Though several chalcogenide quantum materials have been explored extensively for their exotic electronic phenomena such as Shubnikov-de Haas quantum oscillations, [10] weak (anti)localization [11], thermoelectricity, superconductivity, charge-density waves and topological quantum insulating properties, yet the coupling of their topological electrons with phonons is less explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' [12-14] Bi2Se3 is one of the layered chalcogenides which has a fascinating layered crystal structure of five atoms (quintuple layers) stacked with van der Waals (vdWs) gaps and a crystal unit cell is composed of three quintuple layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' [15] Primarily, the topological studies on Bi2Se3 has a focus on investigating surface and bulk electronic structures using magneto-transport and angle- resolved photoemission spectroscopy studies, phonon dispersion, [16-19], but there are imperceptible reports on the anisotropic response of the inelastic light scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' Since the topological quantum phenomena are associated with electrons, electron-phonon and electron- photon interactions [3,20], thus the investigation of the anisotropy of the electron-phonon-photon interaction, dynamics of phonon and evaluation of Raman-tensor are very important to explore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' In this regard, the polarized Raman spectroscopy can provide a significant information about the light sensitive responses of single crystals along various orientations by controlling the polarization of both the incident and scattered photons to acquire the evidences of electron-phonon interactions and anisotropic behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' [21] In this work, we have discussed the angle resolved polarized Raman spectroscopy (APRS) to corroborate the interaction between the polarized light (𝑘𝑖) and the Rawat and Soni et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' 2023 3 crystallographic orientation of the single crystal Bi2Se3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' The isotropic and anisotropic behaviour of phonons are studied with the rotation of crystal along two different configurations in ab-plane (𝑘𝑖||c-axis) and bc-plane (𝑘𝑖||a-axis), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' The observed anisotropic behaviour and polarizability of in-plane (𝐸𝑔) and out-of-plane (𝐴1𝑔) modes are quantified from the Raman tensor’s elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' Our results open the opportunities to understand the role of anisotropic light-matter and electron-phonon interactions by both classical as well as quantum treatment of the Raman tensors obtained from the APRS analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' The experimental details of synthesis and characterization of the single crystal are mentioned in supplemental materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' [22] FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' (a) Electron microscopy image of the fractured cross section of layered Bi2Se3, (b) Powder X-ray diffraction pattern of single crystal showing the typical orientation along the c-axis, (inset: photograph of the grown sample), (c) Schematic of the crystal structure comprises of quintuple layers stacked with a weak Van der Waals gap, (d) Normalized Raman spectra and (e) Schematic of the atomic displacements of the 𝐸𝑔, and 𝐴1𝑔 modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' The layered nature of the grown Bi2Se3 is shown in FESEM image (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' 1 (a)) and the XRD pattern in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' 1 (b), which confirms the orientation of the grown sample along c-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' [23] Rietveld refinement of the XRD pattern of powdered Bi2Se3 provides the lattice parameters a =b ~ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content='13 Å, c ~ 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content='63 Å, and unit cell volume (V) ~ 425 Å3, (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' S1 of supplemental materials [22]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' The residual resistance ratio (RRR ~ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content='11) has been evaluated from the low temperature resistance measurement (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' S2 of supplemental materials [22]), which shows a generate electron transport in a high quality of single crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' [22] Bi2Se3 crystallizes in a rhombohedral crystal structure with (0) (o) (d) ntensity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content='units) 2) (b) 20 160 200 Intensity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content='units) Ramanshift (cm*) (e) 600 ( 20 (deg)Rawat and Soni et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' 2023 4 space group R3̅m (166), which is comprised of quintuple layers (SeI-Bi-SeII-Bi-SeI) separated by weak vdW gap represented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' 1(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' Here, SeI and SeII represents the different chemical environment of Se atoms in the unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' [24,25] The primitive unit cell of Bi2Se3 has fifteen zone- center vibrational modes, three acoustic and twelve optical, which can be represented by: Г = 2𝐸𝑔 + 2𝐴1𝑔 + 2𝐸𝑢 + 2𝐴1𝑢, where 𝐴1𝑔 and doubly degenerate 𝐸𝑔 are Raman active modes, whereas 2𝐴1𝑢, 2𝐸𝑢 are the infra-red active modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' [24] The normalized room temperature Raman spectra, having modes at ~ 37 cm-1 (𝐸𝑔 1), ~ 71 cm-1 (𝐴1𝑔 1 ), ~ 130 cm-1 (𝐸𝑔 2), and ~ 173 cm-1 (𝐴1𝑔 2 ), is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' 1(d) and the corresponding schematics of atomic displacements are illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' 1(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' The modes 𝐴1𝑔 1 (𝐴1𝑔 2 ) and 𝐸𝑔 1 (𝐸𝑔 2) have a different polarizability as they involve the out-of- plane and in-plane displacements in symmetric (anti-symmetric) stretching, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' Thus, angle-resolved polarized spectra (APRS) is an important tool to provide the detailed information on the interaction of the light along the different orientations of the crystal for estimation of elements of Raman tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' Schematic representation of the two configurations used for APRS studies on Bi2Se3 crystal, where polarized laser (ki) incidents along (a) c-axis (on ab-plane) and (b) normal to c- axis (bc -plane).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' Here, ω and θ correspond to the angle between electric polarization vector (𝑒𝑖) of incident light with a-axis (in ab-plane) and b-axis (in bc-plane), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' (a) (inab-plane) (b) (in bc-plane) 532nm Laser 532nmLaser kill = (c-axis) E(e) Sn- E(e) D x(a-axis) y(b-axis) xisRawat and Soni et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' 2023 5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' 2 represents the two configurations used for the APRS measurements, where crystallographic axes a, b, and c are taken as equivalent to x, y, and z axes of rotating stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' For the first configuration (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' 2 (a)), the incident laser (ki) is parallel to the c-axis and electric polarization vector (𝑒𝑖) is making an angle ω with the a-axis (in ab-plane).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' Hence, the scattering configuration is defined as z(xx)𝑧̅, and the corresponding polarization vector of incident and scattered light are 𝑒𝑖⃗⃗ = 𝑒𝑠 ⃗⃗⃗ = (cos ω, sin ω, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' For the second configuration (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' 2 (b)), the incident laser (ki) is parallel to a-axis and electric polarization vector (𝑒𝑖) is making an angle θ with the b-axis (in bc-plane).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' Correspondingly, the scattering configuration is defined as x(yy)𝑥̅ and the polarization vector of incident and scattered light are 𝑒𝑖⃗⃗ = 𝑒𝑠 ⃗⃗⃗ = (0, cos θ, sin θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' Being isotropic in ab-plane, Bi2Se3 crystal does not have any changes in intensity along a and b axes while the anisotropic light-matter interactions along c axis and the details of Raman tensor is not reported in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' Angle dependent polarized Raman spectra (a-b) and corresponding polarized Raman colour plot with the rotation of the Bi2Se3 sample in parallel configuration of polarized incident (ei) and scattered (es) light along ab as well as bc-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' Colour scale on the right side shows the intensity variation of Raman modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' Polarized Raman spectra with the rotation of crystal along both ab(/bc)-plane and corresponding colour plot is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' The intensity of 𝐴1𝑔 1 (𝐴1𝑔 2 ) and 𝐸𝑔 1 (𝐸𝑔 2) modes are not changing along ab-plane (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' 3 (a)), whereas a periodic alteration has been observed along bc- (a) linensity (ab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content='anits) ab-plane 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content='0 Intensity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content='units) 150 61 006 12 600 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content='6 300 01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content='9 30 60 06 120 150 180 210 204060 80100120146160180 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content='00 Ramanshift(cm o (deg) (b) bc-plane Intensity (ab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content='nits) Intensity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content='units) 200 2700 2139 1800 1T 900 30 60 90 120150 180210 Ramanshift(cm) 0 (deg) 10012140106-189Rawat and Soni et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' 2023 6 plane (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' 3 (b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' The results indicate that there is an existence of anisotropy along the bc-plane as compared to ab-plane, which can be examined clearly from polar plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' According to classical treatment of Raman tensor, the inelastic process can be explained by the scattering from an extended medium, where the variations of the polarization can be expressed as a derivative of the susceptibility of the materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' [21] The contribution of such spatial symmetry to the Raman scattering intensity (I) can be expressed as ⟨𝑒𝑖|Ʈ|𝑒𝑠⟩2, where Ʈ is the Raman tensor for a given mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' [24] Thus, the elements of Raman tensor of 𝐴1𝑔 and double degenerate 𝐸𝑔 modes can be represented as: Ʈ (𝐴1𝑔) = [ ƞ𝑒𝑖∅ƞ 0 0 0 ƞ𝑒𝑖∅ƞ 0 0 0 𝛽𝑒𝑖∅𝛽 ], Ʈ (𝐸𝑔) = [ 𝛾𝑒𝑖∅𝛾 0 0 0 −𝛾𝑒𝑖∅𝛾 𝛿𝑒𝑖∅𝛿 0 𝛿𝑒𝑖∅𝛿 0 ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' [ 0 −𝛾𝑒𝑖∅𝛾 −𝛿𝑒𝑖∅𝛿 −𝛾𝑒𝑖∅𝛾 0 0 −𝛿𝑒𝑖∅𝛿 0 0 ], Here the values corresponding to ƞ, β, γ, and δ indicate the amplitudes whereas ∅ƞ, ∅𝛽, ∅𝛾, and ∅𝛿 are the complex phases of the elements of Raman tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' [21] Additionally, the magnitude of each tensor element is related with the specific mode and the crystal symmetry of the material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' The calculated intensities for the estimation of the Ʈ (𝐸𝑔) has contributions from both the doubly degenerate 𝐸𝑔 modes, thus added altogether.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' Using the Raman selection rule, |⟨𝑒𝑖|Ʈ∗|𝑒𝑠⟩|2, under both ab(/bc)-plane, the scattering intensity of all modes have been calculated (Table I), which clearly showed the distinct strength of interaction of polarized light with the crystal’s axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' [5,6,26,27] TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' Mathematically derived intensity of modes using Raman selection rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content='no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' Configuration Raman scattering intensity 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' ab-plane 𝑰𝑨𝟏𝒈 || (ki||c-axis) = |ƞ|𝟐 𝑰𝑬𝒈 || (ki||c-axis) = |𝜸|𝟐 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' bc-plane 𝑰𝑨𝟏𝒈 || (ki||a-axis) = |ƞ|𝟐𝒔𝒊𝒏𝟒𝜽 + |𝜷|𝟐𝒄𝒐𝒔𝟒𝜽 + 𝟏 𝟐 |ƞ||𝜷|𝒔𝒊𝒏𝟐(𝟐𝜽)𝒄𝒐𝒔𝝋ƞ𝜷 𝑰𝑬𝒈 || (ki||a-axis) = |𝜸|𝟐𝒄𝒐𝒔𝟒𝜽 + |𝜹|𝟐𝒔𝒊𝒏𝟐𝟐𝜽 − |𝜹||𝜸| 𝐬𝐢𝐧(𝟐𝜽) 𝒄𝒐𝒔𝟐𝜽 \uf0b4 𝟐𝒄𝒐𝒔𝝋𝜸𝜹 Rawat and Soni et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' 2023 7 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' Intensities of polar plots for 𝐴1𝑔 1 , 𝐴1𝑔 2 , 𝐸𝑔 1, 𝐸𝑔 2 modes in ab-plane (a-b), and in bc-plane (c-f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' Here, solid symbols and green line represent the experimental data fitting of the data using equation in Table I, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' Further, the understanding of the isotropic behaviour along ab-plane of the intensity of 𝐴1𝑔 and 𝐸𝑔 modes are depicted as circular shapes of the polar intensity plots (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' 4 (a-b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' On the other hand, the shape of polar plots for 𝐴1𝑔 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' 4 (c-d)) and 𝐸𝑔 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' 4 (e-f)) modes along bc-plane are different from ab-plane showing the anisotropy of the light matter interaction along crystallographic orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' The intensities of all modes are stronger along bc-plane in comparison to the ab-plane, which advocates the higher differential polarizability along bc-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' Similar observations on the anisotropic light-matter interaction in bc-plane have been reported for Graphene, hBN, 2H- MoSe2, MoS2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' [5,6,28] Fascinatingly, the out of plane modes at ~ 71 cm-1 and ~ 173 cm-1, (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' 4 (c-d)), have 𝐴1𝑔 symmetry but showing considerably different polar pattern at 90o and 270o rotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' The anomalous polarization dependence of the Raman intensities appeared because of the difference in Raman scattering cross-section through the second-order susceptibility or the electron–phonon interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' [21] To understand the discrepancy, the microscopic quantum description of Raman tensor has been adopted, which involved the electron-phonon interaction in addition to the electron-photon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' [29] Here, the total Raman intensity is described by the product of both the electron-photon and in ab-plane inbc-plane (a) 90 Ai fcj 120 120 06 60 (e) 120 90 1200 60 AT 3600 AI 300 50 (Sun 800 150 30 2400 150 30 200 150 30 400 1200 100 Intensity(arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' 0480 10 0180 40 0/180 400 1200 100 008 210 330 2400 210 330 200 210 330 1200 240 300 3600 240 300 240 270 270 270 300 (b) 90 E (p) () 120 60 06 120 60 2400 120 90 360F a E 1800/ (sun 240 150 1800 1200 150 30 1200 150 F 30 120 600 Intensity(arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' 600 0180 0480 0180 120 600 600 240 210 330 1200 210 1200 1800 210 330 360 240 300 1800 270 240 270 300 2400 L 240 270 300Rawat and Soni et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' 2023 8 electron-phonon interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' Hence, the Raman tensor (Ʈ𝑖𝑗 𝑘 ) associated with all modes can be given by: Ʈ𝑖𝑗 𝑘 = 1 𝑉 ∑ ∑ ⟨𝛹𝑣(𝑞 )|𝑒 𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' ∇⃗⃗ |𝛹𝑐′(𝑞 )⟩ ⟨𝛹𝑐′(𝑞 )|𝐻𝑒𝑝 𝑘 |𝛹𝑐(𝑞 )⟩⟨𝛹𝑐(𝑞 )|𝑒 𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' 𝛻⃗ |𝛹𝑣(𝑞 )⟩ (𝐸𝐿 − 𝐸𝑐𝑣(𝑞 ) − 𝑖𝛤𝑐)(𝐸𝐿 − ћ𝜔𝑝ℎ 𝑘 − 𝐸𝑐′𝑣(𝑞 ) − 𝑖𝛤𝑐′) 𝑞′ 𝑣,𝑐,𝑐′ Here, the numerator consists of the product of three matrix elements, (i) the electron-phonon (e-ph) matrix elements (⟨𝛹𝑐′(𝑞 )|𝐻𝑒𝑝 𝑘 |𝛹𝑐(𝑞 )⟩) and two electron-photon matrix elements for incident and scattered light (ii) (⟨𝛹𝑐(𝑞 )|𝑒 𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' 𝛻⃗ |𝛹𝑣(𝑞 )⟩, (iii) ⟨𝛹𝑣(𝑞 )|𝑒 𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' ∇⃗⃗ |𝛹𝑐′(𝑞 )⟩), where 𝑒 𝑖 and 𝑒 𝑠 are the polarization vectors of incident and scattered light, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' [29] The summation is over the electronics branches in conduction (𝑐, 𝑐’) and valance (𝑣) bands along with all wave vectors with first Brillouin zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' 𝛤𝑐 and 𝛤𝑐′ are the broadening factor associated with the lifetime of photo-excited states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' The inclusion of e-ph matrix element gives the major differences among both the out of plane 𝐴1𝑔 modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' Thus, different patterns of polar plots for 𝐴1𝑔 1 , and 𝐴1𝑔 2 modes indicate electron-phonon interactions in Bi2Se3, similar to the observations in other anisotropic layered chalcogenides like WS2, ReS2, GaTe, PdSe2 and black phosphorus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' [21,29-31] In contrast to the 𝐴1𝑔 modes, the polar plots of 𝐸𝑔 modes show four-lobbed polar pattern (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' 4 (e-f)) with the rotation of the crystal, which indicates the maximum strength of anisotropic nature in bc-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' To understand the behaviour of polar plots related to 𝐸𝑔 modes, the spectra have been captured by controlling the polarization of incident light (ei).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' This configuration is done by rotating half wave plate from 0o to 360o while keeping sample stage and analyzer fixed (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' S3 of supplemental materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' [22]) Here, the intensity of both 𝐴1𝑔 modes (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' S3 (a-b) of supplemental materials [22]) showed analogous polar pattern with polarization angle, whereas 𝐸𝑔 modes (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' S3 (c) of supplemental materials [22]) exhibited a low dependency on the rotation of the half wave plate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' This discrepancy of the 𝐸𝑔 modes between the rotation of crystallographic axis and incident laser suggest the anisotropic behaviour along bc-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' [5,6] Anisotropic light-matter interaction has been understood by estimating the amplitude and phase difference of Raman tensor’s element, which mainly contain the information of differential polarizability along different orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' To estimate the Raman tensor elements of all modes, we have fitted the experimental data (in Fig 4) using the intensity’s expressions given in Table I and the obtained details are presented in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' Rawat and Soni et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' 2023 9 TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' Estimated Raman tensor elements obtained from the fitting of experimental data (Fig 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' Modes Raman tensor ab-plane bc-plane 𝑨𝟏𝒈 𝟏 [ 𝟑𝟎 𝟎 𝟎 𝟎 𝟑𝟎 𝟎 𝟎 𝟎 𝜷 ] [ 𝟑𝟓 𝟎 𝟎 𝟎 𝟑𝟓 𝟎 𝟎 𝟎 𝟓𝟕𝒆𝒊𝟎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content='𝟑𝟕𝝅 ] 𝑨𝟏𝒈 𝟐 [ 𝟏𝟕 𝟎 𝟎 𝟎 𝟏𝟕 𝟎 𝟎 𝟎 𝜷 ] [ 𝟐𝟏 𝟎 𝟎 𝟎 𝟐𝟏 𝟎 𝟎 𝟎 𝟒𝟏𝒆𝒊𝟎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content='𝟐𝟒𝝅 ] 𝑬𝒈𝟏 [ 𝟖 −𝟖 𝛅 −𝟖 −𝟖 𝛅 𝛅 𝛅 𝟎 ] [ 𝟖 −𝟖 −𝟏𝟑𝒆𝒊𝟎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content='𝟑𝟗𝝅 −𝟖 −𝟖 𝟏𝟑𝒆𝒊𝟎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content='𝟑𝟗𝝅 −𝟏𝟑𝒆𝒊𝟎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content='𝟑𝟗𝝅 𝟏𝟑𝒆𝒊𝟎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content='𝟑𝟗𝝅 𝟎 ] 𝑬𝒈𝟐 [ 𝟏𝟔 −𝟏𝟔 𝛅 −𝟏𝟔 −𝟏𝟔 𝛅 𝛅 𝛅 𝟎 ] [ 𝟏𝟒 −𝟏𝟒 −𝟑𝟖𝒆𝒊𝟎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content='𝟑𝟐𝝅 −𝟏𝟒 −𝟏𝟒 𝟑𝟖𝒆𝒊𝟎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content='𝟑𝟐𝝅 −𝟑𝟖𝒆𝒊𝟎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content='𝟑𝟐𝝅 𝟑𝟖𝒆𝒊𝟎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content='𝟑𝟐𝝅 𝟎 ] In ab-plane, all modes show isotropic behaviour (Fig 4a and 4b), hence for Ʈ (𝐴1𝑔) and Ʈ (𝐸𝑔), the component of Raman tensor, ƞ (𝐴1𝑔 1 ~ 30 and 𝐴1𝑔 2 ~ 17) and γ (𝐸𝑔 1~ 8 and 𝐸𝑔 2 ~ 16), have been evaluated from the fitting of polar plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' As the propagation vector ki of incident light is along the c-axis, there is no polarization along c-axis, thus, 𝛽 for out of plane 𝐴1𝑔 mode is not evaluated while 𝛽 is zero for in-plane 𝐸𝑔 modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' Here, the phase factor (∅ƞ) is zero due to isotropic responses in ab- plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' On the other hand, in bc-plane (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' 4c and 4d), the component of Raman tensor, ƞ (𝐴1𝑔 1 ~ 35 and 𝐴1𝑔 2 ~ 21) and 𝛽 (𝐴1𝑔 1 ~ 57 and 𝐴1𝑔 2 ~ 41) have been evaluated and the phase factor between ƞ and 𝛽 (∅ƞ𝛽) is ~ 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content='3o (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content='37𝜋) for (𝐴1𝑔 1 ) and ~ 44o (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content='24𝜋) for (𝐴1𝑔 2 ), which is arising due to the anisotropic responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' Additionally, the elements of Raman tensor for in-plane modes are 𝛾 (𝐸𝑔 1 ~ 8 and 𝐸𝑔 2 ~ 14) and 𝛿 (𝐸𝑔 1 ~ 13 and 𝐸𝑔 2 ~ 38) and the phase factor between 𝛾 and 𝛿 (∅𝛾𝛿) is ~ 71o (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content='39𝜋) for (𝐸𝑔 1) and ~ 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content='4o (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content='32𝜋) for (𝐸𝑔 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' Overall, for out of plane 𝐴1𝑔 modes, 𝛽 > ƞ, (57 > 35 for 𝐴1𝑔 1 and 41 > 21 for 𝐴1𝑔 2 ), which indicates that differential polarizability is significantly higher and anisotropic along c-axis (schematic Fig 1e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' By comparing the tensor matrices of out of plane modes, it is clearly evident that symmetric stretching (𝐴1𝑔 1 ) induces larger dipole moment (higher polarizability) than anti- symmetric stretching (𝐴1𝑔 2 ) and the situation is completely otherwise for in-plane modes 𝐸𝑔 1 and 𝐸𝑔 2 as confirmed by the smaller magnitude of Raman tensor elements in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' For both the ab- and bc-plane, the comparison of relative magnitude of Raman tensor elements for of 𝐴1𝑔 1 (|ƞ𝑏𝑐−𝑝𝑙𝑎𝑛𝑒 ƞ𝑎𝑏−𝑝𝑙𝑎𝑛𝑒 ⁄ |~ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content='16) and 𝐸𝑔 2 (|𝛾𝑏𝑐−𝑝𝑙𝑎𝑛𝑒 𝛾𝑎𝑏−𝑝𝑙𝑎𝑛𝑒 ⁄ |~ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content='14), Rawat and Soni et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' 2023 10 which authenticate the estimated elements of the Raman tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' [6] Comparing the APRS estimated Raman tensor elements with studies on MoSe2, MoS2, WSe2, PdTe2, it is clear that the laser polarization dependence Raman spectra demonstrates the anisotropic light-matter interactions in Bi2Se3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' In Summary, the Raman tensor for all modes of single crystal Bi2Se3 corresponds to 𝐸𝑔 1 ~ 37 cm-1, 𝐴1𝑔 1 ~70 cm-1, 𝐸𝑔2 ~ 129 cm-1, and 𝐴1𝑔 2 ~ 172 cm-1 have been systematically studied by APRS measurements along both ab(/bc)-plane under parallel polarization (𝑒𝑖 ∥ 𝑒𝑠) scattering configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' We have estimated the amplitude and phase difference of the tensor elements by fitting the experimental results with the intensity expression obtained by applying Raman selection rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' The different shapes of polar plot of the similar vibrational symmetry (𝐴1𝑔) represents the different interaction of electrons with phonons, which provide the evidence of electron-phonon coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' Among two different orientations (ab(/bc)-plane) of single crystal, strong polarization dependence has been observed along bc-plane for both 𝐴1𝑔 and 𝐸𝑔 modes, which is showing the anisotropic light matter interaction in Bi2Se3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' Acknowledgement We would like to thank IIT Mandi for the instruments and research facilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' A.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=', Science China Materials 63, 1848 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' [31] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=', ACS Nano 10, 8964 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' 1 Supplemental Material Anisotropic Light-Matter Interactions in Single Crystal Topological Insulator Bismuth Selenide Divya Rawat, Aditya Singh, Niraj Kumar Singh and Ajay Soni* School of Physical Sciences, Indian Institute of Technology Mandi, Mandi, 175005, HP India Author to whom correspondence should be addressed: ajay@iitmandi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content='in In this supplemental file, we are providing the details of the synthesis, characterization techniques and selected data complementing the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' (a) Synthesis and characterization details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' Single crystal of Bi2Se3 was synthesized using dual zone vertical Bridgman furnace, by taking a stoichiometric amounts of bismuth ingot and selenium shots (both 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content='999% pure) in a quartz ampoule, which was then vacuum sealed at 10-5 mbar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' The ampoule was kept in a box furnace at 1123 K for 15 hr for homogenization followed by hanging it in Bridgman furnace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' The temperature of the hot zone and cold zone were kept at 1003 K and 953 K, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' The translation rate of the motor for the vertical motion of quartz tube from hot zone to cold zone was fixed at 2 mm/hr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' X-ray diffraction (XRD) was carried out using rotating anode Rigaku SmartLab diffractometer equipped with CuKα radiation (λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content='5406 Å) and in Bragg-Brentano geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' Rietveld refinement of the Powder-XRD pattern was done to determine the crystal structure, lattice parameter, and phase purity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' Resistance measurement was performed in the temperature range of 2 to 300 K using Quantum Design make physical properties measurement system (PPMS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' Raman spectroscopy measurements were carried out using a Horiba LabRAM HR Evolution Raman spectrometer having 532 nm laser excitation, 1800 grooves/mm with the help of a Peltier cooled (CCD) detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' Ultra-low frequency filters were used to access low-frequency spectra, very close to laser line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' To control the polarization state, a (λ/2) half-waveplate and an analyzer were used before objective lens and spectrometer to select the desired polarization component of the incident 2 and scattered light, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' To study the light-matter interaction on the crystallographic axis of Bi2Se3, the sample was kept on the stage rotating from ~ 0o to ~ 360o with a step of ~ 20o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' The linearly polarized laser was directed on the sample and the scattered radiation was collected to the detector in backscattering geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' (b) Rietveld refinement analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' S1:- Rietveld refined XRD pattern of single crystal Bi2Se3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' Black closed circle represents the experimental data point, Solid red line represents the refined data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' The as-synthesized Bi2Se3 crystal was ground into fine powder for XRD analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' The phase purity of Bi2Se3 sample has been confirmed by Rietveld refinements of the powder XRD pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' [1]The Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' S1 shows the Rietveld refined XRD data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' Goodness of fitting was showed by the extracting parameter, χ2 ~ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' Observed Simulated Intensity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content='units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=') Difference Braggposition 10 20 30 40 50 60 70 80 90 20 (deg)3 c) Resistance data of single crystal Bi2Se3: FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content='S2:- Four-probe resistance measurement with the variation of the temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' The electronic transport of the Bi2Se3 has been examined by the four probe resistance (R) and the temperature dependence is consistent with the behavior of degenerate semiconductors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' The longitudinal resistance (R) is fitted using a phenomenological model: R = R0 + λe−θ /T + $T2, where the λ and $ appear for phonon scattering and electron-electron scattering, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' [2,3] The fitting parameter are evaluated and λ ~ 12× 10−3and $ ~ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content='74 × 10−7 𝐾−2, where smaller value of $ suggests negligible electron-electron scattering in Bi2Se3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' The residual resistance ratio (RRR ~ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content='11) shows a high quality of the single crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' (d) APRS spectra of Bi2Se3 in bc-plane with the rotation of polarization vector of incident light while keeping the sample fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content='035 Experimental data pount 9 Fittingdata 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content='020 0 50 100 150 200 250 Temperature (K)4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' S3:- (a) APRS spectra and Polar plot of (b) 𝐴1𝑔 1 (c) 𝐴1𝑔 2 (d) 𝐸𝑔 2 of Bi2Se3 single crystal with the rotation of half-wave plate by keeping the sample fixed in bc-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' Solid symbols represent the experimental data point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' APRS measurements has been performed in in parallel configuration (𝑒𝑖 ∥ 𝑒𝑠), where polarization vector of incident light has varied by rotating the half-wave plate, while keeping the stage of sample and analyzer fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' Here, the intensity of both 𝐴1𝑔 modes showed expected two-lobed analogous polar pattern polar pattern with polarization angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' 𝐸𝑔 mode showed a low dependency on the rotation of the half wave plate, showed isotropic interaction on the rotation of polarization vector of incident light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' [4] References [1] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' Singh, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' Kashyap, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' Soni, Applied Physics Letters 119, 223903 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' [2] T.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content=' (b) 120 90 (a) Ata 00 150 180 210 330 240 270 300 (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} +page_content='units) E 120 90 E: (c) 00 00 150 40° Intensity 1803 800 210 900 240 300 1000 120 90 (d) 150 10 1400 180 1800 210 330 30 60 90 120 150 180 240 270 300 Ramanshift(cm1)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAyT4oBgHgl3EQff_hn/content/2301.00350v1.pdf'} diff --git a/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf b/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..fdfe196653a4f1050835865084a5d7ba931d332a --- /dev/null +++ b/LtAzT4oBgHgl3EQfyv51/content/2301.01758v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid 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Kingdom. +We review conceptual aspects of inflationary scenarios able to produce primordial black holes, +by amplifying the size of curvature fluctuations to the level required for triggering black hole +formation. We identify general mechanisms to do so, both for single and multiple field inflation. In +single field inflation, the spectrum of curvature fluctuations is enhanced by pronounced gradients +of background quantities controlling the cosmological dynamics, which can induce brief phases of +non–slow-roll inflationary evolution. In multiple field inflation, the amplification occurs through +appropriate couplings with additional sectors, characterized by tachyonic instabilities that enhance +the size of their fluctuations. As representative examples, we consider axion inflation, and two-field +models of inflation with rapid turns in field space. We develop our discussion in a pedagogical +manner, by including some of the most relevant calculations, and by guiding the reader through +the existing theoretical literature, emphasizing general themes common to several models. +1Correspondence e-mail: ogan.ozsoy@csic.es +arXiv:2301.03600v1 [astro-ph.CO] 9 Jan 2023 + +Contents +1 +Introduction +2 +2 +PBH formation in the early universe +5 +2.1 +PBH formation as a causal process +6 +2.2 +The relevant quantities for PBH abundance +8 +2.3 +Relating PBH properties with primordial scalar fluctuations +13 +2.4 +Brief summary, and the path ahead +17 +3 +Enhancement of scalar perturbations during single-field inflation +18 +3.1 +The dynamics of curvature perturbation +19 +3.2 +Enhancement through the resurrection of the decaying mode +22 +3.3 +Growth in the power spectrum when the decaying modes are slacking +29 +3.4 +Brief summary +33 +4 +Enhanced primordial power spectrum in multi-field models +34 +4.1 +Enhanced scalar perturbations from axion-gauge field dynamics +35 +4.1.1 +Smooth Axion Inflation +37 +4.1.2 +Bumpy axion inflation +42 +4.1.3 +Spectator axion-gauge field dynamics +46 +4.2 +Strong turns in the multi-scalar field space +50 +5 +Outlook +57 +A Background Cosmology: Mini-Review +58 +B Analytic estimate for the threshold of collapse. +63 +C Solving the Mukhanov-Sasaki equation: Numerical procedure +64 +D Details of the axion-gauge field dynamics +66 +D.1 Gauge field production by rolling scalars +68 +D.2 Scalars sourced by vector fields, the direct coupling case: χ = φ. +73 +D.3 Scalars sourced by vector fields, the indirect coupling case: χ = σ. +75 +E Curvature perturbation +77 +References +79 +1 + +1 +Introduction +Primordial black holes: history of the concept +Inflation, a short period of accelerated expansion in the very early moments of the universe, has +become one of the main pillars of modern cosmology [1, 2]. Leaving aside its success in addressing +the puzzles of the standard hot Big Bang cosmology, inflation provides an explanation for the +quantum mechanical origin of structures such as galaxies (including our own!) and the anisotropies +in the Cosmic Microwave Background (CMB) radiation [3]. In the last two decades, the advances +in the observational cosmology and in particular the observations of the CMB and of the large scale +structure (LSS) of our universe have so far confirmed the predictions of inflation, and arguably +established its status as the main theoretical framework describing the very early universe [4, 5]. +These successes notwithstanding, CMB and LSS probes only provide us information on the early +universe at the largest cosmological scales (10−4 ≲ k [Mpc−1] ≲ 10−1) corresponding to a small +fraction of the early stages of inflationary dynamics. Hence, while inflation provides us with a +consistent, testable framework in understanding the initial conditions in the universe at the largest +scales, we do not have direct access to most of the inflationary dynamics, and to the universe +evolution in the early post-inflationary era. Importantly, these stages could be host to a number of +interesting phenomena, including the production of stable relics such as dark matter (see e.g. [6] +for a historical review on dark matter) that is essential in understanding the world we observe +today, as well as for establishing new physics. Indeed, the existence of non-luminous, cold dark +matter (CDM) that constitutes a quarter of the total energy budget in the universe [7] is one of +the most glaring evidences for beyond the Standard Model physics [8]. The absence of signatures +from collider experiments, along with unsuccessful direct and indirect detection searches, have all +made the DM puzzle particularly compelling [9]. +An intriguing and economical explanation that might account for DM density in our universe +is a scenario where DM is made of compact objects, such as primordial black holes (PBHs). +Pioneered by the works of Y. Zel’dovic and I. Novikov [10] and S. Hawking [11], the initial ideas +in this direction began with the realization that PBHs could form by the gravitational collapse +of over-dense inhomogeneities in the early universe. In the mid 70’s, it was later realized by +the works of B. Carr [12, 13] and G. Chapline [14] that PBHs could contribute to DM density +and provide the seeds for the supermassive BHs populating our universe [15]. Following these +theoretical progresses, the interest of the scientific community on PBHs has risen in the mid +90’s by the reported detection of micro-lensing events from MACHO collaboration [16]. An +immediate interpretation of these results was suggesting on the possibility that a significant +fraction of mass density in our galaxy could be composed of sub-solar mass PBHs. However, +these considerations were later rendered invalid by the findings of EROS [17] and OGLE [18, 19] +collaborations, concluding that only a small fraction of mass in the Milky Way could be in the +form of PBHs. +Stimulated both by the absence of signals for well-motivated particle DM candidates, and the +first detection of gravitational waves (GWs) from merging BHs by the LIGO/VIRGO collaboration +[20], a second surge of interest in PBHs was ignited (see Fig. 1). In particular, different groups +suggested that merging PBHs could be responsible for the observed GW signals, while constituting +a significant fraction of DM density in our universe [21–23]. Since the first appearance of these +2 + +2000 +2005 +2010 +2015 +2020 +Year +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +Number of papers on PBHs ÷ 103 +% of papers on PBHs +Figure 1. Total and relative number of manuscripts appeared on the arXiv from 1996 until today related +to various aspects regarding primordial black holes. Spikes of activity in the literature, particularly after +the mid 90’s due to claimed lensing events by the MACHO collaboration and the GW detection by LIGO +in 2015 is clearly visible. +articles, a significant amount of effort has been pushed forward by the community, to search and +constrain the abundance of PBHs by utilizing their gravitational and electromagnetic effects on +the environment at small scales. Various experiments set stringent constraints on PBH abundance +for solar and sub-solar mass range, leaving a viable window for this scenario for tiny PBH masses +10−17 ≲ Mpbh [M⊙] ≲ 10−12 (M⊙ ≃ 1.98 × 1033 gr) as the totality of DM (see e.g. [24–26]). It +should be noted that some of the constraints derived in the literature make specific assumptions +about the formation process and the subsequent evolution of PBHs (such as monochromatic mass +functions, clustering and accretion processes, etc) and on other model dependent specifics (such +as non-Gaussianity), and therefore they could be relaxed. Since we mostly focus on the subject of +inflationary model building, we will not review these issues and aforementioned constraints, but +the interested reader can find more details in excellent reviews published recently, see e.g. [27–33]. +PBHs are likely to form well before the end of the radiation dominated era (i.e. before the so +called matter-radiation equality), and behave like cold and collision-less matter. Therefore they +constitute an interesting DM candidate, if they are massive enough Mpbh ≳ 1015 g ≃ 10−18M⊙ +to ensure a lifetime comparable with the age of the universe [34]. In this context, a particularly +appealing aspect of PBH dark matter is its economical and minimal structure, in the sense that +this scenario does not require any additional beyond Standard Model (BSM) physics (such as new +particles and interactions), provided that one alters the not-so-well constrained early universe at +small scales by introducing a viable mechanism to account for the production of large density +fluctuations required for PBH formation. +Similar to the generation of CMB anisotropies, a compelling and natural source of these +perturbations in the early universe could be the quantum fluctuations that are stretched outside +the horizon during inflation. However, in order to generate such over-dense regions that can +collapse to form PBHs in the post-inflationary universe, one needs to devise a mechanism to +enhance by several orders of magnitude the inflationary scalar perturbations at small scales +k ≫ kcmb (corresponding to late stages of inflation), far above the values required to match CMB +3 + +observations. As the observed temperature anisotropies prefers a red tilted power spectrum at +CMB scales, this situation generically requires a blue tilted power spectrum, or some specific +features at scales associated with PBH formation. +In the context of canonical single scalar field inflation, the first ideas in this direction appeared +in works by P. Ivanov, P. Naselsky and I. Novikov [35] (see also [36]). In particular, these +authors have shown that if the inflaton potential has a very flat plateau-like region for field ranges +corresponding to the late stages of accelerated expansion, the inflationary dynamics enters a +“non-attractor” regime called ultra slow-roll (USR) [37]. This leads to super-horizon growth of +scalar perturbations [38–40] that can eventually trigger PBH formation in the post-inflationary +universe2. Many explicit single-field inflationary models that exhibit similar local features were +subsequently studied in the literature: for a partial list of popular works see e.g. [43–51] (see also +[52–58] for earlier constructions). In the context of single scalar field inflation, another possibility +to generate an enhancement in the scalar power spectrum is to invoke a variation of the sound +speed of scalar fluctuations, for example through a reduction in the speed of sound c2 +s [49, 59, 60] +or through a rapidly oscillating c2 +s which triggers a resonant instability in the scalar sector [61, 62]. +From a top-down model building perspective, a rich particle content during inflation is not +just an interesting possibility, but appears to be a common outcome of many BSM theories [63]. +Since the early days of research on PBHs, multi-field inflationary scenarios has also attracted +considerable attention as a natural way to realize enhancement in the scalar perturbations at +small scales. For instance, large scalar perturbations may arise through instabilities arising in +the scalar sector, e.g. during the waterfall phase of hybrid inflation [52, 64, 65] or due to turning +trajectories in the multi-scalar inflationary landscape, as reported recently in [66–70]. Another +intriguing possibility in this context is by employing axion-gauge field dynamics during inflation +[71–78]. In these models, particle production in the gauge field sector act as a source for the scalar +fluctuations, and hence can be responsible for PBH formation. +A common feature of all inflationary scenarios capable of producing PBH populations is the +inevitable production of a stochastic GW background (SGWB) induced through higher order +gravitational interactions between enhanced scalar and tensor fluctuations of the metric [79–81]. +Interestingly, this signal may carry crucial information about the properties of its sources including +the amplitude, statistics and spectral shape of scalar perturbations (see e.g. [82–87]) and could +provide invaluable information on the underlying inflationary production mechanism. Furthermore, +since the resulting GW background interacts very weakly with the intervening matter between +the time of their formation and today, it leads to a rather clean probe of the underlying PBH +formation scenario. This allows us to access inflationary dynamics on scales much smaller than +those currently probed with CMB and LSS experiments, through space and ground based GW +interferometers including Laser Interferometer Space Antenna (LISA) [88, 89], Pulsar Timing +Array (PTA) experiments [90, 91] and DECIGO [92, 93]. For a detailed review of induced SGWB +and the dependence of its properties on the post-inflationary expansion history, see [94]. +The structure of this review +If their origin is attributed to the large primordial fluctuations, PBHs may offer us a unique +window to probe inflationary dynamics at sub-CMB scales. In this work, focusing mainly on +2Another inflationary background that exhibit similar features is called constant-roll inflation, see e.g. [41, 42]. +4 + +the activity in the literature within the last few years, we aim to revisit and review different +inflationary production mechanisms of PBHs 3 and their main predictions, in a heuristic and +pedagogical manner. The audience we have in mind are graduate students, or researchers in +related fields who wish to learn about inflation and primordial black holes, and to be guided +through the large literature on the subject by emphasizing common conceptual themes behind +many different realizations. +The review is organized as follows. In Section 2, we present a simplified, intuitive picture of +PBH formation in the inflationary universe and give some approximate estimate for the required +conditions to produce PBHs from the perspective of inflationary dynamics. In Section 3, we +discuss ideas to enhance the curvature power spectrum within single-field inflation, as required +for PBH formation. These mechanisms exploit large gradients in background quantities which +get converted into an amplification of fluctuations. Besides reviewing analytic findings, we also +develop some numerical analysis and provide a link to a code for reproducing our results (see +Footnote 20). In Section 4, we focus on multi-field inflationary scenarios that can generate +PBH populations including particle production during axion inflation, or sudden turns in the +multi-scalar inflationary landscape. Finally, we end with a discussion on future directions in +the concluding Section 5. We supplement this work with several technical Appendices where we +provide useful formulas and calculations used in the main body of this work. +Conventions +Throughout this review, we work with natural units ℏ = c = 1. We will use the reduced Planck +mass defined as M2 +pl = (8πG)−1 and retain it in the equations unless otherwise stated. For time +dependent quantities, over-dots and primes denote derivatives with respect to cosmological time t +and conformal time dτ ≡ dt/a(t) respectively where a(t) is the scale factor of the background +FLRW metric gµν = diag(−1, a2, a2, a2). +2 +PBH formation in the early universe +We start providing a physical description of PBH formation in the early universe, as comprised of +an early stage of inflation, followed by radiation and matter domination (for a mini-review on +background cosmology, see Appendix A). Our aim is to set the stage and relate basic properties +of a PBH population – as their mass and abundance – with the features of primordial curvature +fluctuations originating from inflation. For this purpose, in Section 2.1 we describe the mechanism +of PBH formation in the post-inflationary universe, emphasizing its nature as causal process +controlled by the inflationary quantum fluctuations. In Section 2.2 we discuss relevant concepts +such as the threshold for collapse into black holes, and the corresponding mass and collapse +fraction of PBHs, relevant for a computation of their abundance. Finally, in Section 2.3, we +relate the PBH abundance to primordial physics during inflation, with the aim to determine the +amplitude of scalar fluctuations required for producing a population of PBHs with interesting +3PBHs could also form in the post-inflationary universe through the collapse of cosmic strings [95–97] and domain +walls [98–101], phase transitions [102, 103], bubble collisions [104, 105], scalar field fragmentation via instabilities +[106, 107]. We note that PBHs could also form through the instabilities generated in the final stage of inflation +commonly referred as (p)reheating [108–111]. We will not dwell into this possibility here, for a list of recent works +in this line of research, see [112, 113]. +5 + +consequences for cosmology. All the concepts we discuss form the basis and motivations for our +analysis of inflationary mechanisms for PBH production, which we develop in Sections 3 and 4. +� Main References: In compiling the materials of this Section and to set the main framework +for our discussion, we have benefited from the ideas presented in the reviews by C. Byrnes and P. +Cole [114], M. Sasaki et al. [29] and the Ph.D. thesis by G. Franciolini [115]. +2.1 +PBH formation as a causal process +An important concept in an expanding space-time is the horizon scale, crucial for understanding +the causal properties of the dynamics of perturbations which are responsible for PBH formation. +As an indicator of the rate of our universe expansion, the Hubble rate H(t) ≡ ˙a(t)/a(t) has +dimensions of inverse length (or time−1 in natural units). This makes the quantity H−1 (Hubble +horizon) as the natural candidate for a physical length scale in an expanding universe. Commonly +referred to as the Hubble distance, the quantity 1/H (or c/H, if one wishes to recover physical +units) measures the distance that light travels within one Hubble time. Therefore, it can be +considered as a good proxy for a (time-dependent) length scale controlling the size of a causal +patch in our universe. Bearing in mind that we relate physical quantities to comoving ones by the +scale factor a(t), a useful quantity that guides us in this direction is the comoving Hubble horizon, +(aH)−1, and in particular its time evolution. When expressed in terms of the second derivative of +the scale factor, the time derivative of the comoving horizon can be written as +d +dt +� 1 +aH +� += − +¨a +a2H2 . +(2.1) +Notice that during inflation ¨a > 0: hence, the comoving horizon scale is a decreasing function of +time. Whereas, in a decelerating universe with ¨a < 0 (i.e. in the post-inflationary universe before +dark energy domination), this quantity is an increasing function of time. The property that the +comoving horizon decreases during an accelerated expansion is perhaps the most important element +to understand inflation as a solution of the horizon problem of the Hot Big Bang cosmology, +and a framework for the quantum mechanical origin of structures in our universe 4. The time +dependence of the comoving Hubble horizon is controlled by the value of the background equation +of state w (EoS) as (see Appendix A) +(aH)−1 ∝ a(1+3w)/2. +(2.2) +Therefore, during inflation w ≃ −1 and (aH)−1 ∝ a−1 while, during the subsequent phases of +radiation dominated (RDU) and matter dominated universe (MDU), the comoving horizon evolves +as (aH)−1 ∝ a1 and (aH)−1 ∝ a1/2 respectively. The evolution of the comoving horizon with +respect to logarithm of scale factor ln(a) is illustrated in Fig 2. When we study the statistical +properties of fluctuations in Fourier space, we often label a given perturbation mode with a +comoving length scale k−1, measured in units of megaparsecs (Mpc = 3.26 × 106 light years ≃ +3.1 × 1019 km). 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+AB+nicdVDLSgMxFM3UV62vqS7dBIvgqsyUtuFUHTZQX7gHYMmDU0yQ5JRythPceNCEbd+iTv/xkxbQUP +XDicy/3hPEjCrtOB9Wbm19Y3Mrv13Y2d3bP7CLh10VJRKTDo5YJPsBUoRQTqakb6sSIB4z0gulV5vduiVQ0Ejd6F +hOPo7GgIcVIG8m3i1M/HUoO42AyhxcQwZvl5yY1CrwYy4dc1pNGoVyoN6C4sxymBFdq+/T4cRTjhRGjMkFID14m1ly +KpKWZkXhgmisQIT9GYDAwViBPlpYvT5/DUKCMYRtKU0HChfp9IEVdqxgPTyZGeqN9eJv7lDRId1r2UijRODlojBhUEc +wywGOqCRYs5khCEtqboV4giTC2qRVMCF8fQr/J91K2a2Vq9fVUvNyFUceHIMTcAZcA6aoAXaoAMwuAMP4Ak8W/fWo/Vi +vS5bc9Zq5gj8gPX2CQbhkzg=kpbh = aH +AB+nicdVDLSgMxFM3UV62vqS7dBIvgqsyUltwUXTZQX7gHYMmDU0yQ5JRythPceNC +Ebd+iTv/xkxbQUPXDicy/3hPEjCrtOB9Wbm19Y3Mrv13Y2d3bP7CLh10VJRKTDo5YJPsBUoRQTqak +b6sSIB4z0gulV5vduiVQ0Ejd6FhOPo7GgIcVIG8m3i1M/HUoO42AyhxcQwZvl5yY1CrwYy4dc1pNGo +VyoN6C4sxymBFdq+/T4cRTjhRGjMkFID14m1lyKpKWZkXhgmisQIT9GYDAwViBPlpYvT5/DUKCMYRtKU0HC +hfp9IEVdqxgPTyZGeqN9eJv7lDRId1r2UijRODlojBhUEcwywGOqCRYs5khCEtqboV4giTC2qRVMCF8f +Qr/J91K2a2Vq9fVUvNyFUceHIMTcAZcA6aoAXaoAMwuAMP4Ak8W/fWo/VivS5bc9Zq5gj8gPX2CQVZkzc +=kpbh < aH +Figure 2. A sketch of the time evolution of curvature fluctuations R (labelled by a comoving scale +k−1) with respect to comoving Hubble horizon (dotted lines) (aH)−1 in the early universe. In the post- +inflationary universe, areh denotes the reheating time, aeq refers to the time of matter-radiation equality, +aMDE to matter-dark energy equality, and a0 denotes to value of scale factor today. The blue horizontal +line indicates the comoving size of a representative small scale perturbation responsible for PBH formation. +If the power spectrum associated with these modes is enhanced during inflation, they can transfer their +energy to density perturbations during radiation domination, and ignite PBH formation upon horizon +re-entry at a = aform ≡ af. +of comoving Hubble horizon: (aH)/k. Fluctuations with wavelengths larger than the comoving +horizon are referred to as super-horizon modes k < aH, while sub-horizon perturbations satisfy +k > aH. Each mode crosses the horizon at k = aH. As shown in Fig. 2, a typical fluctuation +with comoving size k−1 (horizontal lines) begins its life deep inside the horizon (typically as a +quantum fluctuation); then it leaves the horizon to become a super-horizon mode, and finally +it re-enters the comoving horizon in the post-inflationary universe. Large scale modes (with +smaller k) exit the horizon earlier than small scale modes, and re-enter the horizon at a later +time in the post-inflationary era. For definiteness, in Fig. 2 we represent the behaviour of the +comoving curvature perturbation R [117, 118] (see [116] for a textbook discussion), which plays +an important role for our discussion. +In fact, apart from providing seeds for the observed cosmic microwave background (CMB) +anisotropies at large scales, the dynamics of curvature fluctuations R may also be at play for PBH +formation, provided that cosmological fluctuations exhibit specific ‘initial conditions’ at small +scales. For this purpose, denoting k = kpbh ≫ kcmb as the comoving momentum associated with +PBH formation, we assume that the curvature power spectrum at these small scales acquires an +amplification well above the level required to match CMB observations: PR(kpbh) ≫ PR(kcmb) ∼ +10−9 [4] (more on this later). Soon after the end of inflation, i.e. after reheating5, the modes +5Throughout this work, we assume an efficient reheating process at the end of inflation such that the universe +become radiation dominated shortly after inflation terminates. For a collage of interesting physics that might arise +through the reheating stage and alternative post-inflationary histories see the recent review [119]. +7 + +associated with the enhancement (e.g. modes with comoving size of k−1 +pbh) become the seeds of +density perturbations in the RDU: +PR(kpbh)1/2 ∼ δρ +ρ ≡ δ . +(2.3) +Since the comoving Hubble scale grows with respect to comoving scales in RDU, the characteristic +scale of perturbations eventually becomes comparable to the comoving horizon at a = af, where +kpbh = (aH)f (the subscript f indicates PBH formation time). +At this point, gravitational +interactions can trigger the collapse of over-dense regions, if the latter have a sufficient over-density +above a certain collapse threshold: δ ≥ δc. +Notice that at sub-horizon scales the radiation pressure can overcome the gravitational collapse: +therefore, the production of PBHs effectively occurs at around horizon re-entry. This implies +that the concept of horizon re-entry is crucial for our understanding PBH formation as a causal +process: in fact, only when the physical wavelength of a perturbation becomes comparable to +the causal distance 1/H, gravity is able to communicate the presence of an over-density, and to +initiate the gravitational collapse. A schematic diagram that summarizes the discussion above is +illustrated in Fig 2. +In what comes next, we introduce relevant quantities such as the threshold for collapse and +the mass and collapse fraction of PBH, which are important for the computation of the current +PBH abundance. +2.2 +The relevant quantities for PBH abundance +The threshold for collapse +The original estimate for the collapse threshold δc for PBH formation was made by B. Carr in +1975, using a Jeans-type instability argument within Newtonian gravity [13]. In Carr’s estimate, +an over-density in RDU would collapse upon horizon re-entry if the fractional over-density of the +perturbation satisfies +δ ≡ δρ +ρ +���� +k=aH += c2 +s, +(2.4) +where cs is the sound speed of density perturbations, which measures how fast a pressure wave +caused by the over-density can travel from the centre to the edge of a local fluctuation. In RDU, +the speed of sound of perturbations satisfies cs = 1/ +√ +3 so that its square is directly related to +EoS during RDU as c2 +s = w = 1/3. Equation (2.4) then implies that a perturbation can collapse +to form PBHs if its over-density is larger than the pressure exerted by the radiation pressure6. +An improved analytical estimate on the threshold is provided in [120], implementing general +relativistic effects, obtaining δc ≃ 0.4 during RDU. +A more precise characterization of the threshold δc requires a dedicated analysis of the evolution +of perturbations in the non-linear regime after horizon re-entry, which can be done with the help +of numerical simulations. Recent efforts in this direction shows that the threshold value of the +density contrast depends on the shape of the density peak, and is given in the range 0.4 ≤ δc ≤ 2/3 +6See Appendix B for a simple analytic argument that leads to this result. +8 + +depending on the shape of density perturbation [121–123]. For the estimates we provide in this +work, we will use a value within the range specified above. +The mass of PBHs +The characteristic mass of PBHs can be related to the mass contained within the Hubble horizon +at the time of formation (a = af) through an efficiency factor γ = 0.2, as suggested by the +analytical model developed in [13]: +M(f) +pbh = γ MH +���� +a=af += γ M(eq) +H +� +M(f) +H +M(eq) +H +� += +� af +aeq +�2 +γ M(eq) +H +. +(2.5) +In this formula, MH(t) ≡ 4πρ(t)/(3H(t)3) is the time-dependent horizon mass, where the sub/super +scripts “f” and “eq” denote quantities evaluated at the time of PBH formation and matter-radiation +equality respectively: we use the standard relations H2 ∝ ρ ∝ a−4 during RDU. Noting that the +horizon mass at the time of equality is given by M(eq) +H +≃ 2.8 × 1017 M⊙ [82], Eq. (2.5) informs us +that PBHs, contrarily to astrophysical black holes, can in principle span a wide range of masses, +depending on their formation time with respect to matter-radiation equality. +Making use of the time-dependent horizon mass as above, we can relate the PBH mass at +formation to the characteristic size of the perturbations that leave the horizon during inflation, +and are responsible for PBH formation. For this purpose, we first rewrite the PBH mass at +formation as +M(f) +pbh = +� ρf +ρeq +�1/2 �Heq +Hf +�2 +γ M(eq) +H +. +(2.6) +Using the property of entropy conservation gs(T) T 3 a3 = constant, and the scaling property of +the energy density with respect to temperature of the plasma during RDU, ρ ∝ g∗(T) T 4, Eq. +(2.6) can then be re-expressed as the following relation +M(f) +pbh(kpbh) = +� g∗ (Tf) +g∗ (Teq) +�1/2 �gs (Teq) +gs (Tf) +�2/3 � keq +kpbh +�2 +γ M(eq) +H +, +≃ +� γ +0.2 +� �g∗ (Tf) +106.75 +�−1/6 � +kpbh +3.2 × 105 Mpc−1 +�−2 +30M⊙ . +(2.7) +In the second line we assume that the effective number of relativistic degrees of freedom in +energy density and entropy are equal, i.e. we set g∗(T) = gs(T) and take g∗(Teq) ≃ 3.38 7 with +keq ≃ 0.0104 Mpc−1, accordingly with the latest Planck results [7]. Equation (2.7) indicates that +for masses of PBHs that could be associated with recent LIGO observations, Mpbh ≃ 30M⊙, +the peak scale of perturbations responsible for PBH formation is much smaller compared to +CMB scales kpbh ≫ kcmb. For sub-solar mass PBHs, the corresponding peak scale for PBH +formation gets progressively smaller. For example, considering the currently allowed sub-lunar +range (Mmoon ≃ 3.7 × 10−8 M⊙) of PBH masses, 10−17 ≲ Mpbh [M⊙] ≲ 10−12, which are objects +that can account for the totality of dark matter, the range of scales associated with PBH formation +7Strictly speaking g∗(T) = gs(T) is only satisfied when species are in thermal equilibrium at the same temperature. +For a nice overview on the thermal history of the universe after inflation, see Chapter 3 of [116]. +9 + +Mpbh [M⊙] +∆N +kpbh [Mpc−1] +Mpbh [M⊕] +Mpbh [MEverest] +106 +14 +103 +1011 +1021 +100 − 102 +18 − 21 +105 − 106 +105 − 107 +1015 − 1017 +10−17 − 10−12 +34 − 40 +1012 − 1015 +10−12 − 10−7 +10−2 − 103 +Table 1. Range of PBH mass(es) vs the corresponding wave-number(s) kpbh (see Eq. (2.7)) of the +primordial modes together with the approximate horizon crossing time measured with respect to e-folding +number the pivot scale kcmb = 0.002 Mpc−1 leaves the horizon during inflation, ∆N ≡ Npbh − Ncmb > 0 +(see Eq. (2.8)). First row refers to the corresponding quantities for a typical Super Massive Black Hole +(SMBH) like the Sagittarius A∗ in the center of our galaxy [124]. The third row refers to astroid-mass +PBHs that can still account for a significant fraction (or all) of DM density today [26]. The corresponding +mass of the PBHs in terms of the earth’s mass M⊙ ≃ 3.33 × 105 M⊕ and the mass of mount Everest +MEverest = 8.1 × 1014 kg ≃ 4.1 × 10−16 M⊙ are given in the last two columns on the right. +is quite small: 1012 ≲ kpbh [Mpc−1] ≲ 1015. See Table 1 for an easier-to-visualize summary of +these considerations. +Elaborating on Eq. (2.7), we can also derive a rough relation between the PBH mass at +formation to the the number of e-folds Npbh at which the PBH-forming modes leave the horizon +during inflation. For this purpose, we first notice that kpbh/kcmb = (aH)pbh/(aH)cmb where the +values of Hubble rate and scale factor should be evaluated at the scales of horizon exit during +inflation (see Fig. 2). Assuming roughly a constant slow-roll parameter ϵ ≡ − ˙H/H2 ≪ 1 between +the horizon-exit time of modes associated with CMB and PBH formation respectively, we can +relate the Hubble and the scale factor as Hpbh = Hcmb e−ϵ(Npbh−Ncmb) and apbh = acmb eNpbh−Ncmb +where Npbh > Ncmb so that we count e-folds forward in time with respect to horizon exit of the +CMB mode8. Using the last two relations we find kpbh/kcmb ≃ e(Npbh−Ncmb)(1−ϵ); once plugged in +Eq. (2.7), assuming kcmb = 0.002 Mpc−1), we find +M(f) +pbh(Npbh) ≈ 7.7 × 1017M⊙ e−2(Npbh−Ncmb)(1−ϵ) � γ +0.2 +� �g∗ (Tf) +106.75 +�−1/6 +. +(2.8) +Modes that leave the horizon much later compared to CMB scales have Npbh−Ncmb ≫ 1, therefore +the exponential in Eq. (2.8) can considerably reduce the overall large normalization, leading to +small PBH masses. +PBH abundance +After discussing possible masses for PBH and how they depend on the dynamics of inflation, we +analyse the notion of abundance of PBHs relative to the energy density of other species. We can +compute this quantity during two epochs: today, and at PBH formation. +When considering the present-day fraction of PBH density, it is a common practice to relate +the PBH abundance to present-day dark-matter density introducing the quantity +fpbh ≡ Ωpbh +Ωdm +, +(2.9) +8We note that another common convention is to count e-folds with respect to the end of inflation denoting the +end point as Nend = 0. +10 + +where for each species i we define Ωi ≡ ρi,0/ρc,0, with subscript “0” denoting quantities evaluated +today and ρc,0 = 3H2 +0M2 +pl is the critical density. Planck measurements provide the following value +for the dark matter abundance [7], +Ωdmh2 = 0.120 ± 0.001, +(2.10) +in terms of h = 0.6736 ± 0.0054, which measures the Hubble rate H0 in units of 100 km s−1 Mpc−1. +We can then relate fpbh today to the density fraction of PBH at the epoch of their formation, +denoting this quantity with β. In fact, since we assume that PBH formation takes place during +RDU, and since after formation the PBH density scales as of like ρpbh ∝ a−3, we can write +β ≡ ρpbh +ρ +���� +a=af += ρpbh,f +ρpbh,0 +ρc,0 +ρf +Ωdmfpbh = ρpbh,f +ρpbh,0 +ρc,0 +ρeq +ρeq +ρf +Ωdm fpbh, +≃ 1 +2 +af +aeq +Ω−1 +m Ωdm fpbh +(2.11) +where Ωm is the current matter density in the universe. In (2.11) we normalize the scale factor today +as a0 = 1, and we use the fact that the total energy density evolves as ρ ∝ a−4 for af < a < aeq, +and ρ(aeq) = 2ρm,0 a−3 +eq . Using the conservation of total entropy, gs(T) T 3 a3 = constant, we can +re-express the factor af/aeq appearing in Eq. (2.11) as follows: +af +aeq += Teq +Tf +�gs(Teq) +gs(Tf) +�1/3 +, +≃ 3.17 × 10−9 � γ +0.2 +�−1/2 �g∗ (Tf) +106.75 +�−1/12 +� +�M(f) +pbh +M⊙ +� +� +1/2 +, +(2.12) +where we make use of Eq. (2.6) to relate Teq/Tf to the mass of PBH at formation, and as before we +assume g∗(T) = gs(T). Finally, plugging (2.12) in (2.11), and implementing Planck measurements +on Ωdm (2.10) and Ωm (see Eq. (A.18)), we directly relate the PBH abundance at formation, β, +to their present-day fraction fpbh, in terms of the PBH mass at formation: +β ≃ 1.33 × 10−9 � γ +0.2 +�−1/2 �g∗ (Tf) +106.75 +�−1/12 +� +�M(f) +pbh +M⊙ +� +� +1/2 +fpbh . +(2.13) +Therefore, we learn that in the case when PBH abundance account for the total DM density +today, fpbh → 1, the fraction of the total density in the form of PBHs (β) at the time of their +formation takes extremely small values, when considering an interesting range of masses M(f) +pbh. +This situation reflects the fact that PBH formation in the early universe is a very rare event. +There is also another way to parametrize β in terms of (relative) number of collapsing regions to +form PBHs. This approach is especially useful to relate the PBH abundance to the statistical +properties of primordial fluctuations as we discuss below. +11 + +Figure 3. Two Gaussian PDFs of over-density field δ with different variances σ2 +2 > σ2 +1. Since the second +distribution have a larger variance, the area under the curve above the critical threshold (δc ≤ δ ≤ ∞) is +larger, leading to larger PBH abundance β (2.14) at formation. +Collapse fraction of PBHs at formation +PBHs can form in RDU, provided that the fractional over-density associated with the characteristic +scale of perturbations is larger than the threshold δ > δc. The PBH abundance can then be +interpreted as the fraction β of such local regions (with respect to the total density) in the +universe at the time of horizon re-entry. The standard treatment of estimating β is based on +the so-called Press-Schechter model of gravitational collapse, widely used in the literature on +large-scale structure formation [125] 9, +β ≡ ρpbh +ρ +���� +a=af +≡ +� ∞ +δc +P(δ) dδ, +(2.14) +where P(δ) is the probability distribution function (PDF), which describes how likely that a given +fluctuation have an over-density δ. Let’s assume that δ follows a Gaussian distribution, +PG(δ) = +1 +√ +2πσe−(δ−µ)2/2σ2, +(2.15) +where µ is the mean and σ2 is the variance of the distribution. In Fig. 3, we represent two +Gaussian PDFs that have the same mean value, and two different variances satisfying σ2 +2 > σ2 +1. +As the second distribution is more “spread” with a larger variance compared to the first one, the +probability of an over-density to be larger than the critical threshold δc, is larger, and so does +PBH abundance β at formation, since the integral in Eq. (2.14) has more support within the +9Contrarily to the original approach by Press-Schechter [125], we do not take into account a symmetry factor +of 2 in the right hand side of (2.14) that accounts for all the mass in the universe, since it is not clear whether +such a factor makes sense when considering non-symmetric PDFs of δ (e.g. non-Gaussian cases). Furthermore, the +error introduced by omitting this factor is comparable with the other uncertainties in the computation of β such as +fraction of horizon mass which collapse to form a PBH (see e.g. [126–128]). +12 + +integration limits δ ∈ [δc, ∞]. Hence, we can expect that the quantity σ plays an important role +for estimating the PBH abundance. +Using (2.13) and (2.14), we can estimate the required variance σ2 of (2.15) that can give rise +to large population of PBH today, as controlled by the quantity β. Focusing on a distribution +with zero mean µ = 0 in (2.15) and integrating (2.14), we have +β = +� ∞ +δc +dδ +√ +2πσ exp +� +− δ2 +2σ2 +� += 1 +2Erfc +� δc +√ +2σ +� +≃ +σ +√ +2πδc +exp +� +− δ2 +c +2σ2 +� +, +(2.16) +where Erfc(x) = 1 − Erf(x) is the complementary error function, and in the last equality we +take δc ≫ σ. As we will learn shortly, this is a good approximation for all practical purposes. +As concrete examples, substituting Eq. (2.16) into Eq. (2.13) we infer that a solar mass PBH +population with fpbh = 10−3 requires δc/σ ≃ 7, whereas for a population with M(f) +pbh = 10−12M⊙ +and fpbh = 1 we need δc/σ ≃ 7.9. Assuming a threshold of δc = 0.4, these results translate into +σ ≃ 0.06 and σ ≃ 0.05 respectively. Notice also from (2.16) that the PBH abundance at the time +of formation is exponentially sensitive to the variance of the distribution. We will dwell more on +this dependence below. But first, we discuss the implications of these findings in terms of the +amplitude of scalar fluctuations generated during the phase of cosmic inflation. +2.3 +Relating PBH properties with primordial scalar fluctuations +We now examine how to relate the notion of PBH abundance with the properties of the comoving +curvature fluctuation R, [117, 118], as produced in the early universe by cosmic inflation. R is +conserved on super Hubble scales as the modes evolve from the inflationary phase to RDU. We +start by connecting the amplitude of R with the fractional over-density δ, which triggers PBH +formation as we learned in our previous discussion. Working in Fourier space, Taylor expanding +at leading order in a gradient expansion (controlled by the small parameter k/aH) and at linear +order in R, one finds [121]: +δ(⃗x, t) ≃ 2(1 + w) +(5 + 3w) +∇2R(⃗x) +(aH)2 ++ . . . +=⇒ +δk ≃ −4 +9 +� k +aH +�2 +Rk, +(2.17) +where used w = 1/3 during RDU and . . . denote terms of higher order O(R2) in the curvature +perturbation10. Defining the power spectrum of a Fourier variable Xk as +⟨XkXk′⟩ = 2π2 +k3 PX(k) δ(3)(⃗k + ⃗k′), +(2.18) +the relation between the power spectrum of over-density and curvature perturbation is then given +by +Pδ(k) ≃ 16 +81 +� k +aH +�4 +PR(k). +(2.19) +In the computation of the density contrast, one should typically use a window function W to +smooth δ(⃗x, t) on a scale R ≈ k−1 ≈ (aH)−1 (e.g. on scales of size k−1 +pbh at horizon re-entry, as +10Non-linearities that we neglect in the expression (2.17) can be important to understand intrinsic non-Gaussianity +present in the PBH formation process, see e.g. [129, 130] and references therein. +13 + +shown in Fig. 2) relevant for PBH formation. Therefore, the variance of density contrast can be +related to the primordial power spectrum as [29, 131] 11 +σ2(R) ≡ ⟨δ2⟩R = +� ∞ +0 +d ln q W2(q, R) Pδ(q) ≃ 16 +81 +� ∞ +0 +d ln q W2(q, R) (qR)4 PR(q) +(2.20) +where W is the Fourier transform of a real space window function. Popular choices of W include +a volume-normalized Gaussian, or a top hat window function, whose Fourier transforms are +respectively given by +W(k, R) = exp +� +−k2R2 +2 +� +, +W(k, R) = 3 sin(kR) − 3kR cos(kR) +(kR)3 +. +(2.21) +When selecting a curvature power spectrum Pδ characterized by a narrow peak around the +wave-number kpbh, the integral in (2.20) can be approximated as σ2 ∼ Pδ(kpbh). Then, utilizing +(2.19) at horizon entry k ≃ aH (i.e. at the time of PBH formation), since 81/16 ∼ 5, we can +roughly relate the variance σ to the primordial curvature power spectrum as +PR(kpbh) ∼ 5 σ2 . +(2.22) +Finally, recall our considerations after Eq. (2.16): a Gaussian PDF of δ requires σ ≃ 0.06 +(σ ≃ 0.05) for M(f) +pbh = M⊙ (M(f) +pbh = 10−12M⊙) to generate a population of fpbh = 10−3 (fpbh = 1) +today. Hence, we can estimate the amplitude of the scalar power spectrum needed at scales +relevant for PBH formation: +PR(kpbh) ∼ 5 σ2 ∼ 10−2 +for Gaussian fluctuations. +(2.23) +This estimate holds for a wide range of sub-solar PBH masses. This implies that we need a +very large amplification of the curvature spectrum between large CMB and small PBH-formation +scales: +∆PR ≡ PR(kpbh) +PR(kcmb) ∼ 107 , +(2.24) +and the task is to produce such amplification in a controllable way by an appropriate inflationary +mechanism. +It is worth pointing out that this estimate does not change much for even smaller mass PBHs +with M(f) +pbh < 10−12M⊙, because the power spectrum has a logarithmic sensitivity to the PBH +fraction β. In order to see this, we can invert the expression (2.16), and use (2.23) to relate the +primordial power spectrum of curvature perturbations to β as +PR(kpbh) ∼ 5σ2 ∼ +5 δ2 +c +2 ln(1/β) . +(2.25) +Now, as an extreme case, we can consider the smallest PBHs M(f) +pbh ≃ 10−18M⊙ that can +11The variance (2.20) can be equivalently written as σ2 = σ2(k) or σ2 = σ2(M) using the relation between the +peak scale of PBH formation and PBH mass (2.6). +14 + +survive until today (not yet eliminated by Hawking radiation) which have the tightest available +observational constraints, restricting their current abundance to fpbh ≲ 10−9 [31]. Plugging these +values in (2.13), PBH fraction at formation gives β ≲ 10−28 which in turn leads to the constraint +PR(kpbh) ≲ 6 × 10−3 in (2.25) for a threshold of δc = 0.4. Therefore, we conclude that for +Gaussian perturbations and for any PBH mass of interest, the amplitude of scalar power spectrum +relevant for PBH formation requires PR ∼ 10−2 for any potentially observable PBH fraction fPBH +today. The discussion above informs us that a small change in the amplitude of power spectrum +leads to many order of magnitude difference in the fraction of regions collapsing into PBHs, as +clearly implied by the exponential dependence of β to PR in (2.25). Similarly, a small change +in the choice of threshold δc could lead to very different estimates in terms of β. For example, +focusing on fixed value of variance σ2 ≃ 0.05 as relevant for PBH formation, β (2.16) can chance +by various orders of magnitude, if we reduce the threshold δc by just about 20%. In fact, +β(δc = 0.4) +β(δc = 1/3) ≃ 10−5 , +(2.26) +demonstrating how tuned the conditions are for producing a cosmologically interesting population +of PBHs. +Collapse fraction vs curvature perturbation +While it is customary to use the smoothed density contrast at horizon crossing to estimate the +number of collapsing regions, it is also possible to work directly with the comoving curvature +perturbation to approximately compute the PBH fraction β at time of formation. In this case, +there is no need of relying on the smoothing procedure of sub-horizon fluctuations provided by +the window functions [131, 132]. Interestingly, as we will see later this approach also provides a +way to assess the effects of large primordial non-Gaussianity that might be present in some of the +PBH-forming inflationary scenarios. +For understanding the role of the primordial curvature fluctuation R, we approximate its +variance with the power spectrum σ2 +R ≈ PR. Using the Press-Schechter approach with a Gaussian +PDF for the curvature fluctuation spectrum, the fraction of collapsing regions at formation can +be estimated as +βG = +� ∞ +Rc +dR e−R2/(2σ2 +R) +√ +2πσR +≃ 1 +2Erfc +� +Rc +√2PR +� +, +(2.27) +where Rc is the threshold. To roughly estimate Rc, we can assume an almost scale invariant power +spectrum of R, for a logarithmic range of wave-numbers relevant for PBH formation. Making +use of a Gaussian window function in (2.20) gives in this case σ2 +R = ⟨R2⟩ ≃ 8PR/81. Finally, +plugging the latter in (2.16), and comparing the resulting expression with (2.27), we obtain [131]: +Rc ≈ +9 +2 +√ +2δc. +(2.28) +For a density threshold of δc = 0.4, the relation above gives Rc ≃ 1.3, which we set as fiducial +value for the estimates below. Following these considerations and using the formulas derived so far +– in particular Eqs. (2.27) and (2.13) – we can then repeat the previous estimates, and determine +the approximate amplitude of power spectrum required for PBH formation. The result is that for +15 + +Gaussian primordial fluctuations a sensible PBH population today requires PR ∼ 10−2. These +findings confirm our earlier results of Eq. (2.23). +The case of non-Gaussian curvature fluctuations +So far, we assumed that primordial fluctuations obey Gaussian statistics in order to estimate the +amplitude of the power spectrum required for PBH formation. Since PBHs are expected to form +through extremely rare large fluctuations (see Fig. 3), any small deviation in the shape of the +tail of the fluctuation distribution – which essentially depend on the amount of non-Gaussianity +(i.e. skewness of the PDF) – can have a significant impact on the PBH abundance [132–138]. +For the sake of obtaining a lower limit on the amplitude of the PBH-forming curvature power +spectrum, we now consider scenarios with large primordial non-Gaussianity. A particularly +interesting case of this type occurs if the main source of the curvature perturbation results from a +higher order interaction, where the distribution of R can be modeled as a χ2 distribution (see +e.g. [71, 132, 139]): +R = g2 − ⟨ g2 ⟩, +(2.29) +where g is a Gaussian random variable (⟨g⟩ = 0) with variance σ2 +g ≡ ⟨ g2 ⟩. The PDF of R in this +case can be determined by making a change of variable PNG(R) = PG(g)|dg/dR|, which takes the +following form +PNG(R) = +e−(R+σ2 +g)/2σ2 +g +2 +� +2π(R + σ2g) σg +. +(2.30) +Making a change of variable to a quantity t through the definition σ2 +g t = R + σ2 +g −→ dR = σ2 +g dt, +the fraction of regions in the universe that can collapse to form PBHs can be estimated as +βNG = +� ∞ +Rc +dR PNG(R) ≃ 1 +2Erfc +�� +1 +2 + +Rc +√2PR +� +, +(2.31) +where in the last step we approximate the variance as ⟨R2⟩ = 2⟨g2⟩2 = 2σ4 +g ≈ PR, in order to +express βNG in terms of the curvature power spectrum12. We can now compute the amplitude of +the curvature power spectrum required for PBH formation, when the statistics of fluctuations is +strongly non-Gaussian. Using (2.13) together with (2.31), a population of solar mass PBHs with +fpbh = 10−3 requires PR ≃ 1.5 × 10−3, whereas for a population of PBHs with M(f) +pbh = 10−12M⊙ +and fpbh = 1, we find PR ≃ 9 × 10−4 ≃ 10−3. Hence we conclude that the required amplitude of +power spectrum is roughly given by +PR(kpbh) ∼ 10−3, +for non-Gaussian fluctuations. +(2.32) +Compared to the case of Gaussian distributed curvature perturbation (see Eq. (2.23)) we learn +that the required amplitude of the power spectrum is reduced by about one order of magnitude. +Therefore, a curvature perturbation with a smaller amplitude can produce the same amount of +12Note that power spectrum is the variance of curvature perturbation per logarithmic interval in k, i.e. ⟨R2⟩ ≡ +� d ln k PR(k). Therefore the approximate signs ≃ in the expressions in (2.27) and (2.31) can be turned into an +equality if we consider the β’s defined in those expressions as the collapse fraction per d ln k in the spectrum, namely +β = β(k). +16 + +10−2 +10−1 +100 +√PR / Rc +10−29 +10−24 +10−19 +10−14 +10−9 +10−4 +βG +βNG +Figure 4. Fraction of the universe that collapses into PBHs as a function of the power spectrum. For +phenomenologically interesting interval of β (see e.g. (2.13)) values, in the non-Gaussian case we need a +smaller amplitude of power spectrum in order to generate the same amount of PBHs at horizon re-entry. +PBH abundance, if non-Gaussianity is present (see Eq. (2.29)). In particular, for R2 +c ≫ PR, +which is typically satisfied to a very good approximation, we can disregard the 1/2 factor in (2.31). +Comparing with Eq. (2.27), the power spectrum required to generate the same collapse fraction +of PBHs in both cases can be related as +PRNG ≃ 2 +R2c +P2 +RG. +(2.33) +To further illustrate these points, in Fig. 4 we show the quantity β both for Gaussian and +non-Gaussian cases, represented as a function of the curvature power spectrum. We learn from +the figure that, in the non-Gaussian case, within a phenomenologically relevant interval of β (see +e.g. (2.13)) a given value of the power spectrum leads to a much larger value of β. We emphasize +that we focused on a specific type of non-Gaussian distribution (namely χ2) to estimate the +amplitude of the power spectrum required for PBH formation, and so the results we derived +could change for milder cases depending on the amplitude and sign of the non-Gaussianity +(i.e. depending on whether the PDF in (2.27) is positively or negatively skewed) [132]. +2.4 +Brief summary, and the path ahead +Let us summarize the arguments we reviewed so far. We computed the required amplitude of +small-scale primordial power spectrum PR(kpbh) to generate a sizeable population of PBHs that +can account for all or a fraction of DM density. The typical small scale of PBH formation kpbh +is related with the BH mass through equation (2.7): see Table 1 for examples. Comparing with +power spectrum at large CMB scales, we need a +∆PR ≡ PR(kpbh) +PR(kcmb) ∼ 106 − 107 +(2.34) +enhancement in the spectrum amplitude between small and large scales, depending on the statistics +obeyed by the primordial curvature perturbation (see Eqs. (2.23) and (2.32)). +17 + +We also learned that the PBH abundance is extremely sensitive to the amplitude of the +primordial curvature spectrum. Notice that the results we reviewed are derived for the case of +PBHs produced during RDU: if early phase transitions or early phases of non-standard cosmology +occur, the corresponding modified equations of state can also considerably influence the properties +of the PBH population [140, 141]. An interesting example is the QCD phase transition, which +can lead to a high peak in the distribution of solar mass PBHs, several orders of magnitude larger +than the corresponding values in RDU [142, 143]. +There are various significant opportunities for improving and elaborating on these results. +In our considerations, we assumed that PBHs form at a particular mass (Eqs. (2.5) and (2.7)), +by means for example of a sharply peaked primordial power spectrum; moreover we ignored +the effects of PBH clustering and mergers. As shown in [144–146], assumptions on the shape +of the primordial spectrum may alter the PBH distribution and the corresponding clustering +properties. Moreover, initial clustering and subsequent mergers may also influence the shape of +initial mass distribution, as well as the abundance of PBHs (see e.g. [147]). Another topic of +debate concerns the use of over-density δ versus the curvature perturbation R when computing +the PBH abundance: see e.g. [148] for a discussion on these issues. In fact, we emphasize that +the calculations we carried on in this Section should be regarded as rough order-of-magnitude +estimates, in need of more precise numerical analysis. Furthermore, in discussing the effects +of non-Gaussianities in PBH formation, we stress that we computed the corresponding power +spectrum only for an extreme example of χ2 non-Gaussian statistics. For a detailed analysis of +the impact of primordial non-Gaussianity on PBH formation and abundance, we refer the reader +to the general discussion in [149]. An additional phenomenological consequence of PBH-forming +scenarios is the production of a stochastic gravitational wave (GW) background. In fact, an +enhanced spectrum of curvature fluctuations, as needed to produce PBH, acts as a source for GW. +The characterization of the GW background, and the corresponding forecasts for its detection, is +an important avenue for the experimental probe of PBH inflationary models. We refer the reader +to [94] for a detailed recent review. +All the topics mentioned above are being actively developed by the PBH community. The +arguments and results we reviewed in this Section are sufficient for introducing our specific purpose, +which is reviewing the theoretical foundation of inflationary scenarios leading to PBH. From now +on, we discuss different conceptual ideas and concrete inflationary mechanisms for obtaining the +enhancement (2.34) of the curvature power spectrum, as needed to generate PBHs. We focus on +the inflationary theory aspects only, without computing the resulting PBH abundance, as well as +other phenomenological properties which are already covered in various recent complementary +reviews [27–33]. +3 +Enhancement of scalar perturbations during single-field inflation +We now focus our attention to inflationary scenarios able to lead to PBH formation. As we learned +in the previous section, they are characterized by a significant enhancement in the curvature +power spectrum at a scale kpbh (which depends on the PBH mass) much smaller with respect to +18 + +CMB scales kcmb. The condition to satisfy is Eq. (2.24), which we rewrite here: +PR(kpbh) +PR(kcmb) ∼ 107 . +(3.1) +We classify inflationary models into single-field (this section) and multi-field type (next section), +depending on whether the mechanism responsible for the enhancement in the scalar fluctuations +respectively relies on a single or multi-field scenario. In general, existing inflationary mechanisms +amplify the spectrum of curvature fluctuations by means of significant gradients in the background +evolution of fields responsible for inflation. In this section we phrase our discussion as model- +independent as possible, mostly focusing on conceptual aspects of the problem. We aim to discuss +the dynamics and the general properties of curvature fluctuations in inflationary models leading +to PBHs, and refer to representative specific scenarios when necessary. +� Main References: Our discussion in this section is based on the papers [38, 39, 49, 59]. +3.1 +The dynamics of curvature perturbation +In order to analyse the behavior of the scalar power spectrum in single-field scenarios, we consider +the second-order action of scalar perturbations around an inflationary phase of evolution. The +background metric corresponds to a (quasi) de Sitter background, with nearly constant Hubble +parameter H. Cosmological inflation is controlled by a slow-roll parameter ϵ ≡ ˙H/H2 satisfying +ϵ ≪ 1, with ϵ = 1 corresponding to the condition to conclude the inflationary process. We work +with conformal time, τ ≤ 0 during inflation. (See e.g. [150] for a classic survey of inflationary +models.) +The dynamics of scalar fluctuations can be formulated in terms of the comoving curvature +perturbation R [117, 118], whose second order action (at lowest order in derivatives)13 takes the +following form (see e.g. [59]) +S(2) +R = 1 +2 +� +dτ d3x 2 a2(τ)M2(τ) ϵ(τ) +c2s(τ) +� +R′2 − c2 +s(τ)(⃗∇R)2 +� +. +(3.2) +In this formula, cs is the sound speed of the curvature perturbation, M is an effective time- +dependent Planck mass, and ϵ the aforementioned slow-roll parameter. +Few initial words for contextualising single-field models aimed to produce PBHs, leading to a +dynamics of curvature perturbation controlled by action (3.2). The simplest option to consider +are PBH-forming models with unit sound speed and constant Planck mass, characterized only by +the shape of the potential V (φ). As mentioned in the Introduction, models in this class require a +potential characterized by flat plateau-like region, see e.g. [43–51] for a choice of works studying +this possibility. (We will discuss its implications for the dynamics of curvature perturbations in +the next subsection.) PBH-forming potentials with the required characteristics can find explicit +realisations for example in models of Higgs inflation [44, 156–158], alpha-attractors [159, 160], +13One can also introduce a time dependent mass in the action (3.2) which may arise through broken spatial +translations as in solid [151] and supersolid [152, 153] inflation. Another possibility is to include higher derivative +terms in the quadratic action to modify the dispersion relation of curvature perturbation as in Ghost inflation [154]. +We will not consider these possibilities here. For a discussion on PBH formation in solid and ghost inflation see +Section 4 and 6 of [59] and [155]. +19 + +and string inflation [48, 49, 161]. Considering more complex possibilities, PBH-generating models +which exploit a time-dependence for the sound speed are based on non-canonical kinetic terms for +the inflaton scalar, as K-inflation [162, 163]: see e.g. [59–62, 164–167] for concrete examples, and +Section 3.3 for some of their implications. Finally, scenarios with a time-dependent effective Planck +mass can be generated by non-minimal couplings of the inflaton scalar with gravity, as in the +Horndeski action [168] and its cosmological applications to G-inflation scenarios [169]. Realisations +of PBH-forming models in set-up with non-minimal couplings belonging to the Horndeski sector +include [170–173]. To the best of our knowledge, early universe models based on the more recent +covariant DHOST actions [174–176], have not been explored so far in the context of PBH model +building. +Interestingly, despite the many distinct concrete realisations, all single-field scenarios rely in +few common mechanisms for enhancing the spectrum of curvature fluctuations, which exploit +the behaviour of background quantities. We are now going to discuss these mechanisms in a +model-independent way. We treat M, cs and ϵ as appearing in action 3.2 as time-dependent +quantities, controlled by the single scalar background profile that drives inflation. To start with, it +is convenient to redefine the time variable in action (3.2), so to adsorb the time-dependent cs into +a re-scaled conformal time and impose an equal-scaling condition of time and space coordinates: +d¯τ = csdτ +=⇒ +S(2) +R = 1 +2 +� +d¯τ d3x z2(¯τ) +� +R′2 − (⃗∇R)2 +� +, +(3.3) +with a prime indicating a derivative with respect to ¯τ, the re-scaled conformal time. Importantly, +we introduce a so-called time-dependent ‘pump field’ z(¯τ) as +z2(¯τ) = 2 a2(¯τ) M2(¯τ) ϵ(¯τ) +cs(¯τ) +. +(3.4) +The dynamics of the curvature perturbation is strongly tied to the time dependence of the pump +field z(¯τ), and more generally to the behavior of the background quantities M, ϵ, cs that constitute +it. +To analyze the evolution mode by mode, we work in Fourier space, and write the Euler-Lagrange +mode equation for curvature perturbation, derived from the action (3.3): +1 +z2(¯τ) +� +z2(¯τ)R′ +k(¯τ) +�′ = −k2Rk(¯τ), +(3.5) +where k ≡ |⃗k| is the magnitude of the wave-number that labels a given mode. This is a differential +equation involving derivatives along the time direction, acting on the function Rk(¯τ) depending +both on time and momentum k. +To express its solution, we implement a gradient expansion approach (see e.g. [38, 39, 49]), +starting from the solution in the limit of small k/(aH), and including its momentum-dependent +corrections which solve (3.5) order-by-order in a k/(aH) expansion. This approach is particularly +suitable for our purpose of describing scenarios where the size of small-scale curvature fluctuations +(k/(aH) large) differs considerably from large-scale ones (k/(aH) small): see condition (2.24). +20 + +Indeed, a gradient expansion allows us to better understand the physical origin of possible +mechanisms which raise the curvature spectrum at small scales. +The most general solution of Eq. (3.5), up to second order in powers of k/(aH), is formally +given by the following integral equation 14 +Rk(¯τ) = R(0) +k +� +�1 + R(0) ′ +k +R(0) +k +� +¯τ +¯τ0 +d¯τ ′ +˜z2(¯τ ′) − k2 +� +¯τ +¯τ0 +d¯τ ′ +˜z(¯τ ′)2 +� +¯τ ′ +¯τ0 +d¯τ ′′ ˜z2(¯τ ′′) Rk(¯τ ′′) +R(0) +k +� +� , +(3.6) +where the sub and super-scripts 0 denote a reference time, and tilde over a time-dependent +quantity indicates that it is normalized with respect to its value at ¯τ = ¯τ0. +Typically, we are interested in relating the late time curvature perturbation at ¯τ to the same +quantity computed at some earlier time τ0. For this purpose, it is convenient to identify ¯τ0 as the +time coordinate evaluated soon after horizon crossing, and R(0) +k +as the mode function computed +at ¯τ0. In order for enhancing the spectrum of curvature fluctuations at small scales (recall the +PBH-forming condition of Eq. (2.24)) we can envisage two possibilities. One option is to exploit +the structure of Eq. (3.6), making sure that its contributions within the square parenthesis become +more and more important as time proceeds after modes leave the horizon. In this way, we generate +a sizeable scale-dependence for Rk(¯τ) after horizon crossing, with the possibility of amplifying +the small-scale curvature spectrum. Alternatively, we can design methods that lead to significant +scale dependence already at horizon crossing, i.e. for the quantity R(0) +k , which then maintains +frozen its value at super-horizon scales. In what follows, we explore both these two options, in +Sections 3.2 and 3.3 respectively. +To develop a quantitative discussion, it is convenient to introduce the so-called slow-roll +parameters as +η ≡ d ln ϵ +dN , s ≡ d ln cs +dN , µ ≡ d ln M2 +dN +(3.7) +where in our definition we make use of the relation between e-foldings and the time coordinate ¯τ: +dN = Hdt = (aH/cs)d¯τ. +In standard models of inflation based on an inflationary attractor dynamics, one imposes +the so-called slow-roll conditions throughout the entire inflationary period, corresponding to the +requirements +η, s, µ, dη +dN , ds +dN , dµ +dN ≪ 1, +(3.8) +which imply that the pump field always grows in time z2 ∝ (−¯τ)−2 as ¯τ → 0 (see Eq. (3.4)). +As a consequence, the second and third terms in the general solution (3.6) decay respectively +as (−τ)3 and (−τ)2 in the late time limit (−τ) → 0. Hence they can be identified as decaying +modes 15 that rapidly cease to play any role in the dynamics of curvature perturbations. This is a +14In fact, if the time evolution of the pump field is known, up to second order in the gradient expansion we can +generate a solution for the curvature perturbation by replacing Rk(¯τ ′′) in the last integral of Eq. (3.6) with the +leading growing mode solution of the homogeneous part of Eq. (3.5), which we can identify as R(0) +k . +15In fact, the standard decaying mode is given by the last term in (3.6) as it decays slowly, i.e. ∝ (−k¯τ)2, +21 + +regime of slow-roll attractor, where soon after horizon crossing the curvature perturbation settles +into a nearly-constant configuration R(0) +k , whose spectrum is almost scale-invariant. In this case, +the momentum-dependent terms in Eq. (3.6) do not have the opportunity to raise the curvature +spectrum at small scales. +Hence, for producing PBH we need to go beyond the slow-roll conditions of Eq. +(3.8), +as first emphasized in [177]. Before discussing concrete ideas to do so, in view of numerical +implementations, as well as for improving our physical understanding, it is convenient to express +the curvature perturbation equation (3.5) in a way that makes more manifest the role of slow-roll +parameters in controlling the mode evolution. We introduce a canonical variable vk defined as +vk(¯τ) = z(¯τ)Rk(¯τ) . +(3.9) +Plugging this definition in Eq. (3.5), we obtain the so-called Mukhanov-Sasaki equation, which +reads +v′′ +k(¯τ) + +� +k2 − z′′ +z +� +vk(¯τ) = 0, +(3.10) +where +z′′ +z = 2 +�aH +cs +�2 � +1 + Θ +� +. +(3.11) +Expanding the derivatives of z (3.4) in terms of the slow-roll parameters of Eq. (3.7), we define +Θ ≡ −1 +2(ϵ + s) + 3 +4(η − s + µ) + 1 +8(η − s + µ)2 − 1 +4(ϵ + s)(η − s + µ) ++1 +4 +� dη +dN − ds +dN + dµ +dN +� +. +(3.12) +Standard slow-roll attractor scenarios correspond to situations where Θ is negligibly small: the +quantity in Eq. (3.11) then reads 2 ¯τ −2, leading to a scale-invariant curvature power spectrum. To +break scale-invariance of curvature perturbation, we need to consider a sizeable time dependent +Θ. We note that the expression (3.12) is exact, and does not assume any slow-roll hierarchy as +Eq. (3.8). Hence it can be used to study the system beyond slow-roll, as we are going to do in +what comes next. +3.2 +Enhancement through the resurrection of the decaying mode +The idea +An interesting mechanism to enhance the curvature perturbation at super-horizon scales is +suggested by the structure of the integrals within the square parenthesis of Eq. (3.6). Suppose +that, for a brief time interval, a given mode k experiences a background evolution during which +the pump field z rapidly decreases after the horizon exit epoch ¯τ0. Then, the would be ‘decaying’ +mode can grow large, and the integrals in the parenthesis of Eq (3.6) can contaminate the nearly +constant solution R(0) +k , eventually leading to a late-time value Rk(¯τ) ≫ R(0) +k +on super-horizon +scales. This situation signals a significant departure from the attractor, slow-roll regime discussed +compared to the second. +22 + +after Eq. (3.8). In fact, in this case the criterion for the enhancement of the curvature perturbation +can be explicitly phrased in terms of the derivative of the pump field, transiently changing sign +during some short time interval during inflation: +z′ +z = aH +cs +� +1 + η − s + µ +2 +� +< 0 . +(3.13) +This condition implies that the combination of the slow-roll parameters, η − s + µ should be of +order O(1) and negative during some e-folds during inflation, violating the slow-roll conditions +(3.8). In particular, we require +η − s + µ < −2 . +(3.14) +If Eq. (3.13) is satisfied, the slow-roll conditions (3.8) are not satisfied, and the contributions +within parenthesis of Eq. (3.6) can grow large. Strong time gradients of homogeneous background +quantities, which lead to condition (3.14), can then be converted into a small-scale amplification +of the curvature power spectrum. As discussed in [59], the expression (3.13), along with the +considerations above, generalizes to a time-dependent sound speed and Planck mass the arguments +first developed in [38, 39]. +Model building, and a parametrization of the non-attractor phase +To illustrate a viable model that can generate a seven-order of magnitude enhancement required +for PBH formation – see Eq. (2.24) – we focus on canonical single-field models, cs → 1, M → Mpl +(and s → 0, µ → 0), in order to simplify our analysis. The background evolution for the single +scalar field driving inflation is +¨φ + 3H ˙φ + V ′(φ) = 0 , +(3.15) +with V (φ) the scalar potential, and the time-derivatives are carried on in coordinate time t. The +non–slow-roll dynamics is controlled by the properties of the potential V , as we are going to +discuss, and by its consequences for the behaviour of the inflaton velocity ˙φ. +Since in these scenarios the pump field can be parametrized purely in terms of the slow-roll +parameter ϵ as z = a +√ +2ϵ Mpl (see e.g. Eq. (3.4)), the linear dynamics of Rk (Eq. (3.6)) is dictated +by the first slow-roll parameter, whose evolution is in turn determined by the sign and amplitude +of the slow-roll parameter η. Hence, the criterion required to realize the desired growth in the +spectrum can be simply parametrized as a condition on the second slow-roll parameter, as η < −2 +in (3.14). +From a concrete model building perspective, scalar potentials V (φ) that can induce this type of +dynamics include a characteristic ‘plateau’ within a non-vanishing field range ∆φ ̸= 0 [35, 36]. This +property gives rise to phases of transient non-attractor dynamics, of ultra slow-roll (abbreviated +USR) [37, 40, 41] or constant roll (CR) [42, 178, 179] evolution, depending on the shape profile of +the potential around the aforementioned feature. In particular, for USR the potential typically +has a very flat plateau with V ′ ≃ 0, whereas for constant roll V ′ < 0, so that the field climbs a +hill by overshooting a local minimum16. As the scalar field, during its evolution, traverses such a +16Here we assume field rolls down on its potential from large to small values ( ˙φ < 0) with V ′(φ) before it +encounters with feature required for the enhancement. Since V ′ < 0 during the feature, there must be a point in +the potential where the second derivative of the field vanishes V ′′ = 0. +23 + +flat region with negligible potential gradient, the acceleration term ¨φ is balanced by the Hubble +damping term in the Klein-Gordon equation (3.15), and the inflaton speed is no longer controlled +by the scalar potential. This phenomenon changes significantly the values of the inflaton velocity +˙φ during the transient non-attractor phase, and inevitably leads to the violation of one of the +slow-roll conditions: +¨φ + 3H ˙φ + V ′(φ) = 0 =⇒ η = −6 − 2V ′ +˙φH ++ 2ϵ, +(3.16) +hence η ≃ −6 (η < −6) for transient USR V ′ = 0 (CR V ′ < 0) phases respectively17. We +emphasize that since the non-slow-roll inflationary era is characterized by a large negative η for a +brief interval of e-folds, the pump field, as well as the first slow-roll parameter ϵ, quickly decay +during this stage as required for the activation of the decaying modes. In fact, +d ln ϵ +dN +≡ η +=⇒ +z2 ∝ ϵ ∝ e−|η|∆N, +(3.17) +where for simplicity we assume a constant η during the non-attractor phase. For explicit inflationary +scenarios that can realize such transient phases in the context of PBH formation, see e.g. [46, 48, +49, 180]. Nevertheless, it is worth pointing out that, although possible, explicit constructions of +suitable inflationary potentials involve a high degree of tuning to render the potentials extremely +flat for a small region in field range, and ensure an appropriate transition for the scalar velocity +among different epochs. See e.g. the discussion in [47], as well as the comments at the end of this +section. +After this general discussion on model building, in the analysis that follows we do not need +to work with an explicit form of potential V (φ) to analyze the enhancement through the non- +attractor dynamics. Instead, we exploit the general idea we are discussing in a model-independent +way, and we model PBH forming inflationary scenarios as a succession of distinct phases which +connect smoothly one with the other, each parametrized by a constant η. (Related approaches +are developed in [181–185]). Our perspective catches the important features of scenarios based +on the idea of transiently resurrecting the decaying mode at super-horizon scales, satisfying Eqs. +(3.13) and (3.14). In order to capture the transitions among phases, we multiply each phase by +the smoothing function [186]: +σ(N, ∆) = 1 +2 +� +tanh +�N − Ni +∆ +� +− tanh +�N − Nf +∆ +�� +, +(3.18) +where N denote e-folds, Ni and Nf are the e-folding numbers at the beginning and end of the +constant η phase, and ∆ signifies the duration of the smoothing procedure. Keeping this smoothing +prescription in mind, the inflationary evolution can be divided into three phases: +• Phase I. The initial phase of inflationary evolution is characterized by a standard slow-roll +(SR) regime, where ϵ, η ≪ 1 and ϵ < η at the pivot scale kcmb = 0.05 Mpc−1 (assuming that +17Note that for single-field inflation we have ϵ ≡ ˙φ2/(2H2M 2 +pl) and using η (3.7) we get η = 2¨φ/( ˙φH) + 2ϵ. Using +the Klein-Gordon equation in the last expression gives the relation on the right hand side of (3.16). +24 + +Phase I +Phase II +Phase III +η +0.02 +−6.30 +0.30 +�� 3.00 +Ni +0.00 +33.2 +35.7 +�� 55.0 +Nj +33.2 +35.7 +55.0 +�� 65.0 +ν +1.00 +0.50 +1.00 +�� 2.00 +Table 2. +Parameter choices that characterize the background evolution of η smoothed by the function +(3.18) at each phase. Note that the final phase of evolution divided into two in order to accommodate the +end of inflation with ϵ = 1 at Nend = 60. +modes at the pivot scale exits the horizon at the beginning of evolution Nb = 0), in order to +match Planck observations [4]. +• Phase II. As the scalar field starts to traverse the flat plateau-like region in its potential, +its dynamics eventually enter the non-attractor era lasting some e-folds of evolution. This +phase is characterized by a large negative η < −6, during which the first slow-roll parameter +ϵ decays exponentially: +ϵ(N) ≡ exp +�� N +60 +η(N′) dN′ +� +. +(3.19) +• Phase III. The final phase of evolution ensures a graceful exit from the non-attractor phase +into a final slow-roll epoch, leading to the end of inflation. Since ϵ decays quickly in the +non-attractor era, this final phase is characterized by a hierarchy between the slow-roll +parameters: +η ≫ ϵ +(3.20) +where η > 0. We typically require a large positive η to bring back ϵ from its tiny values +at the end of the non-attractor era, towards the value ϵ = 1 needed to conclude inflation. +To capture this behavior accurately, we split the final phase of evolution into two parts, +parametrizing η as +η(N) = η(1) +III σ1(N, ∆) + η(2) +III σ2(N, ∆) . +(3.21) +The relevant parameter choices to model the dynamics can be found in the third column in +Table 2. +We note that our choice of η in the initial stage of the Phase III and in Phase II is not a +coincidence: most of the single-field modes there exist a correspondence that relates η’s +in Phase II and Phase III: ηIII = −6 − ηII, which is a consequence of Wands’ duality [187]. +We will elaborate below on the consequence of this correspondence in the context of the +power spectrum, in particular for modes that exit the horizon as the background evolves +from Phase II to Phase III. +Following the discussion above, we can characterize the full background evolution using the +Hubble hierarchy in (3.19) and H(N) = Hend exp [− +� N +60 ϵ(N′)dN′], where Hend denotes the Hubble +rate at the end of inflation, where Nend = 60. +25 + +0 +10 +20 +30 +40 +50 +60 +N +10−10 +10−8 +10−6 +10−4 +10−2 +100 +102 +Phase I +II +Phase III +ϵ +|η| +31 +32 +33 +34 +35 +36 +37 +38 +N +−2 +−1 +0 +1 +2 +—— z′/z +aH +← z′/z < 0 → +Figure 5. Left panel: Evolution of ϵ and η in e-folds through the successive phases outlined in the main +text. The green colored region indicates the range of e-fold numbers where η < 0, corresponding roughly to +the beginning and end of the non-attractor phase. Right panel: the time evolution of z′/z = aH(1 + η/2), +with z′/z < 0 in the region highlighted with red color. +For a representative set of parameter choices (see Table 2), we show in Fig. 5 an example of +background evolution, in which we plot ϵ, η and z′/z. The right panel of the figure makes manifest +that the background evolution leads to z′/z < 0 for a short interval of e-folds (N = 33 − 35.7), +as highlighted by the red region in the plot. In accord with our discussion so far, this behavior +is appropriate for triggering a significant enhancement in the power spectrum of curvature +perturbation through the resurrection of the decaying mode. +Numerical analysis +Having obtained the background evolution, we are ready to describe mode evolution to obtain +power spectrum of curvature perturbation towards the end of inflation 18: +PR(k, Nend) = k3 +2π2 +���� +vk(Nend) +z(Nend) +���� +2 +, +(3.22) +where to study the evolution of curvature perturbations, we make use of the canonical variable +vk and consider the Mukhanov-Sasaki system of equations (3.10)-(3.12) after setting s = µ = 0. +In general, it is not possible to find full analytic solutions for this system of equations, and a +numerical analysis is needed 19. We implement the numerical procedure explained in detail in +the technical Appendix C, which solves the Mukhanov-Sasaki equation with Bunch-Davies initial +conditions, and we provide a Python code that reproduces our numerical findings 20. The resulting +18Note that evaluating the power spectrum at the end of inflation is necessary when modes evolve outside the +horizon, as in the example background we are focusing in this section. +19Although, as we will explain soon, interesting properties of the resulting curvature spectrum can be derived +and understood analytically. +20 In fact, the general procedure outlined in Appendix C can be generalized to accurately solve Mukhanov-Sasaki +equation a broad class of single-field models of inflation. In the context of phenomenological models we discuss in +this and the next section, jupyter notebook files that compute the power spectrum is available at the link github. +We acknowledge the use of the python libraries: matplotlib [188], numpy [189], scipy [190], pandas [191] along with +jupyter notebooks [192]. +26 + +10−2 +101 +104 +107 +1010 +1013 +1016 +k +[Mpc−1] +10−10 +10−8 +10−6 +10−4 +10−2 +PR(k) +PR(k) via eq. (C.10) +107 +109 +1011 +1013 +1015 +k +[Mpc−1] +10−10 +10−8 +10−6 +10−4 +10−2 +PR(k) +PR ∝ k4 +PR ∝ kηII+6 +Figure 6. Power spectrum of curvature perturbation in the three-phase model described in the main text. +Pale green region separated by the vertical lines denote the range of modes that exit the horizon during +the non-attractor era whereas the light blue regions denote range of modes that cross the horizon during +the initial and final slow-roll era respectively. +power spectrum is represented in Fig. 6: it manifestly grows in amplitude towards small scales, +exhibiting a peak at around kpeak ≃ 1012 Mpc−1 ≫ kcmb. Notice that the spectrum grows as k4 +towards its peak, and is characterized by a dip preceding the phase of steady growth [181]. We +will have more to say soon about these features. +Interestingly, for the system under consideration the bulk of the enhancement can be attributed +to the active dynamics of the would-be ‘decaying modes’, the second and third term of Eq. (3.6). +To show this explicitly, we study super-horizon solution of the curvature perturbation in Appendix +C by applying the formula (C.10), a special case of Eq. (3.6), to the canonical single-field scenario +we discuss here. For a grid of wave-numbers that exit the horizon during the initial slow-roll +era, the amplitude of power spectrum obtained in this way is shown by blue dots in Fig. 6. The +accuracy of these locations with respect to the full numerical result (black solid line) confirms our +expectation that decaying modes in (3.6) play a crucial role for the enhancement of the curvature +perturbation for this scenario. In the right panel of Fig. 6, we zoom in to the growth and the +subsequent decay of power spectrum following the peak. +The features of the spectrum: analytic considerations +Besides the numerical findings presented above, we can derive general analytic results for the +spectrum of curvature fluctuations in scenarios activating the would-be decaying modes through a +brief non-attractor era. +We start noticing that for modes that leave the horizon during the initial slow-roll stage +(leftmost region colored by light blue in Fig. 6), the spectrum shows characteristic features such as +the presence of a dip, followed by an enhancement parametrized by a spectral index of ns − 1 = 4 +during the bulk of the growth [86, 181, 193–195]. The dip is physically due to a disruptive +interference between the ‘constant’ mode of curvature fluctuation at super-horizon scales, and +the ‘decaying’ mode that is becoming active and ready to contribute to the enhancement of the +spectrum. The position and depth of the dip is analytically calculable in terms of other features +27 + +of the spectrum, at least in a limit of short duration of the non–slow-roll epoch. It is found +that the position of the dip in momentum space is proportional to the inverse fourth root of the +enhancement of the spectrum, and the depth of the dip is proportional to the inverse square root +of the enhancement of the spectrum [195]. These relations are valid for any single-field models +that enhance the spectrum through a brief deviation from the standard attractor era, including +cases with a time-varying sound speed and Planck mass. They are accompanied by consistency +conditions on the squeezed limit of non-Gaussian higher-order point functions [196–198], as +expected in single-field scenarios. +While in the considerations of the previous paragraph we considered modes leaving the horizon +during the first stage of slow-roll evolution, we can also derive analytic results for what happens +during the non-attractor epoch. In fact, for modes that exit the horizon deep in the non-attractor +era (light green region in the middle of Fig. 6) and the following final slow-roll era, the spectrum +behaves as expected in a standard slow-roll phase, with spectral index +ns − 1 = −2ϵ − η ≃ −ηIII = 6 + ηII . +(3.23) +(Recall that the latin numbers II and III relate with the phases of evolution, see Eqs. (3.19) and +(3.21).) This behavior is a manifestation of the duality invariance of perturbation spectra within +distinct inflationary backgrounds, called Wands duality (see e.g. [187, 199]). Wands duality can be +understood by noticing that the structure of Mukhanov-Sasaki equation, Eq. (3.10), is unchanged +by a redefinition of the pump field that leaves the combination z′′/z invariant: +z(¯τ) → ˜z(¯τ) ≡ z(¯τ) +� +c1 + c2 +� ¯τ +d˜τ +z(˜τ) +� +⇒ +˜z′′ +˜z += z′′ +z +(3.24) +where c1,2 are arbitrary constants. If z(¯τ) controls a phase of slow-roll attractor, ˜z ∝ 1/¯τ, a dual +phase whose pump field z ∝ ¯τ 2 as given by Eq. (3.24) describes a non-attractor era. Although the +statistics of the canonical variable vk is identical in the two regimes, the amplitude of the curvature +perturbation spectrum Rk increases in the non-attractor epoch. In scenarios where the parameter +η is well larger than the other slow-roll parameters, Wands duality (3.24) analytically prescribes +the relation (3.23), in agreement with the numerical findings plotted in Fig. 6. Subtleties can arise +in joining attractor and non-attractor phases, since consistency conditions can be violated [200] +due to the effects of boundary conditions at the transitions between different epochs. All these +considerations are relevant for our topic, given the sensitivity of PBH formation and properties +on the shape of the spectrum near the peak. +For further detailed accounts on the characterization of the interesting features in the power +spectrum of PBH forming single-field scenarios, we refer the reader to [86, 181, 183, 184, 186, 193– +195, 201]. +Stochastic inflation and quantum diffusion +While, so far, we focused on the predictions of the second order action (3.2), non-linearities +and non-Gaussian effects can play an important role in the production of PBHs, as we learned +in Section 2.3. For the case of ultra–slow-roll (USR) models based on non-attractor phases of +inflation, there are sources of non-Gaussianity associated with stochastic effects during inflation. +28 + +The stochastic approach to inflation, pioneered by Starobinsky [202], constitutes a powerful +formalism for describing the evolution of coarse-grained, super-horizon fluctuations during inflation. +It is based on a classical (but stochastic) Langevin equation, which reads in canonical single-field +inflation (N is the number of e-folds, and we assume constant sound speed and Planck mass): +dφ +dN = − V ′ +3 H2 + H +2π ξ(N) . +(3.25) +Here, φ represents a coarse-grained version of super-horizon scalar fluctuations; V ′ is the derivative +of the inflationary potential, which leads to a deterministic drift for the coarse-grained super- +horizon mode; ξ is a source of stochastic noise acting on long wavelength fluctuations, caused +by the continuous kicks of modes that cross the cosmological horizon, and pass from sub to +super-horizon scales during inflation. +Besides the physical insights that it offers, the inflationary stochastic formalism [202–210] offers +the opportunity to obtain accurate results for the probability distribution function controlling +coarse-grained super-horizon modes, beyond any Gaussian approximation. As a classic example, +by solving the Fokker-Planck equation associated with (3.25), the seminal work [206] analytically +obtained the full non-Gaussian distribution functions for certain representative inflationary +potentials, going beyond the reach of a perturbative treatment of the problem. +Returning to the discussion of an USR inflationary evolution for PBH scenarios, we can expect +that stochastic effects can be very relevant in this context, see e.g. [209, 211–219]. In fact, since +the amplitude of scalar fluctuations gets amplified, the stochastic noise can become much larger +than what occurs in slow-roll inflation. Moreover, during USR, the derivative of the potential +V ′ = 0, the classical drift is absent, and the stochastic evolution is driven by stochastic effects +only. Various works studied the topic by solving the stochastic evolution equations, and [220–223] +find that non-Gaussian effects can change the predictions of PBH formation, depending on the +duration of the USR phase. In fact, the stochastic noise modifies the tails for the curvature +probability distribution function, which decays with an exponential (instead of a Gaussian) profile, +and consequently tends to overproduce PBHs. [220–223] set constraints on the duration of the +USR phase, which (depending on the scenarios) can last at most few e-folds before overproducing +PBH. There is a growing activity on these subjects, and we refer the readers to the aforementioned +literature for details on the state of the art on this important topic. +3.3 +Growth in the power spectrum when the decaying modes are slacking +Slow-roll violation without triggering decaying modes +We learned in the previous subsections that a possible way for enhancing the spectrum of +fluctuations at small scales, with respect to its large-scale counterpart, is to amplify the k- +dependent corrections to the constant-mode solution R(0) +k +within the parenthesis of Eq. (3.6). +But, as we anticipated in the paragraph following Eq. (3.6), we can also design scenarios where +an enhanced time-dependence of the slow-roll parameters leads to a scale-dependent curvature +power spectrum at horizon crossing, even without exciting the decaying mode at super-horizon +scales. The idea is to still make sure that the pump field z(τ) increases with time – hence +conditions (3.13) and (3.14) are not satisfied, the decaying mode keeps inactive, and the terms +29 + +within parenthesis of Eq. (3.6) can be neglected. However, at the same time, each individual +slow-roll parameter changes considerably during a short time interval during inflation. The +derivatives of slow-roll parameters can be large: they can contribute significantly to the quantity +Θ controlling the Mukhanov-Sasaki equation, and they can influence the scale-dependence of the +the curvature spectrum at horizon crossing (see Eqs (3.10) and (3.12)). +We start this section by setting the stage, and derive formulas for describing this possibility. +We will then we present an explicit realization of this scenario. It is convenient to work with the +canonical variable vk defined through Eq. (3.9), and solve the Mukhanov-Sasaki system in the form +of the set of equations (3.10)-(3.26). We assume that the pump field z is monotonic and always +increasing with time, and we identify the sound horizon of fluctuations as aH/cs ≃ +� +z′′/(2z) 21. +We can identify two asymptotic regimes for each mode k: i) an early-time regime, when each +mode is deep inside the horizon and ii) a late-time one, when the modes get stretched to become +super-horizon. On the one hand, in the former regime the modes satisfy k2 ≫ z′′/z, and behave +as the standard vacuum fluctuations in Minskowski space-time +vk(¯τ) = e−ik¯τ +√ +2k +, +for +k2 ≫ z′′ +z . +(3.26) +On the other hand, later during inflation the fluctuations get stretched outside the horizon, +entering the second regime, and eventually satisfying k2 ≪ z′′/z, with a solution given by +vk(¯τ) ≃ C1,k z + D2,k z +� +d¯τ ′ +z2(¯τ ′) + O(k2), +(3.27) +where the finite k2 corrections to this solution can be derived in a similar fashion as in (3.6). +Recall that we are now interested in attractor background configurations, hence we can neglect +the last two terms in Eq. (3.27) that rapidly decay. Shortly after horizon crossing, the canonical +variable will settle into the solution vk = z C1,k. Using the field redefinition (3.9) we can identify +the constant mode as the curvature perturbation at late times C1,k = Rk = R(0) +k . In order to +determine its expression, we match the solutions some time around horizon crossing ¯τ = ¯τ0, and +we obtain +|C1,k|2 = |R(0) +k |2 = 1 +2k +1 +z(¯τ0)2 = 1 +2k +cs +a2M22ϵ +���� +¯τ=¯τ0 +. +(3.28) +The horizon-crossing time can be conveniently expressed as leading contribution in a WKB +approximation [59]: +k2 = z′′ +2z +���� +¯τ0 += +�aH +cs +�2 +(1 + Θ) +���� +¯τ0 +, +(3.29) +with Θ given in Eq. (3.12). Collecting there results, we can write the late-time power spectrum +21Notice that if the background dynamics have localized features, phases of slow-roll violation might occur for +some time interval, and this exact identification ceases to be valid. In particular, in this case z′′/z might loose its +monotonic nature for some time interval, leading to multiple horizon crossing for a range of modes, resulting in +oscillations of the spectrum [59]. For the time being, we set a discussion on this possibility aside, and continue +identifying the quantity z′′/z as an effective horizon for slow-roll violating scenarios we are interested in. +30 + +X +na +n∗ +N∗ +σ +ϵ +−2. +−4. +36.0 +2.0 +cs +0. +−2. +35.5 +2.5 +˜ +M2 +0.0 +−3.0 +34.8 +4.0 +Table 3. +Parameter choices that characterize the background evolution of the time dependent parameters +ϵ, cs, ˜ +M = M/Mpl. +0 +10 +20 +30 +40 +50 +60 +N +10−4 +10−3 +10−2 +10−1 +100 +ϵ +cs +˜ +M2 +0 +10 +20 +30 +40 +50 +60 +N +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +1.75 +2.00 +–— +z′/z +(aH/cs) +Figure 7. Left panel: Evolution of ϵ, cs, ˜ +M 2 = M 2/M 2 +pl in e-folds characterized by the expression (3.31) +with the parameter choices presented in Table 3. Right panel: time evolution of (3.13) in units of aH/cs +to illustrate the fact that the decaying modes do not grow for the background presented in the left panel. +for curvature fluctuations as +PR(k) ≡ k3 +2π2 +���� +vk(¯τ) +z +���� +2 += k3 +2π2 +���R(0) +k +��� +2 += +H2 +8π2 ϵ cs M2 (1 + Θ) +���� +¯τ0 +. +(3.30) +From (3.30) we observe that rapid changes in the background quantities ϵ, cs, M and slow-roll +parameters constituting the quantity Θ of Eq. (3.12) as a function of ¯τ0 can then translate into a +scale-dependent amplification of the power spectrum. As we will see, this situation leads to a scale +dependent enhancement in the power spectrum realized through the ‘constructive interference’ of +the time-dependent background parameters ϵ, cs, M. +An explicit realization +We now review a possible realization of this scenario, following closely the discussion of [59]. +We focus on the generalized single-field framework discussed in Section (3.1), and we consider a +background dynamics that includes simultaneous pronounced dips in the time dependent profiles +for the parameters ϵ, cs, M2. We then study the power spectrum by solving numerically the +Mukhanov-Sasaki equation for curvature perturbations (see Appendix C), and we compare the +result with the analytical expressions discussed in Section 3.1 (see e.g. (3.30)). +To analyze a representative scenario in this cathegory, we parameterize the three time dependent +31 + +10−2 +101 +104 +107 +1010 +1013 +k +[Mpc−1] +10−9 +10−8 +10−7 +10−6 +10−5 +10−4 +10−3 +PR(k) +PR ∝ k3/2 +PR(k) via eq. (3.30) +1013 +2 × 1012 +3 × 1012 4 × 1012 +6 × 1012 +2 × 1013 +k +[Mpc−1] +10−9 +10−8 +10−7 +10−6 +10−5 +10−4 +10−3 +PR(k) +k = 6 × 1012 Mpc−1 +k = 7 × 1012 Mpc−1 +Figure 8. The full power spectrum of curvature perturbation for the background model presented in Fig. 7 +(Left). The blue dots represents the accuracy of the formula (3.30) in describing the rise of the PR towards +its peak. As shown by the red-dotted line, the spectral index on the rise satisfies ns − 1 ≲ 3/2. Power +spectrum for scales following the peak to illustrate the oscillations in the amplitude at those scales (Right). +quantities X = {ϵ, cs, M2} as [59] +log10 X(N) = (na − n∗) tanh2 +�N − N∗ +σ +� ++ n∗. +(3.31) +Each of these quantities tend to 10na asymptotically away from N∗ in both directions, |N−N∗| ≫ σ +and become equal to 10n∗ at N∗ while staying around the neighborhood of this value for a number +of e-folds determined by σ. Notice that we are interested in features of the inflationary dynamics +that affect modes at scales much smaller than the CMB pivot scale. Hence we can assume N∗ ≫ 1, +where modes associated with CMB scales leave the horizon at N ≃ 0, while inflation ends at +Nend = 60. A representative set of parameter choices that leads to a localized decrease in the +slow-roll parameters is presented in Table 3. The resulting background evolution together with +the behavior of z′/z is shown in the left and right panels of Fig. 7. We observe from the right +panel of the figure that z′/z > 0 is always satisfied during inflation, suggesting (as expected) that +the decaying modes do not grow in time. Using the parametrization (3.31) we then numerically +solve the Mukhanov-Sasaki equation, following Appendix C. +Our results for PR(k, Nend) are presented in Fig. 8. In the left panel we notice that the +expression (3.30) accurately describes the behavior of the power spectrum towards its peak, +confirming our expectation that the constant growing mode is responsible for the enhancement. +Notice the absence of the dip proceeding the growth (which characterized instead the scenario +of Fig 6). This in agreement with our interpretation in the previous section: the dip is due +to disruptive interference between ‘growing’ and ‘decaying’ modes – while in this context the +decaying mode is not active. On the other hand, as shown in the right panel of the figure, the +power spectrum exhibits oscillations for scales following the peak. As we discussed in footnote +21, this is due to multiple horizon crossing of modes within certain momentum scales, leading to +excited states and oscillations in the spectrum. We illustrate this phenomenon in Fig. 9 —for two +32 + +34 +36 +38 +40 +42 +N +1012 +1013 +k = 6 × 1012 Mpc−1 +k = 7 × 1012 Mpc−1 +� +|z′′/z| +38 +39 +40 +41 +42 +43 +44 +45 +N +10−4 +10−3 +10−2 +–— +� +k3 +2π2 +�1/2 ���vk/z +��� +k2 ≪ z′′/z +←−− Modes are frozen −−→ +Figure 9. Left: The occurrence of multiple horizon crossing for the neighboring modes labeled by red and +purple dot in Fig. 8. Right: Evolution of the modes that exhibit multiple horizon crossing (note the same +color coding with the left panel). The vertical lines illustrates the final horizon crossing time for each mode. +neighboring modes labeled by a red and purple dots in Fig. 8— where we plot k versus +� +|z′′/z| as +a function of e-folds. As shown in the right panel of Fig. 9, although these modes are neighboring, +they exhibit non-trivial behaviour before their final horizon exit N ≃ 40 so that their asymptotic +values (N → Nend) differ considerably, giving rise to the sizeable modulations in the late time +power spectrum spectrum (see Fig. 8), as discussed in [59]. +Besides [59], the ideas discussed in this subsection for enhancing the spectrum at small scales +found further realizations in [60]. Conceptually similar frameworks include sound speed resonance +scenario proposed [61], and realized within a DBI model in [62]. +3.4 +Brief summary +We find it remarkable that many distinct single-field models of inflation, built with the aim +of producing PBH, share common features in the properties of the resulting power spectrum. +The reason being that the enhancement of the curvature spectrum at small scales is due to few +mechanisms common to several scenarios. We identified the idea of resurrecting the decaying +mode of curvature fluctuations at super Hubble scales, by increasing the absolute value of the +slow-roll parameter η – see Section 3.2 – and the idea of having very rapid changes in the values +of slow-roll parameters (keeping them relatively small) – see Section 3.3. +The resulting properties of the curvature spectrum might result essential for testing single-field +models of inflation at small scales, thanks to their predictions for the PBH population properties, +as well as for the spectrum of gravitational waves induced at second order in perturbations [79, 80]. +In fact, the latter is a very interesting topic, relevant and well-studied for single-field models: we +refer the reader to [94] for a detailed review. +Nevertheless, the existing explicit scenarios of single-field inflation, which are able to generate +an appreciable PBH abundance, typically suffer of severe fine-tunings on the choices of the +parameters characterizing their Lagrangians. For example, in producing sufficiently flat, plateau +regions of their potential, and in ensuring regular transitions between attractor and non-attractor +eras during the inflation process. See e.g. the discussion in [47] (but also [224] for a scenario +33 + +that can partially ameliorate the tuning involved). Moreover, as we learned in Section 2.2, the +resulting PBH population can be very sensitive to the details of the spectrum profile, and to +the presence of primordial non-Gaussianities. The latter should at certain extent be generated +in many concrete single-field realizations (see Section 3.2), and render particularly delicate the +task of making precise theoretical predictions. In fact, large non-Gaussianities might not only +change the predictions for PBH production, but can also impose restrictions on single-field model +building, due to large one-loop corrections [225, 226] (see however the recent [227]). +It is then interesting to try to consider PBH inflationary scenarios in contexts where more +than one field acquire dynamics during inflation. The hope being to find qualitatively new ideas +for producing PBH, or alternatively more natural realizations of known mechanisms, in order to +amplify the primordial curvature spectrum at small scales. Possibly, multiple fields affect the +predictions on the statistics of curvature fluctuations with respect to single-field models, with +important implications for PBH formation. We turn to review this topic in what comes next. +4 +Enhanced primordial power spectrum in multi-field models +Despite the remarkable agreement of single-field, slow-roll inflation with the CMB and the LSS +data [4], the fundamental nature of inflation continues to elude us. In particular, it would be +very important to know whether additional fields took part in the dynamics of inflation, besides +the scalar driving cosmic expansion. In fact, while current cosmological observations do not +provide hints of iso-curvature fluctuations associated with extra inflationary degrees of freedom, +it might also be that additional fields play a role during epochs of inflation that are not well +probed by current large-scale surveys. The formation of PBHs, occurring at small scales, can be +sensitive to iso-curvature fluctuations, and represent a valuable probe of inflationary multi-field +dynamics. Multi-field inflation can offer new possibilities for producing PBH, exploiting novel, +distinctive ways for converting large gradients of background quantities into the spectrum of +curvature fluctuations. Moreover, when used in tandem with the techniques discussed in the +previous section, multi-field models might alleviate some of the fine-tuning issues occurring in +single-field scenarios (see the discussion in Section 3.4). For these reasons, in this section we +review a selected choice of theoretical frameworks aimed at enhancing the curvature spectrum +at small scales, by using tools that specifically involve more than one field during inflation, in +particular the tachyonic behaviour of field dynamics in some extra sectors during inflation. +First, in Section 4.1, we review PBH scenarios based on axions interacting with gauge fields +during inflation, exploiting a mechanism based on particle production during inflation. This +possibility is well motivated by particle physics constructions, and it has been much explored in +the literature: we make efforts for carefully reviewing the status of the art, discussing opportunities +and challenges for PBH formation in this context. Then, in Section 4.2 we review multi-field +inflationary scenarios where the rise in the spectrum is a result of large iso-curvature perturbations, +induced by a curved inflationary trajectory traversing a rich multi-field moduli space. +� Main References: In the discussion of Section 4.1 we benefit from the works presented in +[71–74, 77, 78] while the material presented in Section 4.2 is based on [66, 67]. +34 + +4.1 +Enhanced scalar perturbations from axion-gauge field dynamics +Axions [228, 229] are pseudo-scalar particles, theoretically introduced in studies of particle physics +theories beyond the Standard Model. First motivated in scenarios addressing the strong CP +problem in terms of a Peccei-Quinn mechanism [230, 231], they find natural realizations in string +theory (see e.g. [232]), as well as many useful applications to cosmology – see e.g. [233] for a +comprehensive review. The possibility of using the physics of axions for producing PBH is then +very interesting, given their strong theoretical and experimental motivations which allow to make +connections between particle physics, quantum gravity, and cosmology. We do so in this section, +reviewing several representative ideas and their realizations in this context. +Axion fields, denoted here as χ, correspond to Goldstone bosons of the Peccei-Quinn (PQ) +global symmetry, spontaneously broken at a scale dubbed f. Their Goldstone nature equips axions +with a shift symmetry χ → χ+constant, valid at all orders in perturbations. The shift symmetry +makes them interesting inflaton candidates, being their potential naturally very flat [234]. The +PQ symmetry suffers from a chiral anomaly, which breaks the shift symmetry and assign axions a +potential V (χ) through non-perturbative contributions in field theory, or monodromy effects in +string theory (see e.g. [63] for a review). The chiral anomaly implies that axions interact with +gauge vectors through higher-dimensional operators. This property is essential for our arguments. +The typical structure of an axion Lagrangian considered for cosmological purposes reads +L = −1 +2∂µχ ∂µχ − V (χ) − 1 +4Fµν F µν − gcs +4f χ Fµν ˜F µν . +(4.1) +The axion χ is equipped by a kinetic term and a potential term V (χ). The potential is often +a polynomial function of the axion field, at least for small values of χ 22. As anticipated, the +Lagrangian (4.1) also includes a U(1) gauge field Aµ with field strength Fµν, coupling to the axion +through a dimension-5, gauge-preserving operator +Lint = −gcs +4f χ Fµν ˜F µν , +(4.2) +where f is the PQ symmetry-breaking scale, and gcs is a dimensionless coupling constant. The +dual field strength appearing in Eq. (4.2) is ˜F µν ≡ ηµνρσFρσ/(2√−g), and the alternating symbol +η is 1 for even permutations of its indices, and −1 otherwise (starting with η0123 = 1). The +dimension-5 operator (4.2) is especially important for our arguments. As we review below, this +axion-vector coupling leads to a tachyonic instability in the gauge sector, which exponentially +enhances one of the helicities of the gauge vector modes, and produces a large amount of gauge +quanta [239]. The energy adsorbed in the production of vector modes is carried away from the +axion kinetic energy, that slows down its evolution: this is good news for axion inflationary models, +since the axion can slowly roll even along steep potentials [240]. Moreover, the inverse decay of +enhanced gauge quanta into axion fluctuations (schematically, A + A → χ), can amplify curvature +perturbations [241, 242], and lead to large curvature spectrum at small scales, at a level able +to produce PBH [71]. We refer the reader to [243] for a review on axion inflation written one +decade ago, which includes many of the topics mentioned above. We now focus on reviewing +22See e.g. [235–238] for string theory motivated constructions in the context of inflation. +35 + +the consequences of these ideas for scenarios amplifying curvature fluctuations, also covering +opportunities and challenges pushed forward in the most recent literature on this subject. +Amplification of gauge field fluctuations +When considering a dynamical axion field with a time dependent background profile ¯χ(t), the +dimension-5 axion-vector interaction of Eq. (4.2) leads to a copious production of vector fluctu- +ations. This property can be understood by considering the equation of motion (EoM) for the +gauge field mode functions (See Appendix D) [240] +∂2 +xA± + +� +1 ± 2ξ +x +� +A± = 0, +ξ ≡ −gcs ˙¯χ +2Hf +(4.3) +where A± correspond to the gauge vector polarizations, and we defined a dimensionless time +variable x ≡ −kτ = k/(aH), as well as the effective dimensionless coupling ξ between the spectator +axion and the gauge field. Without loss of generality, we work within the conditions +ξ > 0 +and +˙¯χ < 0 . +(4.4) +Hence, the axion rolls along its potential from large positive to small values of ¯χ ≥ 0. +Notice from (4.3) that the dimension five operator of Eq. (4.2) introduces a time dependent +mass term in the dispersion relation of U(1) field, which changes sign depending on the helicity +λ = ±. This property reflects the parity violating nature of the dimension-5 interaction. When the +gauge modes are deep inside the horizon, (x = k/(aH) ≫ 1), the time-dependent correction term +is negligible, and both vector polarizations obey a standard dispersion relation as in Minkowski +space. However, as the modes stretch outside the horizon, the correction becomes dominant for +x = k/(aH) ≲ 2ξ, leading to an instability for one of the circular polarizations of the gauge fields. +In our conventions (4.4), only A− state experiences a tachyonic instability, while A+ is unaffected. +The dynamics of the axion like field, controlled by the axion velocity ˙¯χ ̸= 0, therefore induces a +significant production of helical vector fields. +The nature of gauge field production, and its consequences as a source for scalar perturbations, +is sensitive to the scalar potential V (χ), since this quantity determines the profile of the axion +background velocity ˙¯χ. The dynamics of the curvature perturbations, as generated through the +axion-gauge field dynamics, depends on whether we identify the axion field as the inflaton, or +whether it belongs to some extra spectator sector during inflation. In what comes next, we arrange +our discussion so to clearly distinguish among these possibilities. We consider the following +scenarios: +1. Section 4.1.1 As first possibility, we identify the axion χ with the inflaton φ that drives +inflation: χ = φ. Then, the order parameter ˙¯φ controlling the gauge-field production +increases with time, generating scalar perturbations through an inverse vector decay: δA + +δA → δφ. We refer to this scenario as “Smooth Axion Inflation”, and models that can +be considered in this category are studied in [71–74]. Such scenarios suffer of dynamical +instabilities associated with large back-reaction effects from the gauge fields on the axion +evolution. +36 + +2. Section 4.1.2 In certain axion-inflation models, the axion potential has special features, +located far away from the field range corresponding to scales affecting CMB physics. A +sudden increase in the axion inflaton velocity occurs at their location, with enhanced scalar +perturbations amplified at small PBH scales through inverse vector decay. This possibility +is first discussed in [73], while further developments are studied in [75, 78]. We refer to +scenarios producing PBH by exploiting localized particle production as “Bumpy Axion +Inflation”, and we analyze how they address the instabilities mentioned above. +3. Section 4.1.3 A final possibility corresponds to gauge field production in a hidden sector, +through the dynamics of an axion spectator field. Also this case allows one to address the +aforementioned dynamical instabilites, given that the back-reaction effects from the vector +sector can be placed under control. Depending on the shape of the spectator axion potential, +such a scenario can lead to localized peaks in the scalar curvature power spectrum [73, 77]. +We refer to this scenario as “Spectator axion-gauge field dynamics”. +We build our discussion mainly in terms of representative, concrete examples, presenting the main +results and relegating technical details to Appendixes. +4.1.1 +Smooth Axion Inflation +In the first scenario we consider, we identify the axion χ with the scalar inflaton that drives +inflation, dubbed φ: χ → φ. We study the behaviour of scalar perturbations in a set-up described +by Lagrangian (4.1). We assume that the profile for the scalar potential V (φ) is sufficiently flat, +so to support inflation. In this case, the effective coupling ξ, as defined in Eq. (4.3), adiabatically +increases during the inflationary process: +ξ ≡ gcs +� ϵ +2 +Mpl +f +, +(4.5) +where ϵ is the standard slow-roll parameter. The amplified gauge-field mode function can be +analytically expressed as [240] (recall that the axion speed has negative sign, see Eq. (4.4)) +A−(τ, k) ≃ +1 +√ +2k +�−kτ +2ξ +�1/4 +exp(πξ − 2 +� +−2ξkτ), +ξ ≡ −gcs ˙¯φ +2Hf . +(4.6) +The slowly changing time-dependent parameters ξ and H in Eq. (4.6) are evaluated at the epoch +of horizon crossing. The amplification of gauge field modes is maximized when the size of the +mode is comparable to the horizon, −kτ ∼ O(1). +In fact, the analytic expression in (4.6) is valid within the interval (8ξ)−1 ≪ −kτ ≪ 2ξ. For +the values ξ ∼ O(1) we will be interested on, this range corresponds to a phase during which the +gauge modes grow, and then remain frozen to their maximal value before being diluted away by +the universe expansion. We proceed discussing the time evolution of the relevant quantities in +this set-up. For concreteness, we focus on a representative example. For a more detailed account +on the gauge field amplification by the slowly rolling scalar we refer the reader to Section 2.1 of +[244], or Appendix B of [245]. +37 + +Background evolution in a concrete example +As the effective coupling ξ of Eq. (4.6) increases during inflation, the enhanced vector modes +eventually lead to a sizeable back-reaction on the background evolution of the axionic inflaton +field. This fact can significantly affect the homogeneous dynamics of the system. +The back-reaction is mainly controlled by the vector-dependent friction term in the equation +of motion for the homogeneous inflaton field [240]. This phenomenon implies that the gauge field +amplification occurs at the expense of the inflaton velocity ˙¯φ. We consider a situation characterized +by a transition between a standard slow-roll dynamics in the early stages of inflation, and a new +attractor regime at late times, when the gauge field enhancement dominates over the Hubble +friction [246] (but see the cautionary remark towards the end of this section). +The aforementioned friction effect – induced by the gauge mode production – can be analyzed +considering modified Klein-Gordon and Friedmann equations: +¨¯φ + 3H ˙¯φ + V ′(¯φ) = gcs +f ⟨ ⃗E · ⃗B⟩ , +3H2M2 +pl = 1 +2 +˙¯φ2 + V (¯φ) + 1 +2 +� +⃗E2 + ⃗B2� +, +(4.7) +where we introduced ‘electric’ and ‘magnetic’ field contributions23, as discussed in detail in +Appendix D (see in particular Eq. (D.22)). Using the solutions (4.6) for gauge-field modes, the +expectation values for the electric and magnetic fields can be computed analytically [240, 245], +finding (see Appendix D) +⟨ ⃗E · ⃗B⟩ ≃ 2.1 × 10−4 H4 +ξ4 e2πξ, +ρA ≡ 1 +2⟨ ⃗E2 + ⃗B2⟩ = 1.4 × 10−4 H4 +ξ3 e2πξ . +(4.8) +The quantity ρA corresponds the total energy density contained in the gauge field fluctuations. +Using the evolution equations (4.7), we can identify the conditions corresponding to a small +back-reaction of the vector modes into the background evolution. They are 3H| ˙¯φ| ≫ gcs⟨ ⃗E · ⃗B⟩/f +and ρA ≪ 3H2M2 +pl, which can be expressed as +ξ−3/2eπξ ≪ 79 +˙¯φ +H2 , +−→ +negligible back-reaction on φ equation , +ξ−3/2eπξ ≪ 146Mpl +H , +−→ +negligible back-reaction on the Friedmann equation . +(4.9) +Assuming that inflation starts in a standard slow-roll regime, the relation ˙φ = +√ +2ϵHMpl ≪ HMpl +implies that the first condition in (4.9) is more demanding than the second. When these conditions +are not met, we enter into an inflationary phase characterized by a strong back-reaction of gauge +modes, which goes beyond the standard slow-roll inflationary attractor. +To concretely illustrate the back-reaction of the gauge field production on the homogeneous +background evolution of the inflaton, we focus on a modified version of the linear monodromy +type potential [235] that interpolates between V ∝ φ and V ∝ φ2 from large to small field values +23Be aware that, despite the terminology we adopt, the gauge fields Aµ do not need to correspond to Standard +Model photons. +38 + +0 +10 +20 +30 +40 +50 +60 +N +0 +2 +4 +6 +8 +10 +¯φ +Mpl +Slow-roll +Backreaction +becomes important +ξbr = 4.73 +ξ ≃ 3 +ξ ≃ 3.6 +ξcmb = 2.5 +ξ = 0 +0 +10 +20 +30 +40 +50 +60 +N +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +3H ˙¯φ +V ′(¯φ) +gcs +f +⟨ ⃗E· ⃗B⟩ +V ′(¯φ) +Figure 10. Field profile ¯φ(N) with respect to e-folding number N in smooth axion inflation for the initial +choice ξcmb = ξ(N = 0) = 2.5 (Left). At early times, the dynamics is in the standard slow-roll regime (light +blue region) but as ξ increases, it enters into a stage dominated by the friction generated due to gauge field +production (light red region) which has the effect of prolonging inflation with respect to the background +evolution with ξ = 0. For the choice of ξcmb = 2.5, the behavior of the Hubble damping term 3H ˙¯φ and the +friction induced by the source term ∝ ⟨ ⃗E · ⃗B⟩ are shown in the right panel. +around the global minimum: +V (φ) = λM4 +pl +�� +1 + φ2 +M2 +pl +− 1 +� +, +(4.10) +where λ is a dimensionless parameter that fixes the overall amplitude of the potential. Assuming +negligible back-reaction in the beginning of inflation, λ can determined by enforcing the standard +normalization of power spectrum in the slow-roll regime for a given choice of φin using +PR(kcmb) = +H2 +8π2ϵM2 +pl +≃ +V (φin) +24π2ϵV (φin)M4 +pl +≡ 2.1 × 10−9. +(4.11) +We then solve numerically the equations (4.7) utilizing (4.8) by setting the coupling between the +inflaton and the gauge fields to its maximally allowed value 24 by Planck: gcsMpl/f = 48, see +e.g. [248]. Note that for a given initial value of the scalar field, this number can be translated to +an initial value of the effective coupling which we label as ξcmb: +ξcmb ≃ gcs Mpl +f +1 +2φin +. +(4.12) +For φin = 10.5 Mpl (ξcmb = 2.5), the results of the numerical computation are shown in Fig. 10. +In the left panel we show the inflaton profile as a function of e-folds until the end of inflation +24In fact, at large CMB scales, described by modes leaving the horizon in the early stages of inflation, the value +of the effective coupling ξ is restricted by existing information and constraints on the scalar power spectrum and +non-Gaussianity [241, 242, 247]. Depending on priors and on the shape of the inflaton potential, these constraints +give ξcmb < 2.2 − 2.5 [248]. +39 + +where ˙H = −H2. We can clearly observe that in the absence of coupling to the gauge fields, +gcs = 0 (black dashed line with ξ = 0), inflation lasts for about ≈ 48 e-folds. Turning on the +interactions with gauge fields at the beginning of inflation (solid orange line with ξcmb = 2.5), the +back-reaction induced by the vector fluctuations become noticeable around ≈ 36 e-folds, which in +turn has the overall effect of extending the duration of inflation for about ≈ 12 e-folds compared +to the standard slow-roll case (ξ = 0). +These findings are also supported by the right panel of Fig. 10, where we compare the standard +Hubble damping term 3H ˙φ with the friction term gcs⟨ ⃗E · ⃗B⟩/f, induced by the gauge field +production as a function of e-folds until the end of inflation. We observe that the standard Hubble +friction dominates the dynamics at early times, but as the effective coupling ξ increases the +dynamics gradually evolves into a regime dominated by the back-reaction of the produced gauge +quanta. For the parameter choices we adopt, the onset of this regime starts around Nbr ≃ 36 +e-folds, and at around N ≃ O(50) the system enters into a strong back-reaction regime where the +dynamics becomes completely controlled by the gauge modes [240]. +Therefore, in any viable model of axion-inflation, back-reaction effects induced by gauge field +production become important only towards the latest stages of inflation. For example, for a +rough estimate of the orders of magnitude involved we can adopt a linear potential V ∝ φ. +Assuming the dynamics is initially in the slow-roll regime, we have ξ ≃ gcs +√2ϵV Mpl/f ∝ 1/φ +and φcmb/φbr = ξbr/ξcmb ≃ 2.2 − 1.9. For typical initial field values that can sustain enough +inflationary evolution, we have φcmb ∼ O(10)Mpl, which implies that φbr ∼ O(5)Mpl. Hence +back-reaction becomes dominant in the latest stages, but before the end of inflation corresponding +to φend ∼ O(1). +Scalar fluctuations sourced by gauge fields +As inflation progresses, the particle production becomes more and more efficient. Hence, the +vector modes start to act as an additional source of inflaton fluctuations δφ, due to the presence +of tri-linear coupling δLint ∝ δφF ˜F (See Appendix D.2, and references therein). In particular, +a tachyonic amplification of the vector fields leads – at second order in perturbations – to +an enhancement in the scalar sector. Since these contributions to the scalar fluctuations are +statistically independent from their vacuum counterpart, the resulting curvature power spectrum +acquires two separate contributions (see Appendix E) that can be parameterized as [71, 240] +PR = P(v) +R + P(s) +R , +≃ +H2 +8π2ϵM2 +pl ++ +� +gcs +f +⟨ ⃗E · ⃗B⟩ +3Hβ ˙¯φ +�2 +F2 . +(4.13) +The first term in the second line denotes the standard vacuum contribution to the curvature power +spectrum. The second accounts for the vector part, with the time dependent quantities β and F +defined in Eqs. (D.40) and (E.6). In the weak back-reaction regime, the power spectrum of the +curvature perturbation is dominated by the vacuum contribution in the first term of Eq. (4.13). +As inflation progresses, the effective coupling between the inflaton and the vector fields increases: +the source contribution to the power spectrum starts to kick in, and the power spectrum grows as +PR ∝ ⟨ ⃗E · ⃗B⟩2 ∼ e4πξ (the system is still in the weak back-reaction regime where β ≃ 1). Then +40 + +0 +10 +20 +30 +40 +50 +60 +N +10−9 +10−7 +10−5 +10−3 +PR(N) +ξbr = 4.73 +ξ ≃ 3 +ξ ≃ 3.6 +Figure 11. Total power spectrum (4.13) as a function of e-folds in smooth axion inflation for the potential +in (4.10) and ξ(N = 0) ≡ ξcmb = 2.5 as we discuss in the main text. +the scalar fluctuations grow, they start influencing the gauge fields sources, and lead to an increase +in the damping factor β ∼ ⟨ ⃗E · ⃗B⟩/3H ˙¯φ. Therefore, eventually the sourced contribution to the +power spectrum saturates towards late times during inflation. In fact, the time-dependent factor +F is introduced to capture the modified definition of the curvature perturbation in the strong +back-reaction regime. As discussed in detail in [73], this factor is responsible for an order one +correction to the power spectrum amplitude towards the end of inflation. +We summarize these arguments with a plot, Fig. 11, of the total power spectrum with respect to +the e-folding number. We notice that as the effective coupling ξ between the scalar and the gauge +fields increases along the inflationary trajectory, the power spectrum acquires a large contribution +from the vector source. As desired, it grows towards smaller scales, eventually saturating towards +a constant value. Since the source of the peak in the power spectrum originates from a tri-linear +coupling between the inflaton and the gauge fields, the statistics of the curvature perturbation at +those scales is non-Gaussian (see also [249]). The values of the curvature spectrum amplitude at +the peak appearing in Fig. 11 is sufficiently large to generate a sizeable population of the PBHs +(recall the discussion in Section 2.3). +Cautionary remarks: an instability in the inflaton dynamics +We warn the reader that the findings reviewed in this section have been recently questioned. In +particular, [249–253] study the axion-gauge field dynamics for different choices of potentials and +parameters, focusing in the strong back-reaction regime. By implementing sophisticated numerical +techniques, these works go beyond the constant-velocity approximation assumed in our discussion +above. Their findings indicate that once entering the strong back-reaction regime, the inflaton +velocity is characterized by oscillations of increasing amplitude, in sharp contrast with analytical +studies approximating ˙¯φ (and ξ) as constant. Finally, the recent analytic study of [254] fully +support the numerical results. The physical source of this instability seems to be due to a delayed +response of the vector source to changes in the axion velocity ˙¯φ, whenever the system enters in +a strong back-reaction regime. This phenomenon causes the aforementioned oscillations with +increasing amplitude in the inflaton velocity. +41 + +It is not clear at the moment whether these secular effects can be tamed by adding ingredients +to the set-up discussed in this section. A potential way out is to modify the inflationary dynamics +in such a way to push the strong vector back-reaction regime towards an epoch that does not +affect the predictions on PBH formation in the scalar sector. This possibility can be achieved +in scenarios where the gauge field production is localized, leading to a pronounced peak in the +scalar power spectrum before entering into a strong back-reaction regime (see Section 4.1.2). +Such a way out does not avoid the instability, but at least moves it towards epochs that do not +affect our predictions for the production and properties of curvature perturbations. In Section +4.1.3 we instead discuss the possibility to avoid at all a strong vector back-reaction regime, in +scenarios where the axion is a spectator field and does not drive inflation. Appropriate choices of +the axion potential nevertheless lead to an enhancement of the scalar sector, at a level sufficient +for producing PBH. All the opportunities we review next are possible in scenarios of inflation +including axion-vector couplings, which have very rich ramifications, and are well studied given +their particle physics motivations. +4.1.2 +Bumpy axion inflation +In this section, motivated by the instability arguments discussed above, we consider scenarios +with localized production of gauge fields, able to enhance the curvature spectrum and produce +PBH before entering any strong back-reaction regime. As we will see, the mechanism is based on +the ideas of the previous Section 4.1.1, but also uses tools for the activation of the decaying mode +that we introduced in Section 3.2. +For definiteness, we focus on a representative example, discussing its properties in the main +text and referring to Appendixes for technical details. We assume that the shift symmetry of +the axion is broken both by non-perturbative instanton corrections, as well as by a non-periodic +monomial term in the potential 25 (see e.g. [235, 237, 238, 256–260]), +V (φ) = 1 +2m2φ2 + Λ4 φ +f sin +�φ +f +� +, +(4.14) +where m, Λ, f are parameters of unit mass dimension. In fact, potentials as the above, charac- +terized by modulations on top of a smooth profile, are well motivated and studied in theoretical +constructions of axion inflation models in a variety of situations. For these reasons, we find +interesting to review their useful applications for our aim of producing PBH. +We are interested in obtaining noticeable modulations of the functional form of the potential, +within the regime Λ4 ≲ m2f2 (which we refer as “bumpy” regime in what follows). In particular, +we exploit the aforementioned non-perturbative corrections for generating a roller-coaster profile +to the otherwise smooth potential, where plateau-like regions are connected to each other by steep +cliffs (See Fig. 12). While plateau-like regions are suitable to sustain long enough inflation – and +25The choice of the potential in (4.14) is not unique for the purpose of generating a localized particle production +scenario. In this context, any potential that exhibits step-like feature(s) is sufficient. For example similar to the case +we consider, any potential of the form V (φ) = Vmon(φ)+Vmod(φ) where the modulation part satisfying Vmod ≲ Vmon +is enough for the effects we explore (see e.g. [75, 255]). The choice of the potential in (4.14) does however provide +some useful analytic control over the background evolution of the axion and hence for the resulting amplification of +the power spectrum as discussed in [78]. +42 + +0 +1 +2 +3 +4 +5 +φ/Mpl +0 +20 +40 +60 +80 +100 +120 +140 +V (φ) +m2f 2 +⇐ +β ≡ Λ4/(m2f 2) ≃ 0.996 +β → 0 +Figure 12. The shape of the potential (4.14) in the bumpy regime (orange curve) for Mpl/f = 3.3. The +red stars on the potential denotes the locations in field space where axion velocity and the slope of the +potential is maximal. +to generate nearly scale invariant fluctuations at CMB scales – when reaching the cliff-like regions +the scalar velocity speeds up intermittently. +During such brief stages, a localized production of gauge fields occurs. To illustrate this +phenomenon explicitly, we rely on a numerical procedure along the same lines of what discussed +in single-field models of inflation. We numerically solve the background equations (4.7) in the +bumpy regime of the potential (4.14), neglecting possible back-reaction effects induced by gauge +fields26. The resulting background evolution during inflation is represented in Fig. 13, in terms of +the first two slow-roll parameters ϵ(N), η(N), for the parameter choices used in Fig. 12. Adopting +the initial condition φin ≃ 4.8Mpl, the scalar field completes a total of 60 e-folding of its evolution +when traversing a single bump like region, around which the slow-roll parameter ϵ reaches its +maximal value of ϵ∗ = 0.455 at N∗ = 35.8. Following the stage of maximal velocity for the +scalar field, the background dynamics enters into a short non-attractor phase with η ≲ −6, +during which φ slows down before the dynamics re-enters into a final inflationary slow-roll stage +η ≪ 1. Since the inflationary background proceeds through an epoch of slow-roll violation, in +our calculations we compute the vacuum contribution to the scalar power spectrum using the +numerical methods discussed in Section 3.2, and in Appendix C. In fact, as anticipated above +in developing this example we make use both of ideas based on the conversion into curvature +fluctuations of gauge-field modes (as in the previous section), and on the activation of the would-be +decaying mode of the inflaton scalar sector (as discussed in Section 3.2). +Since while travelling the plateau-like regions the scalar velocity is small, the only contributions +to the scalar power spectrum arise from the cliff-like region of the potential (see Fig 12). Around +26As discussed in detail in [78], this assumption is actually self-consistent, thanks to the localized nature of the +gauge field production. +43 + +0.0 +0.5 +1.0 +(ϵ∗, N∗) = (0.455, 35.8) +ϵ +0 +10 +20 +30 +40 +50 +60 +N +2 +0 +−2 +−4 +−6 +η +Figure 13. Evolution of the Hubble slow-roll parameters ϵ and η with respect to e-fold number, for +the bumpy axion inflation model described by potential (4.14). We numerically evaluate (4.7) using the +parameter choices in Fig. 12 with φin ≃ 4.8, ignoring back-reaction effects as explained in the text. +the cliff, the scalar field velocity and the effective coupling ξ acquire the following profile [78]: +ξ(N) = +gcs δ +1 + δ2 (N − N∗)2 , +−→ +ξ∗ ≡ ξ(N∗) = gcsδ , +(4.15) +where ξ∗ denotes the maximal value of the effective coupling. The dimensionless parameter +δ ≃ +� +2/3(Mpl/f)(m/H) determines the amount of e-folds during which the parameter ξ maintains +its maximal value: i.e. ∆N ∼ δ−1. +Being δ ∝ m, the duration of such epoch is inversely +proportional to the restoring force provided by the potential: hence it inversely depends on +the mass parameter m. A time-dependent profile for the effective coupling gives rise to a scale- +dependent amplification of the gauge fields whose growing solutions correspond to the real part of +the mode functions, see Eq. (D.20), using Eq. (D.21). (See Appendix D.1 for more details). +As expected, the resulting sourced contribution to the power spectrum induced by the δA+δA → +δφ inherits the scale dependence of the gauge fields, and the curvature power spectrum can be +parametrized as [78], +PR(k) = P(v) +R (k) +� +1 + +H2 +64π2M2 +pl +f2,R +� +ξ∗, k +k∗ +, δ +�� +. +(4.16) +The vacuum contribution P(v) +R +can be numerically calculated following some of the methods +explained in Section 3. Due to the presence of a brief non-attractor phase, it is amplified by the +activation of the would-be scalar decaying mode (see Section 3.2). The dimensionless function +f2,R characterizes the contributions from the gauge fields, and has a log-normal shape [78], +f2,R +� +ξ∗, k +k∗ +, δ +� +≃ fc +2,R [ξ∗, δ] exp +� +− +1 +2σ2 +2,R [ξ∗, δ] ln2 +� +k +k∗xc +2,R [ξ∗, δ] +�� +. +(4.17) +As suggested by the expression (4.17), the information about the location, width and height +44 + +0 +5 +10 +15 +20 +25 +30 +35 +40 +45 +50 +N +10−14 +10−12 +10−10 +10−8 +10−6 +10−4 +10−2 +kpeak = k∗ xc +2,R +(k∗, N∗) = +(1.7 × 1012 Mpc−1, 35.8) +PR(N) : ξ∗ = 10.54 (gcs ≃ 6.7) +P(v) +R (N) +Figure 14. The total curvature power spectrum of Eq. (4.16) as a function of N (orange curve), for the +bumpy axion inflation model whose background evolution is shown in Fig. 13. We make the following +parameter choices: δ = 1.57, ξ∗ = 10.54 corresponding to gcs ≃ 6.7 (see e.g. (4.15)). +of the sourced signals in (4.16) depends on the motion of φ through the step-like features of its +wiggly potential, particularly through ξ∗ and δ dependence of the functions xc +2,R, σ2,R, fc +2,R. We +learn from (4.17) that the sourced signal is maximal at k = k∗xc +2,R, where it evaluates to fc +2,R +whereas σ2,R controls the width of the signal. For the background evolution presented in Fig. +13, the fitting functions describing the height, width and the location of the sourced signal is +calculated in [77], using the semi-analytical techniques developed in [261]. +Collecting all the results above, we present in Fig. 14 the full power spectrum in Eq. (4.16) in +terms of the e-folding number, by converting the scale dependence to e-fold dependence using +the horizon crossing condition k = a(N)H(N). Due to non-trivial background evolution, during +which the slow-roll conditions are briefly violated, the vacuum contribution to the power spectrum +(dotted lines) acquires a non-trivial shape, which shares the characteristic features of the single-field +scenarios we discussed in Section 3.2. Notice in fact the presence of the dip due to the interference +between growing and decaying modes of the curvature perturbation, that immediately precedes a +phase of steady growth of the spectrum. +However, in the present example, the duration of the non-attractor era is not sufficient for +providing a prominent peak in the power spectrum solely by the presence of vacuum fluctuations of +the curvature perturbation. At this point, the additional source provided by the gauge fields come +to the rescue, giving rise to an extra steep growth, leading to the required power to generate a +sizeable peak at wave-numbers corresponding to kpeak = k∗xc +2,R ≃ 1.5× 1013 Mpc−1 corresponding +to peak PBH mass at the time of formation M(f) +PBH ≃ 2.2 × 10−13 M⊙ (see Eq. (2.7)). Hence, +in the presence of couplings to the Abelian gauge fields, the roller-coaster architecture of the +potential provides an assisted amplification mechanism of the power spectrum. The curvature +perturbation sourced by the vector fields and its vacuum counterpart help each another, so to +generate a sufficiently pronounced peak to produce PBH. Interestingly, the profile of the curvature +spectrum as function of momenta, in particular its growth rate and the details of the peak, are +45 + +quite different from the corresponding predictions of single-field inflation, as investigated in Section +3. These differences make the two scenarios distinguishable. +To conclude, the localized production scenario we presented in this section might be considered +as a possible way to address the instability issues associated with the background of the smooth +axion inflation of Section 4.1.1. In fact, for the model we considered in this section, back-reaction +effects become prominent for ξ∗ > 15.6 – see [78] for details – a much larger value than the +phenomenological value ξ∗ ≃ 10.5 needed for the amplification of the power spectrum. Hence, we +work and make predictions in a safe region of cosmological evolution. +Although this is promising towards building workable models of PBH production in axion +inflation, the issue regarding the stability of the background will likely to re-appear towards +the end of bumpy axion inflation – in particular towards its final stages when ϵ → 1 where +ξ ≡ gcs +� +ϵ/2(Mpl/f) ≃ O(10). In the next section, we discuss an extension of the model presented +here, in which a spectator axion sector generates a localized scalar enhancement through its +coupling to an Abelian gauge sector. Since in this model the dynamical evolution of the axion +field stops long before inflation ends, we can avoid the strong back-reaction regime which leads to +instabilities of the background configuration. +4.1.3 +Spectator axion-gauge field dynamics +We continue to discuss possible methods to locally enhance the spectrum of scalar fluctuations in +axion-gauge field models, with an eye in solving the dynamical instability problems mentioned +above. We focus on scenarios where the axion is a spectator field, i.e. does not directly drive +inflation. We call σ the spectator axion field, and couple it to a Abelian gauge sector. The +Lagrangian is +Lm = Linf − 1 +2∂µσ ∂µσ − U(σ) − 1 +4Fµν F µν − gcs +4f σ Fµν ˜F µν, +(4.18) +where Linf = −(∂φ)2/2 − V (φ) is the inflaton Lagrangian, with V (φ) its potential. We are +interested on an inflationary set-up where the spectator sector provides sub-leading contribution +to the total energy density during inflation. This implies that the energy densities of the scalar +fields in the model (4.18) satisfy ¯ρσ ≪ ¯ρφ. Assuming small backraction from the gauge field +fluctuations ρA ≪ ¯ρσ (more on this later), the Friedmann equation simplifies to +3H2M2 +pl ≃ ¯ρφ + ¯ρσ +−→ +3H2M2 +pl ≃ V (¯φ), +(4.19) +so that the inflationary background is completely dictated by the inflaton potential. We assume +that V (φ) is flat enough to support sufficient inflation, but otherwise we let it unspecified, as +the fine details of the inflaton dynamics are not essential for what follows. Since we work in a +weak back-reaction system, we can avoid instabilities in the inflaton sector, as the one reviewed in +Section 4.1.1. +We consider, from now on, representative examples of axion potentials and its corresponding +dynamics. The choice of our examples is motivated by their interesting ramifications for PBH +production, and for their motivations from high-energy physics. If the spectator axion σ is +displaced from its global minimum, it can be dynamically active during inflation. Thanks to the +perturbative shift symmetry of the axion sector, we expect the axion potential to be nearly flat, +46 + +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +σ/f +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +M1 +⇐ +U(σ)/Λ4 +0 +2 +4 +6 +8 +10 +12 +σ/f +0 +2 +4 +6 +8 +10 +12 +⇐ +M2 : U(σ)/(µ3f) +Λ4/(µ3f) = 0.95 +Λ4/(µ3f) → 0 +Figure 15. The shape of spectator axion potentials for M1 (left) and M2 (right). For both panels, the red +stars indicate the location of the inflections point(s) at which the slope of the potential U ′(σ) and hence +the background velocity ˙¯σ of σ becomes maximal (see e.g. Eq. (4.20)). +hence the axion dynamics should occurr in a regime of slow-roll ¨¯σ ≪ 3H ˙¯σ. In such a case we can +neglect back-reaction effects induced by gauge field fluctuations27 and the background evolution +of the spectator axion can be studied through the following equation, +¨¯σ + 3H ˙¯σ + U ′(¯σ) = 0, +−→ +3H ˙¯σ + U ′(¯σ) ≃ 0. +(4.20) +In order to realize localized gauge field production through the last term in (4.18), we consider +two class of transiently rolling spectator axion models characterized by the following potentials +[77, 261] +U(σ) = +� +� +� +� +� +� +� +Λ4 +� +1 − cos +�σ +f +�� +, +(M1) , +µ3σ + Λ4 +� +1 − cos +�σ +f +�� +and Λ4 ≲ µ3f +(M2), +(4.21) +where µ and Λ are parameters of mass-dimension one. +The first model, M1, features a standard (discrete) shift symmetric potential akin to natural +inflation [234] where the size of the axion modulations is set by Λ. In this model, the axion +dynamics spans within an interval bounded the maximum (σ = πf) and the minimum (σ = 0) of +the potential. Therefore, for large and small field values (early and late times), the axion rolls +with small velocities. However, ˙¯σ obtains relatively large values at an intermediate time when σ +passes through an inflection point σ∗ = σ(t∗) with U ′′(σ∗) = 0, where the slope of the potential +U ′(σ) becomes maximal. +In the second model, M2, the axion field range is extended via a linear term [235, 237] +proportional to a soft symmetry breaking mass parameter µ. We assume that the axion σ +can probe the corresponding bumpy potential28 in the Λ4 ≲ µ3f regime, similar to the axion +27For the phenomenological examples we will present in this section, this statement can be made precise and +back-reaction can be shown to be negligible [77, 244]. +28In this work, by an appropriate choice of initial conditions and model parameters, we assume that σ traverses +two step-like features on its potential before it settles to its global minimum. +47 + +inflation model we discussed in Section 4.1.3. In the plateau-like region(s), and towards the global +minimum (σ = 0) 29, the spectator axion acquires very small velocities, but obtains a relatively +large transient peak when the slope of the potential U ′(σ) becomes maximal at the cliff-like +region(s), in particular at the inflection point(s) denoted by ∗ in Fig. 15. +The background evolution σ can be analytically derived in the slow-roll regime (4.20), starting +from the expressions for the scalar potentials of Eq. (4.21). For typical field ranges dictated by these +potentials 30, the spectator axion velocity ˙¯σ and the effective coupling strength ξ = −gcs ˙¯σ/(2Hf) +obtain a peaked time-dependent profile [77, 261]: +ξ(N) = +� +� +� +� +� +� +� +2ξ∗ +eδ(N−N∗) + eδ(N∗−N) , +δ ≡ +Λ4 +3H2f2 and ξ∗ ≡ gcs δ +2 +(M1) +ξ∗ +1 + δ2(N − N∗)2 , +δ ≡ +µ3 +3H2f and ξ∗ ≡ gcs δ +(M2) +(4.22) +where N∗ denotes the e-fold number when the axion passes through the inflection point (See Fig. +15) and ξ∗ is the maximal value of the effective coupling parameter at N∗. As in the bumpy +axion inflation model, the width of the time dependent peak in ξ depends on the dimensionless +ratio δ which characterizes the mass of the spectator axion in its global minimum δ ≈ m2 +σ/H2. +For a heavier axion (corresponding to larger δ), the restoring force towards the global minimum +becomes very relevant: the axion σ traverses the inflection point faster, the result being a sharper +peak in ξ. In other words, δ controls the acceleration ( ˙ξ/(ξH) = ¨¯σ/( ˙¯σH) ∼ δ) of σ as it rolls +down on its potential. Notice that given our slow-roll approximation, we require δ ≪ 3. +The peaked structure of ξ profile controls a critical scale k∗ = a∗H∗ characterizing the equation +of motion (D.15), corresponding to the scale of momenta leaving the horizon at an epoch when +the axion velocity is maximal (i.e when ξ = ξ∗). Since the mass of the U(1) field in (D.15) is as +tachyonic as possible around this scale, it gives rise to a scale-dependent growth of the gauge +field fluctuations where only modes whose size is comparable to the horizon size at N = N∗, +i.e k ∼ O(1)a∗H∗, are efficiently amplified. In Appendix D.1 we briefly review the parametric +amplification of the gauge field modes corresponding to both models M1,2 we are presenting in +this section; for more details we refer the reader to [77, 244, 261]. +In the spectator axion models we are focusing, the impact of the vector field production +on the visible scalar fluctuations is encoded only indirectly, by the presence of gravitational +interactions [262]. In fact, although we consider a matter Lagrangian (4.18) where the spectator +axion-gauge field sector is decoupled from the visible inflaton sector, when integrating out the +non-dynamical lapse and shift metric fluctuations we find a mass mixing between inflaton δφ +and spectator axion δσ fluctuations. This phenomenon introduces the possibility that the gauge +fields influence the curvature perturbation R ≃ H δφ/ ˙¯φ through a succession of inverse decays: +δA + δA → δσ → δφ ∝ R (see Appendix E). +Therefore, the dynamics of this sourced contribution can be understood by first studying the +influence of particle production on the spectator axion fluctuations δσ and then by computing +29The roll of σ towards the global minimum can be captured by modifying the monomial term as µ3sigma → +µ3f[ +� +1 + (σ/f)2 − 1], so that the axion potential (4.21) interpolates between µ3σ and (µ3/f)σ2 from large to small +field (σ/f → 0) values respectively. +30For M2, this translates into a single step like region in the potential (see Fig. 15). +48 + +their conversion to curvature perturbation δφ ∝ R. We sketch some of the steps in Appendix D.3; +a more detailed computations can be found in [77, 261]. +Taking into account the vacuum fluctuations within the inflaton sector, the total power +spectrum of curvature perturbation in the spectator axion-gauge field model can be expressed as +[77, 261], +PR(k) = P(v) +R (k) + +� +ϵφP(v) +R (k) +�2 � +i +f(i) +2,R +� +ξ∗, k +k∗ +, δ +� +, +(4.23) +where PR = H2/(8π2ϵφM2 +pl) with ϵφ ≡ ˙¯φ2/(2H2M2 +pl) and the sum over i denotes contributions +from each particle production location. E.g. for model M1 the sum is over only from one of such +regions, while for M2 there are two of them (due to our assumptions about the initial conditions +of σ in the bumpy regime of the potential in (4.21)), as the spectator rolls along the potential +towards its global minimum. Eq. (4.23) teaches us that the part of the spectrum sourced by the +gauge fields has an extra slow-roll suppression ϵφ, in particular with respect to the direct coupling +model we discussed in the previous section (see Eq. (4.16)). However, the part that parametrizes +the scale dependence of the sourced contribution, i.e. f2,R in (4.23), exhibits the same log-normal +scale dependence of the bumpy axion model in (4.17) whose peak height, width and location +(fc +2,R, σ2,R, xc +2,R) is calculated in [77, 261] in terms of ξ∗ for phenomenologically interesting values +of δ that characterize the background evolution of the spectator axion. +To understand the full scale dependence of the power spectrum (and in particular its vacuum +contribution P(v) +R ), we need to specify the scalar potential V (φ) in the inflationary sector. Instead +of specifying V (φ), we take a phenomenological approach to determine the important set of +parameters summarizing the inflationary dynamics. For this purpose, first notice that assuming +the effects introduced by the rolling of σ is negligible at CMB scales (early times during inflation), +we have ns −1 ≃ 2ηφ −6ϵφ and r ≃ 16ϵφ. Considering the latest Planck results, we adopt r ≲ 10−2 +at CMB scales [4], to obtain ϵφ ≃ 6.25 × 10−4. However, the observed value of the spectral tilt +gives ns − 1 ≃ −0.035 [4]. Furthermore, neglecting higher order slow-roll parameters, we assume +that ϵφ remains constant throughout the inflation. These simplifying approximations enable us +the describe total power spectrum in the multi-field scenarios we described above. Denoting the +e-fold dependence of the Hubble parameter during slow-roll inflation as H(N) = Hcmb e−ϵφN +(where the subscript cmb denotes the time at which the CMB pivot scale kcmb = 0.05 Mpc−1 exits +the horizon), we can turn the the scale dependence of the vacuum scalar power spectrum into +e-fold dependence as +P(v) +R (k) ≃ As e−(1−ϵφ)(1−ns)N, +(4.24) +where As ≡ P(v) +R (kcmb) = 2.1 × 10−9 where ϵφ ≃ 6.25 × 10−4 and 1 − ns ≃ 0.035 as we discussed +above. Collecting all the information we presented above, we plot in Fig. 16 the total scalar power +spectrum (Eq. (4.23)) for a representative parameter choices, describing both of the spectator +axion-gauge field models M1 and M2. +We learn from Fig. 16 that since the spectator dynamics do not influence the inflationary +background significantly, the vacuum contribution (dotted lines) has a smooth red tilted power +law form, which should be contrasted with the bumpy axion inflaton model we discussed in the +previous section. However, the transient roll of the spectator axion σ before settling to its global +49 + +0 +10 +20 +30 +40 +50 +60 +N +10−11 +10−9 +10−7 +10−5 +10−3 +10−1 +(ξ∗, N∗) = (4.96, 32.5) +M1 : δ = 0.2 +PR(N) +P(v) +R (N) +0 +10 +20 +30 +40 +50 +60 +N +10−11 +10−9 +10−7 +10−5 +10−3 +10−1 +(ξ(1) +∗ , N (1) +∗ ) = (6.23, 16) +(ξ(2) +∗ , N (2) +∗ ) = (6.23, 32.5) +M2 : δ = 0.3 +PR(N) +P(v) +R (N) +Figure 16. Total scalar power spectrum (black curves) as a function of e-folds during inflation for the +spectator axion-gauge field models M1 (Left) and M2 (Right). In the left panel, we assume ˙¯σ is maximal +at N∗ = 32.5 where the effective coupling reaches ξ(N∗) = 4.96 corresponding to gcs = 49.6 using Eq.(4.21). +In the right panel, spectator traverses two bumps before it settles to its minimum and we choose the +inflection points to occur at N (1) +∗ += 16 and N (1) +∗ += 32.5 where ξ∗ = 6.23 (gcs ≃ 20.7). +minimum triggers gauge field production as it probes the inflection point(s) on its potential (see +Fig. 15). This phenomenon generates an additional Gaussian bump in the scalar power spectrum. +Notice that for larger δ the width of the corresponding bump decreases because ˙¯σ ∝ ξ is maximal +for a shorter time interval, during which it can affect fewer gauge field modes. This explains why +the sourced signals in the second model M2 has a narrower width. As in the direct coupling +cases to the gauge fields we considered earlier, the peak in the power spectrum originates from a +cubic term, i.e. the last term in (4.18) and therefore the statistics of the curvature perturbation is +expected to be non-Gaussian. This explains why the examplifying scenarios we present in Fig. 16 +with a peak power of PR ∼ 10−3 is sufficient to generate large populations of PBHs at masses +corresponding to the peak scales in Eq. (2.8). +To close this section on axion-vector inflationary systems aimed at producing PBHs, we can +conclude that this possibility requires a fair amount of complex model building. Nevertheless, +the physics of axion reached so far a high degree of theoretical sophistication, thanks to the +input and motivations from particle physics, quantum gravity, and cosmology. The distinctive +predictions we explored for what respect to the properties of the curvature power spectrum, in +particular its profile as function of momenta, and its shape around the peak, make these scenarios +distinguishable from single-field inflation, and with relevant ramifications for PBH phenomenology. +It is certainly worthwhile explore and apply existing efforts to explore their consequences for the +physics of PBH formation. +4.2 +Strong turns in the multi-scalar field space +An alternative approach for enhancing scalar fluctuations at small scales is to exploit the dynamics +of multiple field inflation. Multi-field inflation a topic very well studied in the inflationary literature +with a variety of motivations, see e.g. [263] for a review. In reviewing the applications to PBH +production, we focus on a general class of non-linear sigma models that are well motivated when +embedding inflation in high energy physics. Although we go through representative concrete +50 + +examples, the essence of the mechanism is again associated with a transient tachyonic instability +in the scalar sector – along a direction orthogonal to the inflationary one – which is converted +through suitable couplings into an amplification of curvature perturbations. +The generic two-derivative Lagrangian describing the dynamics of such a system of scalar fields +φI minimally coupled to gravity is given by (S ≡ +� +d4x√−g Lm), +Lm = −1 +2GIJ(φ)∂µφI∂µφJ − V (φ), +(4.25) +where the fields may interact through the potential V (φ) and the field field space metric GIJ(φ). +For an FLRW background characterized by the scale factor a(t) and Hubble parameter H(t), the +equation of motion of the homogeneous fields is described by +Dt ˙¯φI + 3H ˙¯φI + GIJV,J = 0, +(4.26) +where the time field space covariant derivative of any field space vector AI is defined as DtAI = +˙AI + ΓI +JK ˙¯φJAK. Considering two field case for simplicity, the background trajectory can be +split into an adiabatic eI +σ = ˙¯φI/ ˙σ and entropic eI +s field bases that are orthogonal to each other. +Here, σ ≡ (GIJ ˙¯φI ˙¯φJ)1/2 which is related to the Hubble slow-roll parameter as ϵ ≡ − ˙H/H2 = +˙σ2/(2H2M2 +pl). +We find it worth mentioning that the vanilla multi-field slow-roll trajectory corresponds to +fields following standard gradient flow of the potential 3H ˙¯φI ≃ −V ,I corresponding to a field +space trajectory that is approximately a geodesic with Dt ˙¯φI ≃ 0 (see e.g. (4.26)). However, a +sufficiently long phase of inflation only requires a small acceleration along the unit vector tangent +to the inflationary trajectory, eσ IDt ˙¯φI ≃ 0 and the acceleration pointing in the perpendicular +direction –which parametrizes the “bending” of the inflationary trajectory – is generically not +constrained and hence can be large. The level of “bending” can be described by the dimensionless +parameter η⊥ so that the orthogonal field bases evolve as +Dt eI +σ = Hη⊥eI +s, +Dt eI +s = −Hη⊥eI +σ, +(4.27) +where η⊥ = −eI +s V,I/(H ˙σ) measures the acceleration of the trajectory perpendicular to its velocity +[264, 265]. Combining it with Hubble rate, one can define a “turning rate” associated with the +trajectory as Ω ≡ η⊥H. Therefore, an inflationary background with a strong turn η⊥ ≫ 1 satisfies +Ω ≫ H. As we will show below, this is the regime of interest for the amplification of the scalar +fluctuations in this multi-field setup. For this purpose, we consider the linear fluctuations around +the background we described above, which is given by the following action [265–267] +S(2) = +� +d4x a3 +� +M2 +plϵ +� +˙R2 − (∂R)2 +a2 +� ++ 2 ˙ση⊥ ˙RQs + 1 +2 +� +˙Q2 +s − (∂Qs)2 +a2 ++ m2 +sQ2 +s +�� +, +(4.28) +where R is the comoving curvature perturbation31, Qs is the instantaneous entropy perturbation +31In the comoving gauge we are operating, the field fluctuations are proportional to entropy fluctuation δφI = QseI +s +while the spatial part of the metric takes the form ˆgij = a2e−2Rδij. +51 + +−15 +−10 +−5 +0 +5 +10 +15 +N − Nf +−100 +−50 +0 +50 +100 +— η2 +⊥ +– – +m2 +s/H2 +−2 +−1 +0 +1 +2 +N − Nf +0 +5 +10 +15 +20 +IN - Region +OUT - Region +Bunch - Davies +−−−−−−−−−→ +Excited State +η⊥(N) +Figure 17. Left: Schematic time evolution of the bending parameter η2 +⊥ and entropic mass m2 +s = b − η2 +⊥ +for a broad turn using a Gaussian η⊥ = ηmax +⊥ +e−y2/(2∆2), (ηmax +⊥ +, ∆2) = (10, 10) and smoothed top-hat profile +η⊥ = ηmax +⊥ +(erf(y − δ/2) − erf(y + δ/2))/2, (ηmax +⊥ +, δ) = (10, 5) with y = N − Nf. Right: Time dependence +of bending parameter η⊥ for representative sharp (rapid) turn examples with Gaussian and top-hat profile. +whose mass is given by +m2 +s = V; ss − H2η2 +⊥ + ϵH2M2 +plRfs +(4.29) +with V; ss = eI +seJ +s V;IJ is the projection of the covariant Hessian of the potential along the entropic +direction and Rfs is the field space curvature. As should be clear from the action (4.28), the +dynamics of the fluctuations in this two field model is completely dictated by three functions +of time N = ln(a): namely Hubble rate H(N), turning rate η⊥(N) and the mass m2 +s(N) of the +entropy mode. To demonstrate the amplification in the power spectrum via strong turns in curved +multi-field space, we follow the effective approach presented in [67] to consider a field space that +undergoes a strong turn η⊥ ≫ 1 around e-folding number Nf during inflation, corresponding +a time well after CMB scales exited the horizon and assume a featureless Hubble rate H(N). +We also assume that the time dependence of entropic mass is mainly controlled by the bending +parameter, taking m2 +s = (b − η2 +⊥)H2 where b is chosen to be a constant for simplicity. +The behavior of the bending parameter (and that of the entropic mass m2 +s) for typical strong +turn cases are shown in Fig. 17. An important quantity that determines the behavior of the +fluctuations is the duration of the turn which leads to either broad (left panel) or rapid turn +(right panel) cases as shown in the figure. Although the dynamics of the scalar perturbations +are qualitatively different for broad and sudden (strong) turn cases, their behavior share some +common characteristics that can be attributed to the transient nature of the strong turn in field +space. First of all, the modes that are still deep in the horizon at the end of the turn do not feel +the presence of the feature and hence their dynamics is standard single-field type where the power +spectrum is given by +P(0) +R (k) = +H2 +8π2ϵM2 +pl +���� +k=aH +, +(4.30) +with a slight red tilted scale dependence. For this purpose, we will utilize the phenomenological +model we discussed in the previous section to parametrize the scale dependence (time dependence) +of the power spectrum as in Eq. (4.24) with a constant ϵ ≃ 6.25 × 10−4 and 1 − ns ≃ 0.035 +52 + +along with Hcmb ≃ 10−5. On the other hand, large scales that leave the horizon well before the +turn experience typical multi-field dynamics resulting with a transfer of power from entropic +to curvature perturbations and enhancement in the power spectrum [263]. The corresponding +amplification is sensitive to the value of the entropic mass ms from the time of horizon exit until +the turn. In the following, we will assume a sizeable ms (i.e. b) before the turn so that the +entropic fluctuations have decayed sufficiently before the onset of the turn so that the power +spectrum is given by the standard result (4.30) for these scales. For scales that exit the horizon +around the time of the turn, the entropic fluctuations exhibit transient (exponential) growth due +to their tachyonic mass m2 +s < 0 around the turn, which is eventually transferred to the curvature +perturbation through the kinetic coupling in (4.28) proportional to the bending η⊥ parameter. +The exponential amplification of PR associated with these scales is a typical observational feature +of strong turns, which we discuss briefly focusing on broad and sharp turn cases separately. +Broad turns +For a strong turn in field space that lasts sufficiently long time so that modes of interest (i.e. modes +that are enhanced) have already exited the Hubble radius at the end of the turn, the dynamics of +the curvature perturbation can be described by the single-field effective theory with an imaginary +sound speed parametrizing the growth of the fluctuations [268, 269]. From the perspective of +two-field description, imaginary sound speed is just a manifestation of the transient instability +induced by the strong non-geodesic motion in field space and the associated transient negative +entropic mass m2 +s [270]. Therefore, the dynamics of mode functions are mainly characterized by +two time scales: i) ˜N corresponding to entropic mass crossing and the onset of instability ii) ¯N +denoting the sound horizon crossing of fluctuations after which the curvature perturbation R +becomes frozen: +k +a( ˜N) += +���ms( ˜N) +��� , +& +k |cs| +a( ¯N) = H( ¯N), +(4.31) +where |cs| denotes the absolute value of the imaginary speed of sound describing the instabili- +ty/growth in the fluctuations (R) in the single EFT language. Since the background dynamics +change mildly between the time of entropic mass crossing ˜N and freeze-out ¯N in the broad turn +case, the exponential enhancement in the power spectrum can be parametrized as [67, 270] +PR(k) = P(0) +R (k) exp +� +πη⊥(2 − +� +3 + b/η2 +⊥) +� ���� ˜ +Nk +, +(4.32) +for all k modes that satisfies k = a( ˜N)|ms| while m2 +s < 0. Notice that exponential enhancement +in the power spectrum with respect to base spectrum is proportional to bending parameter +η⊥ ≫ 1. In Fig. 18, we show the power spectrum profile as a function of e-folds (left panel) +using the analytic formula (4.32) for Gaussian bending profiles η⊥ = ηmax +⊥ +e(N−Nf)2/(2∆2) with +(ηmax +⊥ +, ∆2) = (20, 10) (orange curve - Model 1) and (ηmax +⊥ +, ∆2) = (20, 2) (black curve - Model +2) and Nf = 30. For both models, the range of scales and times affected by the turn where +m2 +s < 0 are highlighted by red and correspond to 2.4 × 108 < k [Mpc−1] < 1.2×15 (Model 1) +1.7 × 1010 < k [Mpc−1] < 1.6 × 1013 (Model 2). As should be clear from the expression (4.32), the +power spectrum reaches its maximal value for the scale whose entropic mass crossing overlaps +53 + +15 +20 +Nf Np +40 +45 +50 +N +10−8 +10−6 +10−4 +10−2 +— PR(N) +−7.5 +−5.0 +−2.5 +0.0 +2.5 +5.0 +7.5 +N − Nf +10−2 +100 +102 +104 +106 +– – – k2/(aH)2 +m2 +s/H2 +Figure 18. Left: Curvature power spectrum (normalized to PR(N = 0) = 2.1 × 10−9) for broad strong +turns and for two representative parameter choices using a Gaussian bending profile (see main text). Right: +Migration of modes k2/(aH)2 for equally spaced ln(k) values around the turn to confirm the expectation +that power spectrum should fall more quickly after the peak (as compared to the left panel) because more +modes experience entropic mass crossing for N < Nf than for N > Nf in the m2 +s < 0 region highlighted by +red. +with the peak of the bending parameter, i.e. when Nf = ˜Nkp corresponding to +kp ≃ kf |(ηmax +⊥ +)2 − b|1/2, +(4.33) +where kf = a(Nf)H(Nf) ≃ 5.24 × 1011 Mpc−1 is the scale that exits the horizon at Nf. To +determine the slope of the power spectrum on the rise, one can compute the spectral index with +respect to the base power spectrum to get +(ns − 1) − (ns − 1)0 ≃ π(2 − +√ +3) +Nf − ˜N +∆2 + Nf − ˜N +η⊥ +(4.34) +which shows that stronger and/or less broad turns in field space lead to a steeper power spectrum. +In particular, the expression (4.34) tells us that for broad and strong turn in field space, the slope +of the power spectrum towards its peak can be much larger than the single-field models where +ns − 1 ≲ 4 [86, 181, 186, 193]. +We would like to note that for scales affected by the turn, the analytic profile presented in +(4.32) is symmetric around the peak however numerical computations carried in [67] shows that +the power spectrum typically exhibit a much quicker fall of behaviour for scales following the +peak compared to ones preceding it. This could be understood by first recalling that the modes +that are enhanced are the ones that goes through entropic mass crossing (4.31) while the entropic +mass is negative m2 +s < 0. Right panel of Fig. 18 can then guide us for an intuitive understanding +for the expected asymmetry in the power spectrum following its peak because a larger range of +modes enjoys entropic mass crossing before the peak of |ms| 32 than afterwards. We therefore +32Recall from our discussion above that entropic mass crossing at Nf corresponds to the peak of the power +spectrum. +54 + +conclude that the expression (4.32) along with the power spectrum shown in the left panel of Fig. +18 only characterize the behavior of the fluctuations qualitatively and for more accurate results +numerical methods are required as shown in [67]. +Sudden turns +When the duration of the turn is shorter (a statement that we will make more precise below), the +scales that are maximally enhanced are still those that experience entropic mass crossing during +the turn and therefore they are of order k ∼ kfηmax +⊥ +(see e.g. Eq. (4.33)). However, differently +from the broad turn case, these modes get caught in the (rapid turn) feature while they are still +deep inside the horizon, i.e. they satisfy k > aH at the end of the turn. This situation effectively +generates an initial excited state for these modes that can be studied analytically in the regime of +sudden and strong (constant) turns η⊥ [66, 271]. Along with a localized exponential amplification, +the resulting power spectrum of curvature perturbation exhibit order one oscillations in k –which +is a common characteristic of sharp features [272, 273] – with a frequency is set by the time of the +turn. We will review these features below by focusing on the analytic model presented in [271] +33. For this purpose, we consider an inflationary trajectory with a top hat34sudden turn profile +(see e.g. the right panel in Fig. 17) characterized by three parameters: Nf the central time of its +location (measured with respect to the horizon exit time of the CMB pivot scale), its duration δ +and typical (constant) value ηmax +⊥ +around Nf: +η⊥(N) = ηmax +⊥ +� +θ +� +N − +� +Nf − δ +2 +�� +− θ +� +N − +� +Nf + δ +2 +��� +, +(4.35) +along with an entropic mass profile during the turn that is given by +m2 +s +H2 = (ξ − 1)η2 +⊥(N) +(4.36) +where ξ < 1 is a constant parameter. Note from the parametrization of the bending parameter +(4.35) that the sudden turn case we consider corresponds to δ > ln(ηmax +⊥ +). In this setup, one +can determine the power spectrum generated by sharp turns, by studying scattering problem of +the two-field system using WKB methods [66]. In particular, this procedure includes matching +operators (and their derivatives) describing the perturbations, from the IN region, where the +Bunch-Davies vacuum is assumed, passing through the turn, and to the OUT region, resulting in +an excited “initial” state as sketched in the right panel of Fig. 17. The behaviour of the entropic +and curvature perturbations in both regions (IN and OUT) are standard where the fields are +dynamically decoupled from each other, while the turn sandwiched between the IN and OUT +region results with their exponential growth. Following this procedure, an analytic expression for +the power spectrum has been derived in [271] for scales satisfying k/kf < √1 − ξη⊥ as: +33For a detailed analytic/numerical analysis on the behaviour of the power spectrum from strong sharp turns see +also [66, 67]. +34More realistic models of the turn are expected to have a smooth time dependence of η⊥ like a Gaussian profile +we considered earlier. As shown by the numerical evaluations in [271], the choice of top hat profile does not introduce +any qualitative difference but appears to be a convenient choice for analytic manipulations. +55 + +100 +101 +k/kf +100 +102 +104 +106 +δ = 0.5, η⊥ = 14 +PR(k)/P(0) +R +100 +101 +k/kf +100 +102 +104 +106 +δ = 0.25, η⊥ = 28 +PR(k)/P(0) +R +Figure 19. Shape of the curvature power spectrum using the analytic formula in Eq. (4.37), for two +representative sharp turn cases where the turn in parametrized with a top-hat bending parameter of Eq. +(4.35). Dashed blue line present the envelope characterized by terms highlighted in Eq. (4.37). +PR(k) +P(0) +R += +e2η⊥δ S +2S2(1 + X + +� +X(1 + X)) +� +�� +� +envelope +× sin2 � +e−δ/2κη⊥ + arctan(κ/S) +� +(4.37) +where +S(k) = +�� +4κ2 + (3 + ξ)2 +4 +− +� +κ2 + (3 + ξ) +2 +� +, +and +X(k) = (3 + ξ)2 +16κ2 +(4.38) +with κ ≡ k/(kfη⊥) and η⊥ denotes the typical constant value of the bending parameter corre- +sponding to ηmax +⊥ +in the top hat profile (4.35). +For the choice of ξ = −3 —which corresponds to an effectively massless entropy mode on +super-horizon scales (see Section 2 of [271] for a detailed discussion on this) — the profile of the +curvature power spectrum 35 for two representative parameter choices characterizing the turn, is +shown in Fig. 19. As described by the analytic expression (4.37), the power spectrum features +characteristic oscillations (sin2 term) modulated by an envelope that is described as highlighted +in the equation. The peak value of the envelope (see e.g. blue dashed lines in Fig. 19) controls +the maximal enhancement of the power spectrum with respect to baseline power spectrum (4.30) +(we assume it to be scale invariant for simplicity) which will occur roughly at the maximum of the +function S(k) defined in (4.38). Arguably, the oscillatory pattern of the power spectrum around +the peak is much more interesting. In particular, as η⊥δ ≪ e−δ/2η⊥ sin2 term changes much faster +than the envelope. On the other hand, in sharp turn regime e−δ/2η⊥ ≫ 1 so that the first term in +the argument of the sin2 term also changes much faster than the second. These two observations +35We also note that the presence of very light entropy mode (for the ξ = −3 case) leads to a non negligible +contribution to the power spectrum for scales that crosses the horizon before the turn, i.e. for k/kf → 0. In +particular, the analytic expression (4.37) do not perform well on these scales. For a more complete analytic formula +including numerical analysis, see [271]. +56 + +imply that oscillations in the power spectrum is periodic to a good degree. The period of the +maxima in the oscillations occurs roughly in ∆κ ≈ πeδ/2/η⊥ corresponding a linear frequency of +∆k ≈ πeδ/2kf +−→ +ω ≡ 2π +∆k ≈ 2e−δ/2 +kf +. +(4.39) +Therefore, effectively the scale kf that exits the horizon at the center of the feature sets roughly +the frequency of oscillations in the power spectrum. +To summarize, strong-sudden turns in field space leads to a power spectrum that exhibits order +one oscillations whose amplitude is modulated by an exponentially enhanced envelope (4.37). The +oscillations are fast, making their peaks almost periodic with a frequency given by (4.39). We note +that these oscillations do not have an impact on the PBH mass spectrum β(M) (see e.g. (2.16)) +because the calculation of the variance σ2(M) (of the density contrast) involves a smoothing +procedure over scales comparable to the width of peaks in the power spectrum PR [67]. Another +issue of interest from the perspective of model building is the influence of non-Gaussianity. The +periods of strong turns during inflation are known to generate non-Gaussianity of flattened type +[269, 274, 275]. The implications of this on the PBH distribution and model building (i.e. the +required peak amplitude of PR) is not known and subject to future research. We therefore would +like to emphasize that in the strong turn examples we discussed in this section (see Fig. 18 +and 19), the peak amplitudes we provided are not chosen guided by a particular bias on the +non-Gaussianity present in these models. +5 +Outlook +Primordial black holes (PBHs), if they exist, can shed light on long-standing questions on the +nature of dark matter, and on the mechanisms driving cosmic inflation. +They can provide +distinctive sources of gravitational waves, potentially detectable with current or forthcoming +gravitational wave experiments. For these reasons, the physics of PBH offer promising opportunities +of collaboration between cosmologists, astronomers, and gravitational wave scientists. +In this review, we focused on theoretical aspects of PBH inflationary model building. We +learned that generating PBH from inflation is hard, but possible. +We reviewed conceptual +ideas for amplifying the primordial curvature spectrum at a level sufficient to trigger black hole +formation. These mechanisms find realizations within single-field inflation, through a conversion of +pronounced gradients of homogeneous quantities into curvature perturbations or within multiple +fields inflation, where curvature fluctuations are instead amplified through appropriate couplings +with additional sectors, which are characterized by tachyonic instabilities. The required tuning +on model parameters, or the degree of model sophistication for realizing these ideas can be +demanding. But it is certainly a worthwhile effort, given that experimental probes of PBH are +sensitive to the details of the curvature power spectrum, for example, through the properties of the +resulting PBH population, or through an induced stochastic gravitational wave background sourced +by curvature fluctuations. In fact, different categories of models lead to distinct, potentially +distinguishable predictions for the statistics of primordial fluctuations. Hence, a detailed analysis +relating theoretical scenarios with cosmological and gravitational wave probes offers new precise +57 + +tests of inflationary mechanisms, complementary to traditional ones associated with the physics +of the cosmic microwave background and of the large-scale structures of our universe. +Although much theoretical work has been done so far by the community, much more is needed +for further investigating and clarifying different aspects of the physics of inflation leading to PBH. +At the level of model building, it will be important to clarify and address challenges associated +with a severe tuning of model parameters, or with the dynamical stability of PBH models requiring +non-attractor, non–slow-roll phases of inflationary evolution. It will also be crucial to continue to +characterize the rich and subtle properties of primordial fluctuations in PBH models of inflation +with large enhancements of the primordial power spectrum at small scales. The to-do list includes +a deeper analytic understanding of the properties of the curvature spectrum profile, as well as +non-linearities and non-Gaussianities and their consequences for observable quantities. Such +theoretical analysis will have relevant ramifications for designing appropriate cosmological probes +of the physics of PBH. Hopefully, a concerted effort of theory and experiments, motivated by a +deeper understanding of PBH physics, will allow us to set new bounds, or possibly make new +discoveries, on the mechanisms driving inflation and on the nature of dark matter. +Acknowledgments +O¨O would is supported by the “Juan de la Cierva” fellowship IJC2020-045803-I, by the European +Structural and Investment Funds and the Czech Ministry of Education, Youth and Sports (Project +CoGraDS-CZ.02.1.01/0.0/0.0/15003/0000437), and by the Spanish Research Agency (Agencia +Estatal de Investigaci´on) through the Grant IFT Centro de Excelencia Severo Ochoa No CEX2020- +001007-S, funded by MCIN/AEI/10.13039/501100011033. GT is partially funded by the STFC +grant ST/T000813/1. For the purpose of open access, the authors have applied a Creative +Commons Attribution licence to any Author Accepted Manuscript version arising. +A +Background Cosmology: Mini-Review +Space-time metric. The central premise in modern cosmology is that as we look at the Universe +on large enough scales, it appears to be simpler and more uniform compared to the small scales. In +other words, if we focus on sufficiently large scales, clumpy regions like the distribution of galaxies +appear to be isotropic and homogeneous. The large scale spatial homogeneity and isotropy of +the universe has been tested by a variety of observations such as the Large Scale Structure (LSS) +surveys [276, 277] but perhaps the most important evidence supporting this claim comes from +the almost uniform temperature of the CMB originating from different parts of the sky. To first +approximation, we can therefore assume the Universe to be isotropic and homogeneous. The +high degree of spatial symmetry uniquely determines the structure of space-time geometry where +physical distances are measured by the so called Friedmann [278], Lemaˆıtre [279], Robertson [280] +and Walker [281] (FLRW) line element 36, +ds2 = −dt2 + a2(t) +� +dr2 +1 − Kr2 + r2 dΩ2 +� +, +(A.1) +36Note that for flat spatial hyper-surfaces K = 0, we can define new coordinates by x = r cos φ sin θ, y = r sin φ sin θ +and z = r cos θ to turn the metric into a commonly used form ds2 = −dt2 + a2(t)δijdxidxj in the literature. +58 + +where dΩ2 = +� +dθ2 + sin2 θ dφ2� +is the line element on the two dimensional sphere S2 and K = +{−1, 0, 1} represents negative, zero and positive curvature of constant-time hyper-surfaces respec- +tively. Note that the symmetries of the Universe allow us to describe the metric by just a single +function of time a(t) and a constant parameter K. The function a(t) is called the scale factor +which parametrizes the size of the spatial slices at a given moment in time and the Hubble rate +H(t) ≡ ˙a(t)/a(t) describes the speed of expansion at a given moment of time. Here, an expanding +universe H(t) > 0 corresponds to monotonically increasing scale factor a(t). +Dynamics of the universe. Dynamics of space-time is governed by the Einstein field equation, +Rµν − 1 +2gµνR = +1 +M2 +pl +Tµν, +(A.2) +where Rµν ≡ ∂λΓλ +µν −∂νΓλ +µλ +Γλ +λρΓρ +µν −Γρ +µλΓλ +νρ is the Ricci tensor build out of Christoffel symbols, +R ≡ Rµ +µ = gµνRµν is the Ricci scalar and in terms of the metric, Christoffel symbols is given by +Γσ +µν ≡ 1 +2gσρ (∂µgρν + ∂νgµρ − ∂ρgµν) . +(A.3) +The Einstein equation in (A.2) relates the geometry of space-time on the l.h.s to the matter +content in the universe through the appearance of energy momentum tensor Tµν on the r.h.s. As +in the case of the space-time metric, homogeneity and isotropy restrict the possible choices matter +content and enforce the energy-momentum tensor to take the perfect fluid form, +¯Tµν = (¯ρ + ¯P) Uµ Uν + ¯P gµν, +(A.4) +where ¯ρ and ¯P are the background energy density and the pressure in the rest frame of the fluid +and Uµ = (−1, 0, 0, 0) is its time like 4-velocity of the fluid relative to the observer. The evolution +equation for the energy density deriving the expansion of the universe can be derived from the +ν = 0 component of covariant conservation law of energy momentum tensor: +∇µ ¯T µ +0 = 0 +⇒ +˙¯ρ + 3 ˙a +a (¯ρ + ¯P) = 0. +(A.5) +On the other hand, using the FLRW metric (A.1), Einstein field equations gives us information +about how space-time reacts to the matter content in the universe through the Friedmann, +H2 = +¯ρ +3M2 +pl +− K +a2 , +(A.6) +and acceleration equation, +¨a +a = − +1 +6M2 +pl +(¯ρ + 3 ¯P). +(A.7) +The equations (A.5)-(A.7) are the key equations in determining the evolution of the universe +and its constituents at large scales, namely a(t) and ¯ρ(t), ¯P(t) (assuming knowledge on spatial +curvature K). An important aspect of these equations is that they are not independent, for +example it is possible to combine the last two to obtain the first one. In practice, this implies that +59 + +we need another ingredient to solve for three variables ¯ρ(t), ¯P(t) and a(t) (or equivalently H(t)). +A quantity that comes to the rescue in this context is the equation of state (EoS) parameter +which provides a linear relation between pressure and energy density of the fluid(s) constituting +the universe: +¯P(t) = w ¯ρ(t), +(A.8) +where w = constant for each fluid contributing to the total pressure. Although this relation is not +the most general form of P(ρ) that is available to us, this parametrization is perfectly adequate +in providing an accurate course grained description of our universe through most of its history. +In particular, it provides a simple analytic control in determining the dynamics of the different +components of the universe and the evolution of the Hubble parameter (scale factor). In order +to describe the history of the universe in a continuous manner, we typically consider multiple +fluids that contributes to the energy density of the universe while satisfying the relation (A.8). +For example labeling each EoS by wi and assuming a single component dominates the energy +density for a given moment of time in the universe, one can easily integrate (A.5) to obtain +¯ρi(t) ∝ a(t)−3(1+wi). +(A.9) +Cosmological inventory. We can classify different sources of cosmological evolution by their +contribution to the pressure: (i) For relativistic gas of particles (radiation) pressure is about one +third of energy density, with wr = 1/3. Popular constituents of such a fluid are photons, neutrinos, +gravitons. (ii) Non-relativistic pressureless dust (matter) wm = 0 such as dark matter (e.g. PBHs) +and baryons, (iii) Cosmological constant wΛ = −1 such as vacuum energy. From Eq. (A.9), energy +density of each of these fluids therefore evolve as ¯ρr ∝ a−4, ¯ρm ∝ a−3, ¯ρΛ ∝ a0. Therefore in the +future, cosmological constant will dominate, while in the far past a → 0 the universe was radiation +dominated and in between these two stages there is a period of matter dominated stage. Due to +inadequacy of explaining initial conditions of the observed CMB anisotropies, cosmologist tend to +complete the picture we described above with a very early phase of accelerated expansion ¨a > 0 +called inflation (see e.g. [282] for a comprehensive review). During this phase, EoS parameter also +closely mimics the behavior of a cosmological constant winf ≃ −1. More precisely, the departure +of the EoS from the value −1 during inflation is characterized by a time dependent slow-roll +parameter ϵ(t) = − ˙H/H2 ≪ 1 where winf = −1 + 2ϵ(t)/3. In this framework, during inflation +ϵ ≪ 1 while inflation terminates when ϵ = 1. After this stage, it is generically assumed that the +constituent(s) (i.e. a scalar field or fields) that drives inflation is considered to decay to relativistic +particles in a process called (p)reheating [110, 111]. In this review, we will assume this process +proceeds very efficiently so that soon after inflation ends the universe is filled with relativistic +particles and consequently, evolves through the radiation dominated (RD), matter dominated +(MD) and finally dark energy dominated (DED) phases. On the other hand, we refer the full +picture that arise by the inclusion of an early accelerated expansion as the inflationary universe +which is composed of following consequent phases: Inflation → RD → MD → DED. +Evolution of the comoving Hubble horizon. As we described in the main text, the evolution +of the comoving horizon is crucial to understand causal evolution of fluctuation modes in the +inflationary universe. To understand its time evolution in the picture we described above, we +60 + +focus on its time derivative +d +dt(aH)−1 = − 1 +aH2 (H2 + ˙H) = − +¨a +a2H2 = 1 +a +(1 + 3w) +2 +(A.10) +where we used the acceleration equation (A.7) together with Friedmann equation (A.6) focusing +on flat FLRW slicing K = 0.37 The expression above can be shaped into a form suitable for +integration as d ln((aH)−1) = [(1 + 3w)/2] d(ln a) which immediately gives +(aH)−1 = H−1 +0 +a(1+3w)/2, +(A.11) +where we normalized the scale factor today as a0 = 1. Notice that during inflation winf ≃ −1, +comoving horizon decreases while during the subsequent RD (wr = 1/3), MD (wm = 0) phases it +grows with scale factor. +Density parameters Ω. To describe different energy components in the universe, cosmologist +often parametrize radiation, matter and dark energy density relative to the critical energy density +of spatially flat hyper-surfaces using the following definitions (and dropping the over-bar notation +to describe background quantities) +ρc,0 ≡ 3H2 +0M2 +pl +→ +Ωi ≡ ρi,0 +ρc,0 +(A.12) +with subscript “0” denoting quantities evaluated today and “i” labels different kinds of fluids. +Focusing on the post-inflationary era, we can then re-write the Friedmann equation in terms of +dimensionless density parameters as +3H(a)2M2 +pl = ρr,0 a−4 + ρm,0 a−3 + ρΛ,0 → H2(a) +H2 +0 += Ωr a−4 + Ωm a−3 + ΩΛ. +(A.13) +From the latest observations of the CMB anisotropies by the Planck collaboration, matter and +dark energy density is determined to be [7], +ΩΛ = 0.6847 ± 0.0073, +Ωm = 0.3153 ± 0.0073. +(A.14) +On the other hand, we can infer radiation density today by utilizing the transition time of the +universe from RD to MD. This moment in our universe is commonly referred to as matter-radiation +equality which is defined by +ρr(aeq) = ρm(aeq) +⇒ +aeq = Ωr +Ωm +. +(A.15) +37Indeed, CMB data informs us that spatial geometry of our universe is flat on large cosmological scales (see +(A.18)). On the other hand, since it dilutes slowly ∝ a−2 compared to the radiation and matter, we would expect it +dominate the energy density before DED but this did not happen. This implies that initial value of the curvature +must either be tuned to be extremely small or it should relax to small values through a dynamical mechanism. +Inflation could be also a solution to this puzzle, because during such an exponential expansion any initially large +curvature would be inflated away. +61 + +Noting the relation between the red-shift parameter 38 z(t) and scale factor a(t) = (1 + z(t))−1 +and Planck’s prediction zeq = 3402 ± 26 [7] together with Eq. (A.14) therefore gives +Ωr ≃ 9.26535 × 10−5, +(A.16) +where we used the central values of the quantities Ωm, zeq determined by Planck. +Making a two component fluid approximation around the time of matter radiation equality, +a ≈ aeq, the total energy density of the universe at a = aeq can be related to matter density today +via +ρ(aeq) ≃ ρr,0 +a4eq ++ ρm,0 +a3eq +≃ 2 ρm,0 +a3eq +. +(A.17) +Cosmological parameters. Other cosmological parameters determined by the latest Planck +2018 data (TT,TE,EE + low E + lensing, % 68 CL) are as follows [7]: +ΩK = 0.001 ± 0.002, +H0 = 67.36 ± 0.54 [km s−1Mpc−1], +Ωmh2 = 0.1430 ± 0.0011, +keq = 0.010384 ± 0.000081 [Mpc−1]. +(A.18) +So far we did not mention thermodynamical properties such as temperature and entropy in the +universe following inflation. In what follows, we will note some of the key formulas we will use in +Section 2 without getting into the details on the derivation of these formulas. For an in depth +discussion on the contents we will introduce below, we refer the reader to chapter 3 of the seminal +books by Kolb and Turner [283] and Mukhanov [284] or to Baumann’s recent book [116]. +The total energy density of relativistic degrees of freedom that are in thermal equilibrium with +the plasma, can be related to the temperature of the plasma T (i.e. temperature of the photon +gas) as +ρr = π2 +30g∗(T) T 4, +(A.19) +where g∗(T) is the effective number of relativistic degrees of freedom at the temperature T which +is defined as +g∗(T) = +� +bos. +gb + 7 +8 +� +fermi. +gf, +(A.20) +with gb and gf denoting the intrinsic degree of freedom (i.e. spin) for bosonic and fermionic +species, respectively. The total entropy density of relativistic species is defined as +s(T) ≡ 2π2 +45 gs(T) T 3, +(A.21) +where gs is the effective number of degrees of freedom in entropy. For species in thermal equilibrium +(i.e. species that have the same temperature Tb = Tf = T), gs(T) = g∗(T) because for each +bosonic/fermionic degree of freedom, entropy density is defined to be s ≡ (ρ + P)/T with a +common denominator. +A very useful formula can be derived by using the conservation of total entropy in the universe, +38The red-shift parameter can be defined as the fractional shift in the physical wavelength λ of a photon emitted +at a distant point and time t in the universe until today, i.e. z(t) ≡ (λ0 − λ(t))/λ(t) = a(t0)/a(t) − 1 = 1/a(t) − 1. +62 + +S = sV ∝ sa3 ≃ constant. which can be utilized to relate the scale factor and the temperature of +the plasma as +gs(T) T 3 a3 ≃ constant. +=⇒ +a(t1) +a(t2) = T2 +T1 +�gs(T2) +gs(T1) +�1/3 +. +(A.22) +B +Analytic estimate for the threshold of collapse. +In this short appendix, we provide an analytic estimate on the characteristic value of collapse +threshold for PBH formation during RDU, closely following [29] (see also [115]). For this purpose, +we take the background space-time after inflation to have the spatially flat (K = 0) FLRW form +(A.1) and so the evolution of the scale factor described by the Friedmann equation: 3H2M2 +pl = ρ(t). +Now consider a locally perturbed, spherically symmetric region in the universe that is initially +outside the horizon and could eventually collapse to form a PBH upon horizon re-entry. The +metric describing such a region can be written as +ds2 = −dt2 + a2(t) e−2R(ˆr) � +dˆr2 + ˆr2 dΩ2 +� +(B.1) +where a(t) is the scale factor and R < 0 is the non-linear generalization of the conserved comoving +curvature perturbation defined on a super-Hubble scales [285]. At large distances ˆr → ∞, curvature +perturbation assumed to vanish (R → 0) so that the universe is described by the spatially flat +FLRW metric. By making a coordinate redefinition, r = ˆre−R(ˆr), the metric describing the +spherical perturbed region (B.1) can be transformed into the one describing a closed universe +with positive spatial curvature (as in (A.1)), +ds2 = −dt2 + a2(t) +� +dr2 +1 − K(r) r2 + r2 dΩ2 +� +, +(B.2) +where the relation between perturbations of the two metric is given by K(r) r2 = ˆrR′(ˆr)(2−ˆrR′(ˆr)) +showing that the local spatial curvature is given in terms of the first order spatial (leading order) +derivatives of the curvature perturbation R(ˆr). Ignoring higher order spatial derivatives of K +(i.e. K′ ∼ R′′) on sufficiently large scales, the evolution of the spherical region is given by the 00 +component of the Einstein equations (A.2) +H2 ≡ ρtot +3M2 +pl += ρ(t) +3M2 +pl +− K(r) +a2 +(B.3) +which is equivalent to the evolution of a closed universe (see e.g. (A.6)) with a small perturbation +δρ induced by spatial curvature K(r): +δ ≡ ρtot − ρ +ρ += δρ +ρ = +K +H2a2 , +(B.4) +where we make use of ρ(t) = 3H2M2 +pl. Considering (B.3), since radiation density dilutes faster +ρ ∝ a−4, a local spherical region with K > 0 will eventually stop expanding and collapse to form +a PBH. This happens precisely when the right hand side of (B.3) becomes negative, i.e. at t = tc +where δ = 1. In RDU, since perturbation modes that have a length scale smaller than the Jeans +63 + +Length (kJ ≡ aH/cs) cannot collapse, the smallest comoving scale that can undergo collapse at +t = tc is given by k2 = (aH)2/c2 +s. Therefore, we have +δ(tc) = K +k2 +k2 +a2H2 = +K +c2sk2 = 1, +(B.5) +which suggests us to identify K = c2 +sk2. Therefore, at the time of horizon re-entry (k = afHf), +the perturbations relevant for PBH formation should have a density contrast larger than +δc = +K +(afHf)2 = c2 +s +� +k +afHf +�2 += c2 +s. +(B.6) +C +Solving the Mukhanov-Sasaki equation: Numerical procedure +In this Appendix, we discuss important steps for the numerical evaluation of the Mukhanov-Sasaki +(MS) equation. +The aim is to obtain the power spectrum of curvature perturbation in the +inflationary scenarios we discussed in Sections 3.2 and 3.3. Note that scenarios discussed in those +sections can be captured by the generalized MS equation (3.10) utilizing (3.11) and (3.26) as we +describe below. +We start by scaling away the highly oscillatory contributions from the Bunch Davies (BD) +initial conditions (3.26). For this purpose, we define a dimensionless variable ¯vk through +vk(¯τ) = ¯vk(¯τ) +√ +2k +e−ik¯τ, +(C.1) +so to rewrite the MS equation (3.10) as +¯v′′ +k(¯τ) − 2ik ¯v′ +k(¯τ) − z′′ +z ¯vk(¯τ) = 0. +(C.2) +For mode by mode evaluation, the e-folds N is a more suitable variable for numerical integration +so we turn the derivatives w.r.t ¯τ in (C.2) to e-folds using dN = (aH/cs)d¯τ, and obtain +d2 ¯vk +dN2 + +�� +1 − ϵ − s +� +− 2i csk +aH +� d ¯vk +dN − +c2 +s +(aH)2 +z′′ +z ¯vk = 0 . +(C.3) +We parametrize the scale factor and Hubble rate as +a(N) = aend eN−60, +H(N) = Hend e− +� N +60 ϵ(N′)dN′ , +(C.4) +and note that +z′′ +z = +�aH +cs +�2 � +(1 − ϵ − s) +� +1 + η − s + µ +2 +� ++ +� +1 + η − s + µ +2 +�2 ++ 1 +2 +� dη +dN − ds +dN + dµ +dN +�� +. +(C.5) +In terms of the new variable, the Bunch-Davies initial conditions simplifies considerably and can +64 + +described as +¯vk(N) +�� +in = 1, +¯v′ +k(N) +�� +in = 0. +(C.6) +Provided that the background evolution (i.e. ϵ, η, µ etc) is known in terms of e-fold number, the +MS equation can be solved numerically for each k mode, in terms of the rescaled variable (C.3), +using the initial conditions (C.6). In this respect, some care must be taken for the initialization +time of individual modes because deep inside the horizon k2 ≫ z′′/z, the solution to (C.3) is +highly oscillatory which would be costly for the numerical computation. Typically, it is enough +to initialize the modes at some time such that they are sufficiently inside the horizon. For this +purpose, we choose to evolve each mode by setting the initial time as +N(k) +in += N(k) +0 +− 4, +(C.7) +where the “k” super-script indicates the intrinsic mode dependence for the choice of Nin and +N(k) +0 +denotes the horizon crossing time 39. As argued in the main text (see the discussion around +(3.29)), the latter can be obtained via +k2 = z′′ +2z +���� +N(k) +0 +(C.8) +using (C.5) (or equivalently (3.11)) provided the background solution is known. Having obtained +the individual mode evaluation from N(k) +in +to Nend = 60, the power spectrum (3.22) can be +described as +PR(k, Nend) = +H2 +end +8π2ϵendcs,endM2 +pl +�cs,end +˜ +Mend +�2 � k +kend +�2 ��¯vk(Nend) +��2, +(C.9) +where we defined ˜ +M ≡ M(N)/Mpl, while kend = aendHend is the mode that exits the horizon at +the end of inflation. +In order to set the overall normalization of the power spectrum we need to determine Hend +w.r.t Mpl. We do so by requiring the normalization of the power spectrum indicated by Planck +at the pivot scale kcmb = 0.05 Mpc−1 using PR(kcmb, Nend) ≃ 2.1 × 10−9. Notice that, from the +general formulas we presented above, canonical single-field scenarios (see Section 3.2) can be +recovered by making the following replacements cs → 1, s → 0, M → Mpl, µ → 0 and d¯τ → dτ +and dN = (aH) dτ. A pedagogical Python notebook file that calculates the power spectrum using +the prescription described above can be found through the github link. +The role of decaying modes in the canonical single-field scenarios. We can now verify +if the general expression (3.6) provides an accurate description for the enhancement of the power +spectrum in the canonical single-field model discussed in Section 3.2. For this purpose, we assume +that the leading solution to the curvature perturbation at super-horizon scales is given by the +constant solution at horizon crossing Rk(τ) = R(0) +k +and plugging it into the last term in (3.6), we +39Unless modes undergo resonance and get excited deep inside the horizon, the choice of initial conditions in +Eqs. (C.7) and Eq. (C.6) provide an accurate prescription for the initialization of the numerical evaluation. For the +models we consider in Sections 3.2 and 3.3 this is indeed the case. For a model that leads to excited states inside +the horizon see [286, 287]. +65 + +can then generate an iterative solution for the curvature perturbation as +Rk(Nend) = R(0) +k +� +�1 + vR(k) +� +Nend +N0 +dN′ +˜z2 (˜a ˜H) +− k2 +� +Nend +N0 +dN′ +˜z2 (˜a ˜H) +� +N′ +N0 +dN′′ +˜z2 +(˜a ˜H) +� +� , +(C.10) +where we switched to e-folds as time variable, vR ≡ R′ +k(N0)/Rk(N0) is the fractional velocity +of curvature perturbation at horizon crossing epoch N0, and tilde over quantities denotes a +normalization with respect to their values at N0, i.e. ˜X ≡ X(N)/ ˜X(N0). Using the standard +solution to the curvature perturbation during the initial slow-roll era, an analytic formula for +the fractional velocity of curvature perturbation is obtained [86], valid for modes that leave the +horizon during the initial slow-roll era: +vR(k) = − +x2 +0 +1 + x2 +0 +− i +x3 +0 +1 + x2 +0 +, +with +x0 ≡ +k +a0H0 +. +(C.11) +Similarly, for modes that crosses the horizon during the initial slow-roll era, the amplitude of the +curvature perturbation at horizon crossing results +��R(0) +k +��2 = +H2 +0 +8π2ϵ0M2 +pl +� +1 + +k2 +(a0H0)2 +� +. +(C.12) +Combining these results together with a given background evolution (i.e. ˜z, ˜a ˜H), the power +spectrum of curvature perturbation for modes that leaves the horizon can be described via the +definition +PR(k, Nend) = k3 +2π2 +��Rk(Nend) +��2 , +(C.13) +using (C.10), (C.11) and (C.12). For a set of modes that exit the horizon during Phase 1 (the +initial slow-roll era), the amplitude of power spectrum obtained using the procedure we outlined +is shown by blue dots in Fig. 6. +D +Details of the axion-gauge field dynamics +In this appendix, we focus on the dynamics of the gauge fields by the presence rolling scalar +χ(t) and the influence of this dynamics on the scalar perturbations for the models we discuss in +Sections 4.1.1, 4.1.2 and 4.1.3. The part of the action (4.1) relevant for this purpose is given by +SGF = +� +d4x√−g +� +−1 +4FµνF µν − gcs χ +8f +ηµνρσ +√−g FµνFρσ +� +, +(D.1) +where using the definition of the field strength tensor Fµν = ∂µAν − ∂νAµ and the totally +antisymmetric nature of the symbol ηµνρσ, we note the following identities +FµνF µν = 2 (gµρgνσ − gνρgµσ) ∂µAν ∂ρAσ, +(D.2) +ηµνρσFµνFρσ = 4 ηµνρσ ∂µAν ∂ρAσ. +(D.3) +66 + +Notice that apart from the gauge fields, these expressions involve the metric. Therefore, it is con- +venient to characterize the metric fluctuations in its most general form using ADM decomposition +as +ds2 = −N2 dt2 + ˆgij +� +dxi + Ni dt +� � +dxj + Nj dt +� +, +(D.4) +where ˆgij is the spatial 3-metric on constant time hyper-surfaces, N is the lapse function and Ni +is the shift vector. In terms of these variables, the component of the metric with the upper indices +can be expressed as +g00 = − 1 +N2 , +g0i = gi0 = Ni +N2 , +gij = ˆgij − NiNj +N2 , +(D.5) +where ˆgij is the inverse of the spatial metric. Including the Einstein-Hilbert term LEH = M2 +plR/2 +to the action (D.1) and the matter action associated with the scalar field(s) χ, one can show +that the second order terms that include N, Ni do not contain more than one time derivative, +implying that they can be identified as Lagrange multipliers to be solved in terms of the physical +fluctuations of χ and Aµ [245, 288]. +To characterize the dynamics of gauge fields we assume vector fields start linear order in +perturbations (so that they do not exhibit a time dependent background vev ⟨ ¯Aµ(t)⟩ = 0) and +adopt the Coulomb gauge 40 ∂iAi = 0 in the gauge sector along with the flat gauge choice in the +scalar - gravitational sector +χ(t, ⃗x) = ¯χ(t) + δχ(t, ⃗x), +ˆgij = a2(t) [ δij + hij ] , +∂iAi = 0, +(D.6) +where hij is the transverse, traceless tensor fluctuation of the metric, ∂ihij = hii = 0. In this +appendix, we will present the dynamics of the gauge fields by expanding the action (D.1) up to +third order in fluctuations. Noting that the shift vector is order one Ni in perturbations and +expanding the lapse N = 1 + δN, we compile all the information above to obtain the second and +third order action involving gauge field fluctuations as +S(2) +GF = +� +d4x a3 +� 1 +2a2 ˙Ai ˙Ai − +1 +2a4 ∂jAi ∂jAi + gcs ˙¯χ +a3f ϵijk Ai ∂jAk +� +, +(D.7) +S(3,1) +GF += +� +d4x a3 +� +gcs +δχ +f +� +⃗E. ⃗B − ⟨ ⃗E. ⃗B⟩ +�� +(D.8) +S(3,2) +GF += +� +d4x a3 +� +−δN +2 +� +⃗E2 + ⃗B2 − ⟨ ⃗E2 + ⃗B2⟩ +� +− Ni +a2 ˙AjFij +� +(D.9) +S(3,3) +GF += +� +d4x a3 +� +−hij +2 +� +EiEj + BiBj +�� +, +(D.10) +40We note that at linear order in fluctuations, the Coulomb gauge condition ∂iAi = 0 is equivalent to setting the +temporal component of the gauge field to zero, A0 = 0. This can be seen by explicitly expanding the gauge field +action (D.1) to second order in A0 and Ai and then solving for the non-dynamical A0 mode (see also the discussion +in Section 3 of [289]). +67 + +where we introduced electric and magnetic field notation using Ei = − ˙Ai/a, Bi = ϵijk∂jAk/a2. +Note that we artificially introduced tadpole terms proportional to the expectation values involving +⃗E and ⃗B fields in the expressions (D.8) and (D.9). Later we will subtract these terms in the action +describing the inflation (gravity plus inflaton sector) in (D.31) to consistently take into account +the modifications that might arise to the background equations of the scalar field(s) χ = {φ, σ} +and the Friedmann equation (see e.g. Eq. (4.7)). +The action labeled by S(3,2) +GF +parametrizes the cubic interactions (see e.g. the first and the +third term in (D.9)) induced by the gravitational fluctuations. The influence of these terms on +the observable scalar sector has been studied in [242, 290] with the conclusion that they can be +neglected compared to the direct interaction term in (D.8). This is because the vertices associated +with the gravitationally induced cubic terms in (D.9) is suppressed L(3,2) +GF +∝ (√ϵ/Mpl) δχ O(A2) +with respect to the one in (D.8) unless f ≃ Mpl and gcs ≪ 1 41 and therefore can be safely ignored +compared to the latter. Finally, the cubic action in (D.10) parametrizes the influence of the gauge +fields on the tensor part of the metric. In this review, we will not study these effects. For the +impact of gauge field production on the tensor fluctuations during inflation and its interesting +parity violating effects, see [77, 78, 261, 291–294] and the references therein. +To summarize, in the presence of rolling axion-like fields during inflation and the interaction +Lint ∝ χF ˜F, the dynamics of the gauge fields can be studied by focusing on the second order +action S(2) +GF (D.7). On the other hand, the influence of the gauge fields on the scalar sector depends +on whether we identify χ as the inflaton (Sections 4.1.1 and 4.1.2) or as a spectator scalar rolling +during inflation (Section 4.1.3). In the following, we will first introduce the basics of the gauge +field production in the presence of a rolling scalar discussing the different cases we cover in this +review. The nature of the subsequent sourcing of (visible sector) scalar fluctuations by the vector +fields depends whether the scalar χ directly interacts with U(1) fields or not. We will cover each +case following our discussion on the vector field production. +D.1 +Gauge field production by rolling scalars +To understand the gauge field production by a rolling scalar (either an inflaton or a spectator +scalar), we focus on the second order action (D.7) and decompose the gauge field into its helicity +modes λ = ± in Fourier space using conformal time dτ = dt/a as +ˆAi(τ, ⃗x) = +� +d3k +(2π)3/2 ei⃗k·⃗x � +λ=± +ϵ(λ) +i +(⃗k) ˆAλ(τ,⃗k), +(D.11) +where the polarization vectors obey +ki ϵ± +i (⃗k) = 0, +ϵijk kj ϵ± +k (⃗k) = ∓i |⃗k| ϵ± +i (⃗k), +ϵλ +i (⃗k) ϵλ′ +i (⃗k)∗ = δλλ′, +ϵλ +i (⃗k)∗ = ϵλ +i (−⃗k) = ϵ−λ +i +(⃗k) +(D.12) +and we defined +ˆAλ(τ,⃗k) = +� +Aλ(τ, k) aλ(⃗k) + A∗ +λ(τ, k) a† +λ(−⃗k) +� +, +(D.13) +41This regime is not interesting from a phenomenological point of view as the gauge field production will be very +weak in this case. +68 + +which satisfies ˆAλ(τ,⃗k)† = ˆAλ(τ, −⃗k) so that ˆAi(τ, ⃗x) is a hermitian operator. Finally, annihilation +and creation operators satisfy the standard commutation relations +� +ˆaλ(⃗k), ˆa† +λ′(⃗k′) +� += δλλ′δ(⃗k − ⃗k′). +(D.14) +Plugging the decomposition in (D.7) and varying the action, the mode functions of the gauge field +can be shown to satisfy +A′′ +± + k2 +� +1 ± aH +k 2ξ +� +A± = 0, +ξ ≡ −gcs ˙¯χ +2Hf +(D.15) +It is clear from Eq. +(D.15) that the dispersion relation of the gauge fields are modified in +the presence of the last term in (D.1). More importantly, the negative helicity modes A− can +experience an tachyonic instability for modes satisfying k/(aH) < − ˙¯χ/(Hf) while positive helicity +modes stay in their vacuum. These facts reflect the parity violating nature of the interaction +Lint ∝ χF ˜F. The behavior of the solutions to the Eq. (D.15) is sensitive to the velocity profile ˙¯χ +of the rolling scalar. In what follows we will discuss the different cases and the corresponding +solutions within the context of Sections 4.1.1, 4.1.2 and 4.1.3. +Production by a slowly-rolling scalar. For a slowly rolling scalar field, we can treat gcs ˙χ/(Hf) +as constant per Hubble time. In terms of the effective coupling ξ, this adiabaticity condition can +be parametrized as +˙ξ +ξH = +¨¯χ +˙¯χH − +˙H +H2 ≪ 1. +(D.16) +In this case, the solution to the Eq. (D.15) that reduces to the Bunch Davies vacuum solutions +A± = e−ikτ/ +√ +2k deep inside the horizon k ≫ aH can be written in terms of the Coulomb +functions +A−(τ, k) ≃ +1 +√ +2k +[G0(ξ, −kτ) + iF0(ξ, −kτ)] +(D.17) +where ξ = −gcs ˙¯χ/(2Hf) and −τ = (aH)−1. Another simplification can be made focusing on the +ξ ≫ −kτ regime (as we will see, particle production is only efficient for ξ ∼ O(1) and takes place +as −kτ → 0) +A−(τ, k) ≃ +� +−kτ +2k +� +2eπξπ−1/2K1( +� +−8ξkτ) + ie−πξπ1/2I1( +� +−8ξkτ) +� +, +(D.18) +where I1 and K1 are modified Bessel functions of the first and second kind. From the solution +above, one can realize that for the interesting case of ξ ∼ O(1), field amplification occurs shortly +after horizon crossing −kτ ∼ O(1). Therefore for a final simplification we can take the large +argument limit of the Bessel functions in (D.18) to get +A−(τ, k) ≃ +1 +√ +2k +�−kτ +2ξ +�1/4 +eπξ−2√−2ξkτ +� +1 + i +2 e−2πξ+4√−2ξkτ +� +, +(8ξ)−1 ≪ −kτ < 2ξ. +(D.19) +The real part of the solution (D.19) is the growing solution as −kτ → 0 and encodes the physical +amplification of the negative helicity mode by the presence of a slowly rolling axion-like field. +69 + +Within the approximations we undertake, the imaginary part of the solution (D.19) represents the +UV divergent part as −kτ grows which should precede the vacuum solution A− = e−ikτ/ +√ +2k deep +inside the horizon. Therefore, ignoring the imaginary part typically amounts to throwing away the +standard divergent (also present in flat space) peace of quantities like ⟨ ⃗E. ⃗B⟩ and ⟨ ⃗E2 + ⃗B2⟩ (see +below). For a detailed discussion on these issues we refer the reader to [244] and the Appendices +of [77]. +Production by a transiently rolling scalar. For the models we discuss in Sections 4.1.2 and +4.1.2, the potential of the scalar χ has a feature around which the background velocity ˙¯χ and the +effective coupling ξ in the equation of motion of the gauge field modes (D.15) have a transient +peak with a maximal value ξ∗ at τ = τ∗. A peaked time dependent profile of ξ = ξ(τ) in turn +translates into a scale dependent growth of the mode functions A− where only modes that are +in the vicinity of the scale k∗ = a∗H∗ = (−τ∗)−1 that exits the horizon at τ = τ∗ are maximally +amplified. For the time dependent profiles we study in Section 4.1.2 and 4.1.3, it is hard to obtain +a fully analytic solution describing the amplification of the gauge modes. However, an accurate +description of the mode function at late times can be obtained employing WKB approximation +methods supplemented with numerical analysis [261] as we mention below. In particular, at late +times τ/τ∗ < 1, the amplification of the mode functions can be parametrized in terms of a (real +and positive) normalization factor as +A−(τ, k) ≃ +1 +√ +2k +� −kτ +2ξ(τ) +�1/4 +NA(ξ∗, x∗, δ) e−2E(τ)√−2ξ∗kτ +� +1 + i e4E(τ)√−2ξ∗kτ +2NA(ξ∗, x∗, δ)2 +� +(D.20) +where we defined x∗ = −kτ∗ = k/k∗ and E(τ) is a time dependent function that asymptotes to +zero at late times τ/τ∗ → 0 whose functional form depends on the model under consideration. +For example for the bumpy axion inflation of Section 4.1.2 and its cousin spectator model (see +Section 4.1.3), it is given by E(τ) = 1/(δ| ln(τ/τ∗)|). On the other hand, for the transiently rolling +axion model with the standard cosine potential one gets E(τ) = +√ +2(τ/τ∗)δ/2/(1 + δ). Notice +that the solution (D.20) reduces to (D.19) of constant ξ if we make the following replacements +E(τ) → 1 and NA → eπξ. An important point that should be observed from the form of the +solution (D.20) is the dependence of the normalization (amplification) factor on the dimensionless +parameter δ which can be derived in terms of the physical model parameters (see Sections 4.1.2 +and 4.1.3). As we explain in the main text this parameter is a measure of the scalar’s mass around +global minimum δ ≈ m2 +χ/H2 and hence determines the rate at which the field rolls towards its +minimum. In this sense it determines the width of gauge field modes that takes part in the particle +production: the faster the scalar traverses the region in its potential where the velocity is maximal +(i.e. with larger acceleration ˙ξ/(ξH) ∝ ¨¯χ/( ˙¯χH) ∼ δ), each gauge field mode spends less time in +the tachyonic region and fewer of them will be excited by the rolling scalar, leading to a sharper +distribution of excited modes. These arguments clarifies the dependence of the Normalization +factor NA in (D.20) on the parameter δ along with the ξ∗ and x∗ = k/k∗ that characterize the +amplification and scale dependence of the particle production process. +At fixed ξ∗ and δ, the scale dependence of the normalization factor can then be obtained by +solving numerically (D.15) for a grid of x∗ = k/k∗ values and matching these solutions to the +WKB solution given at late times. In this way, one can confirm that NA is given by a log-normal +70 + +distribution for the models we consider in Sections 4.1.2 and 4.1.3: +NA (ξ∗, x∗, δ) ≃ Nc +A [ξ∗, δ] exp +� +− +1 +2σ2 +A [ξ∗, δ] ln2 +� +x∗ +qc +A [ξ∗, δ] +�� +, +(D.21) +where the functions Nc +A, qc +A and σA parametrizes the background dependence of gauge field +production through their dependence on ξ∗ and δ. In particular, accurate fitting formulas for +these quantities can be obtained at fixed δ in terms of ξ∗. For the parameter choices we adopt in +this review, these formulas can be found in [77, 78, 244]. +The “Electric” and “Magnetic” fields as sources. For future reference we also note the +electric and magnetic fields which are related to the auxiliary potential Ai as +Ei(τ, ⃗x) = − 1 +a2 ∂τAi(τ, ⃗x), +Bi(τ, ⃗x) = 1 +a2 ( ⃗∇ × ⃗A(τ, ⃗x) )i = 1 +a2 ϵijk ∂jAk(τ, ⃗x). +(D.22) +Utilizing the decomposition (D.11) together with (D.12) and (D.13), we then take into account +only the growing part (i.e. real part) of the solutions (corresponding to the physical amplification +of vector fields caused by a rolling scalar) we derived in (D.19) and (D.20) to express the Fourier +decomposition of the ⃗E and ⃗B fields as follows +ˆEi(τ, ⃗x) = +� +d3k +(2π)3/2 ei⃗k·⃗x ˆEi(τ,⃗k), +ˆBi(τ, ⃗x) = +� +d3k +(2π)3/2 ei⃗k·⃗x ˆBi(τ,⃗k), +(D.23) +where +ˆEi(τ,⃗k) = − +� +k +2 +ϵ− +i (⃗k) +a(τ)2 +�2ξ(τ) +−kτ +�1/4 +NA(ξ∗, −kτ∗, δ) exp +� +−E(τ) +� +−2ξ∗kτ +� +ˆO−(⃗k), +ˆBi(τ,⃗k) = − +� +k +2 +ϵ− +i (⃗k) +a(τ)2 +� −kτ +2ξ(τ) +�1/4 +NA(ξ∗, −kτ∗, δ) exp +� +−E(τ) +� +−2ξ∗kτ +� +ˆO−(⃗k), +(D.24) +where we defined the short-hand notation ˆOλ(⃗k) = [aλ(⃗k)+a† +λ(−⃗k)]. Note that the case of particle +production through a slowly-rolling scalar with ξ ≃ cons. can be recovered from the formulas +by making the following replacements ξ∗ → ξ, E(τ) → 1 and NA → eπξ. Having studied the +particle production in the gauge field sector, we now study the expectation values including +electromagnetic fields before we discuss how these sources influence the fluctuations of the scalar +sector. +Expectation values involving gauge fields. Noting again the decomposition of the vector +fields (D.11) (along with (D.12) and (D.13)) and the definition of their electromagnetic counterparts +(D.22), the energy density of the gauge fields ρA and ⟨ ⃗E. ⃗B⟩ can be expressed as +ρA ≡ 1 +2⟨ ⃗E2 + ⃗B2⟩ = +� +d ln k dρA +d ln k, +⟨ ⃗E. ⃗B⟩ = +� +d ln k d⟨ ⃗E. ⃗B⟩ +d ln k . +(D.25) +Taking into account only the amplified mode function A− of the gauge field, the energy density +71 + +ρA and ⟨ ⃗E. ⃗B⟩ per logarithmic wave-number is given by +dρA +d ln k ≃ H4 +8π2 x4 +�2ξ +x + 1 +� ��� ˜A−(x) +��� +2 +, +d⟨ ⃗E. ⃗B⟩ +d ln k +≃ − H4 +8π2 x4 d +dx +��� ˜A−(x) +��� +2 +, +(D.26) +where we utilized [244], +A′ +− = +� +2kξ +−τ A∗ +− +→ +d ˜A− +dx += − +� +2ξ +x +˜A∗ +− +(D.27) +defining the the following dimensionless variables: x = −kτ and +√ +2k A−(τ, k) ≡ ˜A−(x). Em- +ploying the solutions for A− we derived in (D.19) and (D.20), one can use the formulas (D.26) +and (D.25) to obtain ρA and ⟨ ⃗E. ⃗B⟩. In particular for the localized gauge field production models +presented in Section 4.1.2 (see [78]) and (4.1.3) (see [77]), these formulas can be used to justify +the negligible back-reaction of the gauge fields on the background evolution. We will not repeat +these calculations here for the localized production case, however below we will derive the relevant +formulas for the slowly-rolling smooth axion inflation of Section 4.1.1. For this purpose, we focus +on the real part of the solution (D.19) that corresponds to the physical amplification of the gauge +field fluctuations. Plugging the solution in (D.26) and (D.25), we obtain +ρA = +H4 e2πξ +8π2(2ξ)1/2 +� 2ξ +0 +dx x7/2 +�2ξ +x + 1 +� +e−4√2ξx, +⟨ ⃗E. ⃗B⟩ = H4 e2πξ +4π2 +� 2ξ +0 +dx x3 +� +1 − +1 +4√2ξx +� +e−4√2ξx, +(D.28) +where we send the lower limits of the integrals to zero as the integrands converge in the x → 0 +limit. Similarly, since we are only focusing on the physical amplification of the gauge fields by +throwing away the imaginary part of the solution (D.19), the integrands vanish quickly deep in +the UV x → ∞ and so we can also send the upper limit of the integrals in (D.28) to infinity, +2ξ → ∞. Finally, by making a change of variable to 4√2ξx = y, one can realize that the resulting +integrals can be carried analytically and in fact they are proportional to Gamma functions with +integer arguments. In particular, we get +ρA = H4 +ξ3 e2πξ Γ(7) +219π2 +� +1 + +1 +26ξ2 +Γ(9) +Γ(7) +� +, +⟨ ⃗E. ⃗B⟩ = H4 +ξ4 e2πξ [ Γ(8) − Γ(7) ] +221π2 +. +(D.29) +Inserting the numerical values of the Gamma functions, these expressions give rise to the Eq. +(4.8) we provide in the main text. +72 + +D.2 +Scalars sourced by vector fields, the direct coupling case: χ = φ. +To understand the sourcing of scalar fluctuations by the gauge fields, we should consider the +gravitational and inflaton (that we refer to Sinf) action in addition to SGF : Stot = Sinf + SGF +where +Sinf = +� +d4x√−g +� +M2 +pl +2 R − 1 +2∂µφ∂µφ − V (φ) +� +. +(D.30) +Expanding the action in terms of the scalar fluctuations δN, Ni and δφ around a background +solution, one gets the following linear and second order actions, +S(1) +inf = +� +d4x a3 +� � +3H2M2 +pl − 1 +2 +˙¯φ2 − V (¯φ) − 1 +2⟨ ⃗E2 + ⃗B2⟩ +� +δN +(D.31) +− +� +¨¯φ + 3H ˙¯φ + V ′(¯φ) − gcs +f ⟨ ⃗E. ⃗B⟩ +� +δφ +� +, +S(2) +inf = 1 +2 +� +d4x a3 +� +δ ˙φ2 − (∂iδφ)2 +a2 +− V ′′(¯φ) δφ2 − 3H2M2 +plδN 2 − 2HM 2 +pl∂iNiδN +(D.32) +− 2V ′(¯φ) δφ δN − 2 ˙¯φ δ ˙φ δN − 2 ˙¯φ Ni∂iδφ + ˙¯φ2δN 2 +� +. +Notice that in S(1) +inf (D.31), we subtracted the tadpole terms ∝ ⟨ ⃗E. ⃗B⟩, ⟨ ⃗E2+ ⃗B2⟩ that we introduced +earlier in Eqs. (D.8) and (D.9). By virtue of the background equations presented in Eq. (4.7), +these terms precisely cancel. The terms that contains gravitational fluctuations δN and Ni in the +second order action S(1) +inf (D.32) induces additional mass contribution to the inflaton fluctuations +which can be seen by varying this action with respect to δN and Ni and solving them in terms of +δφ as +δN = − +� ϵ +2 +δφ +Mpl +, +∂iNi = +� ϵ +2 +1 +Mpl +� +δ ˙φ − ηH +2 δφ +� +, +(D.33) +where we defined the Hubble slow-roll parameters as +ϵ = − +˙H +H2 = +˙¯φ2 +2H2M2 +pl +, +and +η = +˙ϵ +ϵH . +(D.34) +Similar to our discussion regarding the cubic terms induced by gravity within the gauge field +sector (see e.g. (D.9)), the influence of the lapse and shift on the second order action of the +inflaton fluctuations can be typically ignored in the slow-roll regime (as well as for the non- +attractor era associated with the model in Section 4.1.2 where ϵ → 0, |η| ≃ O(1)) which is a +situation that is generically referred as the decoupling limit of gravity within the literature. For +completeness however we will keep them here. Plugging (D.33) in the action (D.32), we perform +several integration by parts and taking into account the source terms induced by the gauge fields +(δχ → δφ in Eq. (D.8)), the equation of motion obeyed by the inflaton fluctuations can be written +as +δ ¨φ + 3Hδ ˙φ − +� +⃗∇2 − m2 +eff(t) +� +δφ = gcs +f +� +⃗E. ⃗B − ⟨ ⃗E. ⃗B⟩ +� ++ gcs +f +∂⟨ ⃗E. ⃗B⟩ +∂ ˙¯φ +δ ˙φ +(D.35) +73 + +where ⃗∇2 = ∂i∂i is the Laplacian in flat Euclidean space and the effective time-dependent mass is +defined as m2 +eff = V ′′(¯φ) − (2ϵη + 6ϵ − 2ϵ2)H2. Noting the second derivative of the potential in +terms of the slow-roll parameters: +V ′′(¯φ) = H2 +� +−3η +2 + 5ϵη +2 +− 1 +4η2 − +˙η +2H − 2ϵ2 + 6ϵ +� +, +(D.36) +the effective time dependent mass can be described fully in terms of the slow-roll parameters as +m2 +eff = H2 +�9 +4 − 1 +4(η + 3)2 + ϵη +2 − +˙η +2H +� +. +(D.37) +In (D.35), the last term is introduced to parametrize the additional sources that might arise in the +strong back-reaction regime. In particular, as ⟨ ⃗E. ⃗B⟩ grows during inflation (i.e. as the effective +coupling ξ increases during inflation, see Eq. (4.8)), this will first have the effect of sourcing +the inflaton perturbations through the first term in the right hand side of (D.35). As a result +the inflaton perturbations starts to grow, and eventually the solution of the gauge field modes +obtained by assuming a homogeneous inflaton will no longer be valid, and expected to go from +the solution (D.19) to a more general solution A−[¯φ + δφ]. The additional term in the right hand +side of (D.35) precisely introduced to capture the influence of this modified solution of the gauge +fields on the inflaton perturbations. Since gauge field production (and ⟨ ⃗E. ⃗B⟩) is sensitive to +the velocity of the homogeneous inflaton mode, it is reasonable to expect the influence of this +additional source to be proportional to (∂⟨ ⃗E. ⃗B⟩/∂ ˙¯φ)δ ˙φ as shown in (D.35) (see also [71, 73, 240] +for a detailed discussion on this point). Recalling ξ = −gcs ˙¯φ/(2Hf) and (4.8), we note +gcs +f +∂⟨ ⃗E. ⃗B⟩ +∂ ˙¯φ +≃ gcs +f +⟨ ⃗E. ⃗B⟩ +˙¯φ +2πξ, +(D.38) +and so the influence of the source term can be parametrized as an additional damping term in the +equation of motion of the inflaton fluctuations as +δ ¨φ + 3Hβ δ ˙φ − +� +⃗∇2 − m2 +eff(t) +� +δφ = gcs +f +� +⃗E. ⃗B − ⟨ ⃗E. ⃗B⟩ +� +, +(D.39) +where +β = 1 − gcs +f +⟨ ⃗E. ⃗B⟩ +3H ˙¯φ +2πξ. +(D.40) +Assuming β = 1 amounts to neglecting the influence back-reaction effects of the inflaton fluctuations +on the gauge field solutions. This is for example the approach taken in [78], for the bumpy axion +inflation we discuss in Section 4.1.2 with the reasoning that back-reaction effects are mild due +to the localized nature of gauge field production (and so does the resulting increase in inflaton +fluctuations) in the presence of a transiently increasing effective coupling ξ between scalar and +gauge field sector. In the smooth axion inflation (Section 4.1.1) however, gauge field sources and the +resulting scalar fluctuations continuously grow which can eventually influence the dynamics of the +fluctuations through an additional (positive) friction term in β (D.40) as soon as the homogeneous +dynamics of the inflaton enters into the back-reaction regime with gcs⟨ ⃗E. ⃗B⟩/(3H| ˙¯φ|f) ∼ O(1). +74 + +Armed with the equations of motion in real space, one can study vacuum and sourced solutions +of the inflaton perturbations in Fourier space by using the splitting δφ = δφv + δφs and utilizing +the standard Green function methods. We will not repeat these computations here, for the +calculation of the scalar power spectrum relevant for PBH formation, interested readers can follow +the works of [71, 73, 240, 242] in the context of smooth axion inflation and [78] in the context of +bumpy axion inflation we discuss in Sections 4.1.1 and 4.1.2. +D.3 +Scalars sourced by vector fields, the indirect coupling case: χ = σ. +To capture the dynamics of scalar fluctuations in the models studied in Section 4.1.3, we extend +the inflationary action with a spectator sector σ that interacts with the gauge fields as in (D.1). +The action that describes inflation is therefore given by +Sinf = +� +d4x√−g +� +M2 +pl +2 R − 1 +2∂µφ∂µφ − V (φ) − 1 +2∂µσ∂µσ − U(σ) +� +, +(D.41) +where we assume that the spectator axion-like field do not contribute significantly to the background +evolution during inflation which is mainly driven by a flat enough inflaton potential V that we +will leave unspecified. +Expanding the action in terms of the scalar fluctuations δN, Ni and δφ, δσ around a background +solution, we obtain the following linear and second order actions, +S(1) +inf = +� +d4x a3 +� � +3H2M2 +pl − +� +a +� 1 +2 +˙¯φ2 +a − Va(¯φ) +� +− 1 +2⟨ ⃗E2 + ⃗B2⟩ +� +δN +(D.42) +− +�¨¯φ + 3H ˙¯φ + V ′(¯φ) +� +δφ − +� +¨¯σ + 3H ˙¯σ + U ′(¯σ) − gcs +f ⟨ ⃗E. ⃗B⟩ +� +δσ +� +, +S(2) +inf = 1 +2 +� +a +� +d4x a3 +� +δ ˙φ2 +a − (∂iδaφ)2 +a2 +− V ′′ +a (¯φa) δφ2 +a − 3H2M2 +plδN 2 − 2HM 2 +pl∂iNiδN +(D.43) +− 2V ′ +a(¯φa) δφa δN − 2 ˙¯φa δ ˙φa δN − 2 ˙¯φa Ni∂iδφa + ˙¯φ2 +aδN 2 +� +, +where the summation over a runs over fluctuations and background values of the two fields: +φa = {φ, σ} and Va = {V (¯φ), U(¯σ)}. Varying D.43 with respect to Lagrange multipliers δN and +Ni, we obtain +2HM 2 +pl δN = +� +a +˙¯φaδφa, +(D.44) +−2HM 2 +pl ∂iNi = +� +a +� ˙¯φaδ ˙φa + V ′ +a(¯φa)δφa +� ++ +� +6H2M2 +pl − +� +a +˙¯φ2 +a +� +δN. +(D.45) +Plugging these solutions back in the actions, we obtain the following second order action for scalar +fluctuations [245, 262], +S(2) +inf = 1 +2 +� +d4x a3 +� � +δ ˙φ2 +a − (∂iδφa)2 +a2 +� +− m2 +abδφaδφb +� +, +(D.46) +75 + +m2 +ab = V (tot) +,ab +− 1 +a3 +d +dt +� +a3 +H +˙¯φa ˙¯φb +M2 +pl +� +. +(D.47) +where summation over repeated indices a, b is implied and V (tot) +,ab +≡ ∂2V (tot)/(∂ ¯φa∂ ¯φb) with +V = � +a Va(¯φa) = V (¯φ) + U(¯σ). +We note that in deriving the expressions above we used +background equations of motion ignoring the artificially introduced mean field expectation values +involving gauge fields (noticing that these terms that appear in (D.8), (D.9) and (D.42) sums up +to zero): +¨¯φa + 3H ˙¯φa + V ′ +a(¯φa) = 0, +(D.48) +−2 ˙HM 2 +pl = +� +a +˙¯φ2 +a. +(D.49) +Indeed for the spectator models discussed in Section 4.1.3, due to localized nature of the gauge +field production back-reaction effects is small and can be ignored [73, 77, 244], and therefore the +background evolution can be studied in a consistent way by focusing on the equations in (D.48) +along with the Friedmann equation 3H2M2 +pl = � +a ˙¯φ2 +a + Va(¯φa). Taking into account the effects of +gauge field sources (δχ → δσ in Eq. (D.8)) on the spectator scalar fluctuations, the equations of +motion for the scalar perturbations read as +δ ¨φa + 3Hδ ˙φa − +� +⃗∇2 − V ′′ +a +� +δφa − +� +b +1 +a3 +d +dt +� +a3 +H +˙¯φa ˙¯φb +M2 +pl +� +δφb = Ja( ⃗E, ⃗B), +(D.50) +where δφa = (δφ, δσ)T and Ja = (0, gcs +f ⃗E. ⃗B)T . Notice that since we consider sum separable +potentials V (tot) = V + U, the first term in the mass matrix (D.47) makes a diagonal contribution +to the mass of the each field in the equations of motion (D.50). However, the presence of the second +term in the mass matrix (D.47), which is induced by the presence of gravitational fluctuations +δN and Ni, introduces a mass mixing between scalar fluctuations δφa − δφb (a ̸= b)(i.e. through +the last term in (D.50)). In other words, although we consider a Lagrangian (D.41) where the +two scalar fields appear to be decoupled from each other, gravitational interaction will inevitably +introduce a minimal communication channel between the physical fluctuations of the two scalar +sectors. At leading order in the slow-roll expansion, the mass mixing (a ̸= b) originates from the +following terms in the Lagrangian +Lmix = +1 +2a3 +d +dt +� +a3 +H +˙¯φa ˙¯φb +M2 +pl +� +δφaδφb +−→ +Lmix ≃ 6√ϵφϵσ H2δφδσ, +(D.51) +where ϵa = ˙¯φ2 +a/(2H2M2 +pl) is the slow-roll parameter of the each field. +Therefore as long as +both fields roll with a non-vanishing velocity ˙¯σ, ˙¯φ ̸= 0 during inflation, their fluctuations can be +converted to one another. In the presence of particle production in the gauge field sector, the +mixing between the two scalar sectors is crucial in understanding the influence of the gauge field +sources on the visible scalar sector fluctuations δφ. In order to see this explicitly, we focus on the +76 + +leading order mixing in the slow-roll expansion to rewrite the system of equations in (D.50) as +δ ¨φ + 3Hδ ˙φ − +� +⃗∇2 − m2 +φ +� +δφ ≃ 6√ϵφϵσ H2δσ, +δ¨σ + 3Hδ ˙σ − +� +⃗∇2 − m2 +σ +� +δσ ≃ gcs +f +⃗E. ⃗B, +(D.52) +where m2 +a ≃ V ′′ +a − 6ϵaH2 at leading order in the slow-roll expansion. We note that in the second +line of (D.52), we ignored mixing terms that can source spectator fluctuations δσ through δφ which +is a sub-leading effect compared to sourcing of δσ by the enhanced gauge fields. This amounts +to considering the main production channel of the observable scalar fluctuations schematically +as: δA + δA → δσ → δφ. Using the equations of motion in real space (D.52) and considering its +sources in (D.24), the observable power spectrum of scalar fluctuations can be carried by using +Green’s function methods in Fourier space following detailed calculations presented in [77, 261] +along with the prescription we present in Appendix E regarding the curvature perturbation. +E +Curvature perturbation +In this appendix, we define the curvature perturbation R on comoving slices and study its +parametric dependence on the matter fluctuations for some of the scenarios we consider in the +main text. We begin by noting the definition of comoving curvature perturbation in flat gauge +which reads as [282, 295]: +R = − +H +(¯ρ + ¯P) δqflat +(E.1) +where ¯ρ and ¯P is the total background energy density and pressure (see Appendix A) and δqflat is +the scalar momentum density in flat gauge. In terms of the perturbed energy momentum tensor, +δqflat is given by +δT 0 +i = ∂iδqflat +←− +Tµν ≡ − +2 +√−g +δSm +δgµν , +(E.2) +where the definition of the unperturbed energy momentum tensor provided on the right hand +side of (E.2) can be used for a given matter action describing the system. We note that in this +definition δSm/δgµν represents the variation of the matter action with respect to the metric field. +Anticipating that we will focus on different limits of a more complicated/general model, we start +by parametrizing perturbed δT 0 +i for the spectator axion model we discuss in Section 4.1.3 with +the matter action given by the sum of (D.41) and (D.1). Noting that the last term in (D.1) is +topological and does not gravitate, we have +δT 0 +i = g0µ (∂µφ∂iδφ + ∂µσ∂iδσ + gρσFµρFiσ) , +(E.3) +where the last term characterize the contribution from gauge fields which is second order in fluc- +tuations. Following (E.3), (E.2) and (E.1), it is therefore clear that for a multi-sector inflationary +model, curvature perturbation can in principle obtain contributions from all fields present in the +matter Lagrangian. In what follows, we take (E.3) as the main reference point to study different +limits of it to derive an expression for the curvature perturbation R relevant for some of the +models we study in the main text. +77 + +Canonical single-field inflation. This case corresponds to neglecting terms proportional to +spectator and gauge field fluctuations in (E.3). Noting that ¯ρ + ¯P → ˙¯φ2 for single-field canonical +inflation, to linear order in inflaton fluctuations, the curvature perturbation (E.1) is given by +R = H +a ˙¯φ +(aδφ) +↔ +R = v +z , +(E.4) +where we made the connection between the Mukhanov-Sasaki variable (see Section 3.1) and inflaton +fluctuations clear v = aδφ using the definition of pump field in this case z = −a ˙¯φ/H = a +√ +2ϵMpl. +Upon canonical quantization of δφ (and hence R), their corresponding Fourier space variables +satisfy the same relation in (E.4) and the dimensionless power spectrum at the end of inflation +can be computed via (C.13). +Smooth Axion Inflation and Bumpy Axion Inflation. For the models we study in Sections +4.1.1 and (4.1.2), we instead focus on the curvature perturbation ζ on uniform density gauge +which can be related to the density perturbation in flat gauge δρflat and comoving curvature +perturbation R on super-horizon scales as [295] +ζ ≃ −R = −H +˙¯ρ δρflat, +(E.5) +where ¯ρ is the total energy density of the axion gauge field system. Using background equations +of the system, time derivative of the total energy density is given by ˙¯ρ = −3H ˙¯φ2 − 4HρA [296] +where ρA is the energy density of the gauge field as defined in (4.8). Noting this relation, to linear +order42 in the cosmological fluctuations, the comoving curvature perturbation R can be obtained +as [73] +R = H +˙¯φ +δφ F, +F ≡ +− ˙¯φ V ′(¯φ) +H(3 ˙¯φ2 + 4ρA) +, +(E.6) +where we used δρflat ≃ V ′(¯φ)δφ neglecting contributions to the total energy proportional to the +kinetic energy of the inflaton as they should be small both in the slow-roll and the inflationary +regime dominated by the friction provided by the gauge fields. In (E.6), the factor F parametrizes +the correction to the definition of the curvature perturbation in the strong back-reaction regime +which must be taken account in the smooth axion inflation model we present in the main text. +On the other hand, the standard relation in (E.4) applies in the regime of negligible back-reaction +of the gauge field on the evolution of the inflaton and expansion of the universe, i.e. when ρA +is negligible and the relation −3H ˙¯φ ≃ V ′(¯φ) is satisfied. This situation applies both at early +times during the smooth axion model we consider as well as within the bumpy axion inflation +model of Section 4.1.2 where back-reaction effects have shown to be small around the peak of the +scalar power spectrum. To summarize, in the strong back-reaction regime, the power spectrum of +42We note that gauge fields also contribute to the curvature perturbation at second order in perturbation theory +simply because of their influence on the energy density. It turns out that this contribution is proportional to the +absolute value of the Poynting vector R(AA) ∝ a| ⃗E × ⃗B|. Therefore on general grounds, we expect this contribution +to be negligible at the end of inflation (i.e. at the time we are interested in the correlators of R) because particle +production in the gauge field sector saturates on super-horizon scales and the corresponding electromagnetic fields +decay as ⃗E, ⃗B ∝ a−2. See e.g. the discussion presented in [77, 297] within similar contexts. +78 + +curvature perturbation can bu computed using the standard relation (E.4) times a factor of F +correction. +Spectator axion-gauge field model. In the model presented in 4.1.3, in principle we also +need to take into account all the contributions to the curvature perturbation using the formulas +(E.3), (E.2) and (E.1). However, as explicitly checked in [77], the contribution to the curvature +perturbation that is bilinear in the gauge fields can be safely ignored at late times during which +we are interested in the correlators of R. Further simplifications on the functional form of R arise +due to the spectator nature of the axion sector σ and due to the assumption that it settles back +to its global minimum long before the end of inflation where ˙¯σ → 0. 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Sorbo, “Inflationary magnetogenesis with added helicity: +constraints from non-gaussianities,” arXiv:1707.09750 [astro-ph.CO]. +95 + diff --git a/MdE2T4oBgHgl3EQfBQZc/content/tmp_files/load_file.txt b/MdE2T4oBgHgl3EQfBQZc/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..dce72901b796f31d751fcc4c69b4d3f41b93042a --- /dev/null +++ b/MdE2T4oBgHgl3EQfBQZc/content/tmp_files/load_file.txt @@ -0,0 +1,5005 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf,len=5004 +page_content='Inflation and Primordial Black Holes Ogan ¨Ozsoy †,‡, Gianmassimo Tasinato⋆,× † CEICO, Institute of Physics of the Czech Academy of Sciences, Na Slovance 1999/2, 182 21, Prague.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' ‡ Instituto de F´ısica T´eorica UAM/CSIC, Calle Nicol´as Cabrera 13-15, Cantoblanco, 28049, Madrid, Spain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' ⋆ Dipartimento di Fisica e Astronomia, Universit`a di Bologna, via Irnerio 46, Bologna, Italy × Department of Physics, Swansea University, Swansea, SA2 8PP, United Kingdom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' We review conceptual aspects of inflationary scenarios able to produce primordial black holes, by amplifying the size of curvature fluctuations to the level required for triggering black hole formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' We identify general mechanisms to do so, both for single and multiple field inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' In single field inflation, the spectrum of curvature fluctuations is enhanced by pronounced gradients of background quantities controlling the cosmological dynamics, which can induce brief phases of non–slow-roll inflationary evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' In multiple field inflation, the amplification occurs through appropriate couplings with additional sectors, characterized by tachyonic instabilities that enhance the size of their fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' As representative examples, we consider axion inflation, and two-field models of inflation with rapid turns in field space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' We develop our discussion in a pedagogical manner, by including some of the most relevant calculations, and by guiding the reader through the existing theoretical literature, emphasizing general themes common to several models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' 1Correspondence e-mail: ogan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content='ozsoy@csic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content='es arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content='03600v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content='CO] 9 Jan 2023 Contents 1 Introduction 2 2 PBH formation in the early universe 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content='1 PBH formation as a causal process 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content='2 The relevant quantities for PBH abundance 8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content='3 Relating PBH properties with primordial scalar fluctuations 13 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content='4 Brief summary, and the path ahead 17 3 Enhancement of scalar perturbations during single-field inflation 18 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content='1 The dynamics of curvature perturbation 19 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content='2 Enhancement through the resurrection of the decaying mode 22 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content='3 Growth in the power spectrum when the decaying modes are slacking 29 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content='4 Brief summary 33 4 Enhanced primordial power spectrum in multi-field models 34 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content='1 Enhanced scalar perturbations from axion-gauge field dynamics 35 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content='1 Smooth Axion Inflation 37 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content='2 Bumpy axion inflation 42 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content='3 Spectator axion-gauge field dynamics 46 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content='2 Strong turns in the multi-scalar field space 50 5 Outlook 57 A Background Cosmology: Mini-Review 58 B Analytic estimate for the threshold of collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' 63 C Solving the Mukhanov-Sasaki equation: Numerical procedure 64 D Details of the axion-gauge field dynamics 66 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content='1 Gauge field production by rolling scalars 68 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content='2 Scalars sourced by vector fields, the direct coupling case: χ = φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' 73 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content='3 Scalars sourced by vector fields, the indirect coupling case: χ = σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' 75 E Curvature perturbation 77 References 79 1 1 Introduction Primordial black holes: history of the concept Inflation, a short period of accelerated expansion in the very early moments of the universe, has become one of the main pillars of modern cosmology [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' Leaving aside its success in addressing the puzzles of the standard hot Big Bang cosmology, inflation provides an explanation for the quantum mechanical origin of structures such as galaxies (including our own!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=') and the anisotropies in the Cosmic Microwave Background (CMB) radiation [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' In the last two decades, the advances in the observational cosmology and in particular the observations of the CMB and of the large scale structure (LSS) of our universe have so far confirmed the predictions of inflation, and arguably established its status as the main theoretical framework describing the very early universe [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' These successes notwithstanding, CMB and LSS probes only provide us information on the early universe at the largest cosmological scales (10−4 ≲ k [Mpc−1] ≲ 10−1) corresponding to a small fraction of the early stages of inflationary dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' Hence, while inflation provides us with a consistent, testable framework in understanding the initial conditions in the universe at the largest scales, we do not have direct access to most of the inflationary dynamics, and to the universe evolution in the early post-inflationary era.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' Importantly, these stages could be host to a number of interesting phenomena, including the production of stable relics such as dark matter (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' [6] for a historical review on dark matter) that is essential in understanding the world we observe today, as well as for establishing new physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' Indeed, the existence of non-luminous, cold dark matter (CDM) that constitutes a quarter of the total energy budget in the universe [7] is one of the most glaring evidences for beyond the Standard Model physics [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' The absence of signatures from collider experiments, along with unsuccessful direct and indirect detection searches, have all made the DM puzzle particularly compelling [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' An intriguing and economical explanation that might account for DM density in our universe is a scenario where DM is made of compact objects, such as primordial black holes (PBHs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' Pioneered by the works of Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' Zel’dovic and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' Novikov [10] and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' Hawking [11], the initial ideas in this direction began with the realization that PBHs could form by the gravitational collapse of over-dense inhomogeneities in the early universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' In the mid 70’s, it was later realized by the works of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' Carr [12, 13] and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' Chapline [14] that PBHs could contribute to DM density and provide the seeds for the supermassive BHs populating our universe [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' Following these theoretical progresses, the interest of the scientific community on PBHs has risen in the mid 90’s by the reported detection of micro-lensing events from MACHO collaboration [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' An immediate interpretation of these results was suggesting on the possibility that a significant fraction of mass density in our galaxy could be composed of sub-solar mass PBHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' However, these considerations were later rendered invalid by the findings of EROS [17] and OGLE [18, 19] collaborations, concluding that only a small fraction of mass in the Milky Way could be in the form of PBHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' Stimulated both by the absence of signals for well-motivated particle DM candidates, and the first detection of gravitational waves (GWs) from merging BHs by the LIGO/VIRGO collaboration [20], a second surge of interest in PBHs was ignited (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' In particular, different groups suggested that merging PBHs could be responsible for the observed GW signals, while constituting a significant fraction of DM density in our universe [21–23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' Since the first appearance of these 2 2000 2005 2010 2015 2020 Year 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content='25 Number of papers on PBHs ÷ 103 % of papers on PBHs Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' Total and relative number of manuscripts appeared on the arXiv from 1996 until today related to various aspects regarding primordial black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' Spikes of activity in the literature, particularly after the mid 90’s due to claimed lensing events by the MACHO collaboration and the GW detection by LIGO in 2015 is clearly visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' articles, a significant amount of effort has been pushed forward by the community, to search and constrain the abundance of PBHs by utilizing their gravitational and electromagnetic effects on the environment at small scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' Various experiments set stringent constraints on PBH abundance for solar and sub-solar mass range, leaving a viable window for this scenario for tiny PBH masses 10−17 ≲ Mpbh [M⊙] ≲ 10−12 (M⊙ ≃ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content='98 × 1033 gr) as the totality of DM (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' [24–26]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' It should be noted that some of the constraints derived in the literature make specific assumptions about the formation process and the subsequent evolution of PBHs (such as monochromatic mass functions, clustering and accretion processes, etc) and on other model dependent specifics (such as non-Gaussianity), and therefore they could be relaxed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' Since we mostly focus on the subject of inflationary model building, we will not review these issues and aforementioned constraints, but the interested reader can find more details in excellent reviews published recently, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' [27–33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' PBHs are likely to form well before the end of the radiation dominated era (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' before the so called matter-radiation equality), and behave like cold and collision-less matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' Therefore they constitute an interesting DM candidate, if they are massive enough Mpbh ≳ 1015 g ≃ 10−18M⊙ to ensure a lifetime comparable with the age of the universe [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' In this context, a particularly appealing aspect of PBH dark matter is its economical and minimal structure, in the sense that this scenario does not require any additional beyond Standard Model (BSM) physics (such as new particles and interactions), provided that one alters the not-so-well constrained early universe at small scales by introducing a viable mechanism to account for the production of large density fluctuations required for PBH formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' Similar to the generation of CMB anisotropies, a compelling and natural source of these perturbations in the early universe could be the quantum fluctuations that are stretched outside the horizon during inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' However, in order to generate such over-dense regions that can collapse to form PBHs in the post-inflationary universe, one needs to devise a mechanism to enhance by several orders of magnitude the inflationary scalar perturbations at small scales k ≫ kcmb (corresponding to late stages of inflation), far above the values required to match CMB 3 observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' As the observed temperature anisotropies prefers a red tilted power spectrum at CMB scales, this situation generically requires a blue tilted power spectrum, or some specific features at scales associated with PBH formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' In the context of canonical single scalar field inflation, the first ideas in this direction appeared in works by P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' Ivanov, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' Naselsky and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' Novikov [35] (see also [36]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' In particular, these authors have shown that if the inflaton potential has a very flat plateau-like region for field ranges corresponding to the late stages of accelerated expansion, the inflationary dynamics enters a “non-attractor” regime called ultra slow-roll (USR) [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' This leads to super-horizon growth of scalar perturbations [38–40] that can eventually trigger PBH formation in the post-inflationary universe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' Many explicit single-field inflationary models that exhibit similar local features were subsequently studied in the literature: for a partial list of popular works see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' [43–51] (see also [52–58] for earlier constructions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' In the context of single scalar field inflation, another possibility to generate an enhancement in the scalar power spectrum is to invoke a variation of the sound speed of scalar fluctuations, for example through a reduction in the speed of sound c2 s [49, 59, 60] or through a rapidly oscillating c2 s which triggers a resonant instability in the scalar sector [61, 62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' From a top-down model building perspective, a rich particle content during inflation is not just an interesting possibility, but appears to be a common outcome of many BSM theories [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' Since the early days of research on PBHs, multi-field inflationary scenarios has also attracted considerable attention as a natural way to realize enhancement in the scalar perturbations at small scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' For instance, large scalar perturbations may arise through instabilities arising in the scalar sector, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' during the waterfall phase of hybrid inflation [52, 64, 65] or due to turning trajectories in the multi-scalar inflationary landscape, as reported recently in [66–70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' Another intriguing possibility in this context is by employing axion-gauge field dynamics during inflation [71–78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' In these models, particle production in the gauge field sector act as a source for the scalar fluctuations, and hence can be responsible for PBH formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' A common feature of all inflationary scenarios capable of producing PBH populations is the inevitable production of a stochastic GW background (SGWB) induced through higher order gravitational interactions between enhanced scalar and tensor fluctuations of the metric [79–81].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' Interestingly, this signal may carry crucial information about the properties of its sources including the amplitude, statistics and spectral shape of scalar perturbations (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' [82–87]) and could provide invaluable information on the underlying inflationary production mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' Furthermore, since the resulting GW background interacts very weakly with the intervening matter between the time of their formation and today, it leads to a rather clean probe of the underlying PBH formation scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' This allows us to access inflationary dynamics on scales much smaller than those currently probed with CMB and LSS experiments, through space and ground based GW interferometers including Laser Interferometer Space Antenna (LISA) [88, 89], Pulsar Timing Array (PTA) experiments [90, 91] and DECIGO [92, 93].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' For a detailed review of induced SGWB and the dependence of its properties on the post-inflationary expansion history, see [94].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' The structure of this review If their origin is attributed to the large primordial fluctuations, PBHs may offer us a unique window to probe inflationary dynamics at sub-CMB scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' In this work, focusing mainly on 2Another inflationary background that exhibit similar features is called constant-roll inflation, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' [41, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' 4 the activity in the literature within the last few years, we aim to revisit and review different inflationary production mechanisms of PBHs 3 and their main predictions, in a heuristic and pedagogical manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' The audience we have in mind are graduate students, or researchers in related fields who wish to learn about inflation and primordial black holes, and to be guided through the large literature on the subject by emphasizing common conceptual themes behind many different realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' The review is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' In Section 2, we present a simplified, intuitive picture of PBH formation in the inflationary universe and give some approximate estimate for the required conditions to produce PBHs from the perspective of inflationary dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' In Section 3, we discuss ideas to enhance the curvature power spectrum within single-field inflation, as required for PBH formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' These mechanisms exploit large gradients in background quantities which get converted into an amplification of fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' Besides reviewing analytic findings, we also develop some numerical analysis and provide a link to a code for reproducing our results (see Footnote 20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' In Section 4, we focus on multi-field inflationary scenarios that can generate PBH populations including particle production during axion inflation, or sudden turns in the multi-scalar inflationary landscape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' Finally, we end with a discussion on future directions in the concluding Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' We supplement this work with several technical Appendices where we provide useful formulas and calculations used in the main body of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' Conventions Throughout this review, we work with natural units ℏ = c = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' We will use the reduced Planck mass defined as M2 pl = (8πG)−1 and retain it in the equations unless otherwise stated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' For time dependent quantities, over-dots and primes denote derivatives with respect to cosmological time t and conformal time dτ ≡ dt/a(t) respectively where a(t) is the scale factor of the background FLRW metric gµν = diag(−1, a2, a2, a2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' 2 PBH formation in the early universe We start providing a physical description of PBH formation in the early universe, as comprised of an early stage of inflation, followed by radiation and matter domination (for a mini-review on background cosmology, see Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' Our aim is to set the stage and relate basic properties of a PBH population – as their mass and abundance – with the features of primordial curvature fluctuations originating from inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' For this purpose, in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content='1 we describe the mechanism of PBH formation in the post-inflationary universe, emphasizing its nature as causal process controlled by the inflationary quantum fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' In Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content='2 we discuss relevant concepts such as the threshold for collapse into black holes, and the corresponding mass and collapse fraction of PBHs, relevant for a computation of their abundance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' Finally, in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content='3, we relate the PBH abundance to primordial physics during inflation, with the aim to determine the amplitude of scalar fluctuations required for producing a population of PBHs with interesting 3PBHs could also form in the post-inflationary universe through the collapse of cosmic strings [95–97] and domain walls [98–101], phase transitions [102, 103], bubble collisions [104, 105], scalar field fragmentation via instabilities [106, 107].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' We note that PBHs could also form through the instabilities generated in the final stage of inflation commonly referred as (p)reheating [108–111].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' We will not dwell into this possibility here, for a list of recent works in this line of research, see [112, 113].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' 5 consequences for cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' All the concepts we discuss form the basis and motivations for our analysis of inflationary mechanisms for PBH production, which we develop in Sections 3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' � Main References: In compiling the materials of this Section and to set the main framework for our discussion, we have benefited from the ideas presented in the reviews by C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' Byrnes and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' Cole [114], M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' Sasaki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' [29] and the Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' thesis by G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' Franciolini [115].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content='1 PBH formation as a causal process An important concept in an expanding space-time is the horizon scale, crucial for understanding the causal properties of the dynamics of perturbations which are responsible for PBH formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' As an indicator of the rate of our universe expansion, the Hubble rate H(t) ≡ ˙a(t)/a(t) has dimensions of inverse length (or time−1 in natural units).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' This makes the quantity H−1 (Hubble horizon) as the natural candidate for a physical length scale in an expanding universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' Commonly referred to as the Hubble distance, the quantity 1/H (or c/H, if one wishes to recover physical units) measures the distance that light travels within one Hubble time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' Therefore, it can be considered as a good proxy for a (time-dependent) length scale controlling the size of a causal patch in our universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' Bearing in mind that we relate physical quantities to comoving ones by the scale factor a(t), a useful quantity that guides us in this direction is the comoving Hubble horizon, (aH)−1, and in particular its time evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' When expressed in terms of the second derivative of the scale factor, the time derivative of the comoving horizon can be written as d dt � 1 aH � = − ¨a a2H2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content='1) Notice that during inflation ¨a > 0: hence, the comoving horizon scale is a decreasing function of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' Whereas, in a decelerating universe with ¨a < 0 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' in the post-inflationary universe before dark energy domination), this quantity is an increasing function of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' The property that the comoving horizon decreases during an accelerated expansion is perhaps the most important element to understand inflation as a solution of the horizon problem of the Hot Big Bang cosmology, and a framework for the quantum mechanical origin of structures in our universe 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' The time dependence of the comoving Hubble horizon is controlled by the value of the background equation of state w (EoS) as (see Appendix A) (aH)−1 ∝ a(1+3w)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content='2) Therefore, during inflation w ≃ −1 and (aH)−1 ∝ a−1 while, during the subsequent phases of radiation dominated (RDU) and matter dominated universe (MDU), the comoving horizon evolves as (aH)−1 ∝ a1 and (aH)−1 ∝ a1/2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' The evolution of the comoving horizon with respect to logarithm of scale factor ln(a) is illustrated in Fig 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' When we study the statistical properties of fluctuations in Fourier space, we often label a given perturbation mode with a comoving length scale k−1, measured in units of megaparsecs (Mpc = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content='26 × 106 light years ≃ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content='1 × 1019 km).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' Therefore, a crucial quantity to conceptualize the behavior of perturbations in the inflationary universe is the ratio of the wavelength of a given mode with respect to the size 4A detailed discussion on these topics can be found in Chapter 4 of Baumann’s book [116].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} +page_content='AB7XicbVBNSwMxEJ2tX7V+VT16CRahXsquFPVY9OKxgv2AdinZNvGZpMlyQpl6X/w4kERr/4fb/4b0+0etPXBwO9GWbmBT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE2T4oBgHgl3EQfBQZc/content/2301.03600v1.pdf'} 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Furthermore, Enriquez [5] generalize GRT1 to an intermediate closed +2010 Mathematics Subject Classification. 11M32. +1Later, we will define a Lie algebra tn,N and its sub-Lie algebra t0 +n,N for general n ≥ 2 and N ≥ 1. +2In this paper, we will use a convention such that the path γ ∈ π1(P1 \ {0, 1, ∞}, p, q) corresponds to a power series whose +coefficients of eak · · · ea1 is +� +0 0 +1 +d = 0, +θ′(k) := dist ◦ reg +� ◦ ς ◦ θ(k), +θ1(k) := θ′(k) + 2 +� +k=(k′,{1}2m) +m≥1 +e−1e2m−1 +0 +� θ′(k′), +w⋆(k1, . . . , kd) := +� +−e1ek1−1 +0 +(e0 − e1)ek2−1 +0 +· · · (e0 − e1)ekd−1 +0 +d > 0 +1 +d = 0. +Note that {̺(w⋆(k))} is a basis of C. Now, by using the above notations, we define a Q-linear map ℘ : C → +A0({0, 1, −1}) by +℘(̺(w⋆(k1, . . . , kd))) := θ1(k1, . . . , kd). +In [11, Lemma 2.17], the author proved that +Lm(reg +�(ς(w)) = Lm(℘(w)) +(3.1) +for w ∈ C. The properties of Lm used in the proof of [11, Lemma 2.17] is the regularized double shuffle +relations and duality relation for Lm |A0({0,1}), and (the special case of) the distribution relation (1 − +21−2m)Lm(e1e2m−1 +0 +) = −Lm(e−1e2m−1 +0 +). These properties are also satisfied by +Zϕ : A0({0, 1, −1}) → k +; +u �→ ⟨ϕ, u⟩ +for ϕ ∈ PENT(2, k) by Proposition 12. Thus (3.1) also holds for Zϕ, i.e., we have +⟨ϕ, reg +�(ς(w)⟩ = ⟨ϕ, ℘(w)⟩ +(3.2) +for ϕ ∈ PENT(2, k) and w ∈ C. +Define regz→0 : B → A0({0, 1, −1}) by +B +e−1�→0 +−−−−→ B′ +u�v�→u⊗v +−−−−−−−→ +≃ +B′′ ⊗ B′′′ +regz→0⊗ǫ +−−−−−−→ A0({0, 1}) ⊗ C +u⊗v�→dist(u)�℘(v) +−−−−−−−−−−−−→ A0({0, 1, −1}). +where B′, B′′, B′′′ and C are Q-modules defined by +B′ := B ∩ Q ⟨e0, ez, e−z2⟩ , +B′′ := B′ ∩ (Q ⊕ ezQ ⟨e0, ez, e−z2⟩) , +B′′′ := B′ ∩ Q ⟨e0, e−z2⟩ , + +THE CYCLOTOMIC GROTHENDIECK-TEICHMÜLLER GROUP AND THE MOTIVIC GALOIS GROUP +6 +regz→0 is the ring homomorphism from Q⟨e0, ez, e−z2⟩ to A({0, 1}) defined by regz→0(e0) = regz→0(e−z2) = +e0 and regz→0(ez) = e1, and ǫ is the ring homomorphism from Q⟨e0, e−z2⟩ to A({0, 1}) defined by ǫ(e0) = e0 +and ǫ(e−z2) = e1. +For c ∈ {0, 1, −1}, define ∂c : B → B by +∂c(ea1 · · · eak) = +k +� +i=1 +(ordz=c(ai − ai+1) − ordz=c(ai − ai−1)) ea1 · · · eai−1eai+1 · · · eak +where a0 = 0, ak+1 = z, and ordz=c(0) = 0. Define ϕ : B → A0({0, 1, −1}) by +ϕ(u) = +∞ +� +l=0 +� +c1,...,cl∈{0,1,−1} +regz→0(∂c1 · · · ∂clu) +� reg +�(ec1 · · · ecl). +Now, ICF ⊂ A0({0, 1, −1}) is defined by +ICF := {u |z→1 −ϕ(u) | u ∈ B} +where the map u �→ u |z→1 is the ring morphism from Q ⟨e0, e−1, ez, e−z2⟩ to A({0, 1, −1}) defined by +e0 |z→1= e0, e−1 |z→1= e−z2 |z→1= e−1, and ez |z→1= e1. The following is the main theorem of [11]. +Theorem 14. Let Lm : A0({0, 1, −1}) → H2 be the Q-linear map defined by Lm(ea1 · · · eak) = Im(0; a1, . . . , ak; 1). +Then +ker(Lm) = ICF. +4. The dual vector space of Ut4,N(Q) +Let Utn,N(k) be the completion of the universal enveloping algebra of tn,N(k). Recall that the mixed +pentagon equation is formulated as an equality in exp t4,N(k) ⊂ Ut4,N(k) = Ut4,N(Q)ˆ⊗k. Let Rn,N be the +ring of formal power series over Q in free variables ˜tij (2 ≤ j ≤ n) and ˜t(a)ij (i ̸= j, 2 ≤ i, j ≤ n, a ∈ Z/NZ). +Then we can regard Utn,N(Q) as a quotient of Rn,N by a surjective map +Rn,N → Utn,N(Q) +; +˜t1j �→ t1j, ˜t(a)ij �→ t(a)ij. +By an obvious way, the (topological) dual vector space Homcont +Q +(Rn,N(Q), Q) of Rn,N(Q) can be regarded +as the non-commutative polynomial ring over Q in n − 1 + (n − 1)(n − 2)N variables. Then the dual space +of Ut4,N is embedded as the Q-vecor subspace of this non-commutative polynomial ring. In this section, we +propose a way to construct elements of this subspace. +4.1. Definition of Ω⋆ +F . Let F be a field, +Ω• +F := +� +Ω0 +F → Ω1 +F → Ω2 +F → · · · +� +the complex of Kähler differential forms on Spec(F), +B•(Ω• +F ) := (B0(Ω• +F ) → B1(Ω• +F ) → · · · ) +the reduce bar complex of Ω• +F , and +H0(B•(Ω• +F )) := ker(B0(Ω• +F ) → B1(Ω• +F )) +the first cohomology group of B•(Ω• +F ). Define a Q-linear subspace Ω⋆ +F of T (Ω1 +F ) := �∞ +k=0(Ω1 +F )⊗k by +Ω⋆ +F := T (¯ΩF ) ∩ H0(B•(Ω• +F )) +where +¯Ω1 +F := {da +a | a ∈ F ×} ⊂ Ω1 +F . +More explicitly, Ω⋆ +F is the kernel of the Q-linear map from �∞ +k=0(¯Ω1 +F )⊗k to �∞ +i,j=0(¯Ω1 +F )⊗i ⊗ Ω2 +F ⊗ (¯Ω1 +F )⊗j +defined by +ω1 ⊗ · · · ⊗ ωk �→ +k−1 +� +i=1 +ω1 ⊗ · · · ωi−1 ⊗ (ωi ∧ ωi+1) ⊗ ωi+2 ⊗ · · · ⊗ ωk. + +THE CYCLOTOMIC GROTHENDIECK-TEICHMÜLLER GROUP AND THE MOTIVIC GALOIS GROUP +7 +4.2. The characterization of Utn,N. Put QN := Q(µN) and Fn,N := QN(z2, . . . , zn). Fix a generator +ζN ∈ µN. Let Ω1 +n,N the Q-submodule of Ω1 +Fn,N spanned by +dzi +zi +(2 ≤ i ≤ n) +and +dzi +zi − ζa +Nzj +(2 ≤ i, j ≤ n, i ̸= j, a ∈ Z/NZ). +We regard T (Ω1 +n,N) := �∞ +k=0 Ω⊗k +n,N ⊂ T (Ω1 +Fn,N) as the Q-algebra by u · v = v ⊗ u.3 Then we identity Rn,N(k) +with the dual vector space of T (Ω1 +n,N) by an element +∞ +� +m=0 + +� +i +dzi +zi +· s1i + +� +i,j,a +dzi +zi − ζa +Nzj +· s(a)ij + + +m +∈ T (Ωn,N)ˆ⊗Rn,N. +Now, we can give a characterization of the dual vector space of Utn,N(k). +Proposition 15. For n ≥ 2 and N ≥ 1, Utn,N(Q) is the dual vector space of +Ω⋆ +Fn,N ∩ T (Ω1 +n,N) +(⊂ T (Ω1 +n,N)), +i.e., +Utn,N(k) = HomQ(Ω⋆ +Fn,N ∩ T (Ω1 +n,N), k), +Ω⋆ +Fn,N ∩ T (Ω1 +n,N) = Homcont +Q +(Utn,N(Q), Q). +Proof. It follows from definition. +□ +4.3. Construction of elements of Ω⋆ +F . For k ≥ 0, we denote by Ak +F the Q-vector space generated by +formal symbols I(a0; a1, . . . , ak; ak+1) with a1, . . . , ak ∈ F and a0, ak ∈ F ∪ {∞}. Let us define a Q-linear +map ∂ : Ak +F → Ak−1 +F +⊗ ¯Ω1 +F by +∂I(a0; a1, . . . , ak; ak+1) = +� +r∈{±1} +r +k +� +i=1 +I(a0; a1, . . . , �ai, . . . , ak; ak+1) ⊗ d log(ai+r − ai), +where we put d log(a) = 0 for a ∈ {0, ∞}. +Definition 16. For k ≥ 0, we define a Q-linear map ψk : Ak +F → (¯Ω1 +F )⊗k as follows. For the case k = 0, we +put ψ0(I(a0; a1)) = 1. For k ≥ 1, define ψk recursively as the composite map +Ak +F +∂−→ Ak−1 +F +⊗ ¯Ω1 +F +ψk−1⊗id +−−−−−−→ (¯Ω1 +F )⊗(k−1) ⊗ ¯Ω1 +F ≃ (¯Ω1 +F )⊗k. +Lemma 17. The composite map +Ak +F +∂−→ Ak−1 +F +⊗ ¯Ω1 +F +∂⊗id +−−−→ Ak−2 +F +⊗ ¯Ω1 +F ⊗ ¯Ω1 +F +u⊗ω1⊗ω2�→u⊗(ω1∧ω2) +−−−−−−−−−−−−−−−→ Ak−2 +F +⊗ Ω2 +F +is zero. +3The standard definition of the multiplication is u · v = u ⊗ v, but, we change the order here to reconcile the differences in +the orders of multiplications used in the definitions of associator and ICF. + +THE CYCLOTOMIC GROTHENDIECK-TEICHMÜLLER GROUP AND THE MOTIVIC GALOIS GROUP +8 +Proof. By definition, we have +(∂ ⊗ id) ◦ ∂(I(a0; a1, . . . , ak; ak+1)) += +� +1≤i≤k +� +r∈{±1} +∂(I(a0; a1, . . . , �ai, . . . , ak; ak+1)) ⊗ d log(ai+r − ai) += +� +1≤j 0. Hence, for u ∈ B, we have +� +exp(−α(t13 + t23 ++ + t23 +− ))(h1,2,3)−1(h1,23,4)−1 exp(−α(t23 ++ + t24 ++ + t34 ++ )), ψ(g(u)) +� += +∞ +� +n=0 +(−α)n +n! +⟨(h1,2,3)−1(h1,23,4)−1 exp(−α(t23 ++ + t24 ++ + t34 ++ )), ψ ◦ g ◦ (∂1)n(u)⟩ += +∞ +� +n=0 +(−α)n +n! +⟨h, ((∂1)n(u))|z→1⟩ +(Lemma 27) += ⟨h, u |z→1⟩ +(by (6.4)), +which completes the proof. +□ +Lemma 28. Let λ : Q ⟨e0, e−1, ez, e−z2⟩ → h2 be a ring homomorphism defined by +λ(e−1) = λ(ez) = 0, λ(e0) = e0 − e1, λ(e−z2) = −e1. +Then, for u ∈ Q⟨e0, e−1, e−z, e−z2⟩, we have +� +δ(h)2,3,4, ψ(g(u)) +� += ⟨h, λ(u)⟩ . + +THE CYCLOTOMIC GROTHENDIECK-TEICHMÜLLER GROUP AND THE MOTIVIC GALOIS GROUP +14 +Proof. First, note that +δ(h)2,3,4 = h(t23 ++ , t34 ++ , 0). +Since +� +t23 ++ , d log((−z−1) − a) +� += 0, +� +t34 ++ , d log((−z−1) − a) +� += −1 +for a ∈ {0, 1, −z, w}, we have +� +δ(h)2,3,4, f(ea1 · · · eak) +� += 0. +if aj = −z−1 for some j. Furthermore, since +� +t23 ++ , d log(1 − a) +� += +� +t34 ++ , d log(1 − a) +� += 0 +for a ∈ {0, −z, w}, we have +� +δ(h)2,3,4, g(ea1 · · · eak) +� += 0 +if aj ∈ {−z−1, 1} for some j. Since +� +t23 ++ , d log(0 − (−z)) +� += 0 +� +t23 ++ , d log(0 − w) +� += 1 +� +t23 ++ , d log((−z) − w) +� += 0 +� +t34 ++ , d log(0 − (−z)) +� +− 1 = 0 +� +t34 ++ , d log(0 − w) +� +− 1 = −1 +� +t34 ++ , d log((−z) − w) +� +− 1 = −1, +we have +� +δ(h)2,3,4, g(u) +� += ⟨h, λ(u)⟩ , +which completes the proof. +□ +Lemma 29. For u ∈ B′′, we have +� +exp(αt12)T (h1,2,34)δ(h)2,3,4, ψ(g(u)) +� += ⟨h, regz→0(u)⟩ . +Proof. Note that we have +T (h1,2,34) = h(t23 ++ + t24 ++ , t12, −t12 − t23 ++ − t24 ++ − t23 +− − t24 +− ) +and +∂(t23 ++ +t24 ++ )(g(ea1 · · · eak)) = (δak,0 + δak,−z2)g(ea1 · · · eak−1), +∂(t12)(g(ea1 · · · eak)) = δak,zg(ea1 · · · eak−1), +∂(−t12−t23 ++ −t24 ++ −t23 +− −t24 +− )(g(ea1 · · · eak)) = 0. +Thus, for ea1 · · · eak ∈ B, we have +� +exp(αt12)T (h1,2,34)δ(h)2,3,4, ψ(g(ea1 · · · eak)) +� += +� +T (h1,2,34)δ(h)2,3,4, ψ(g(ea1 · · · eak)) +� +(ak ̸= z) += +k +� +j=0 +� +δ(h)2,3,4, ψ ◦ g(ea1 · · · eaj) +� +· +� +h, regz→0(eaj+1 · · · eak) +� += +k +� +j=0 +� +h, λ(ea1 · · · eaj) +� +· +� +h, regz→0(eaj+1 · · · eak) +� +(by Lemma (28)). +If ai = z for some 1 ≤ i ≤ k, then +k +� +j=0 +� +h, λ(ea1 · · · eaj) +� +· +� +h, regz→0(eaj+1 · · · eak) +� += 0. +(6.5) + +THE CYCLOTOMIC GROTHENDIECK-TEICHMÜLLER GROUP AND THE MOTIVIC GALOIS GROUP +15 +If ea1 · · · eak ∈ B′′ then we have +k +� +j=0 +� +h, λ(ea1 · · · eaj) +� +· +� +h, regz→0(eaj+1 · · · eak) +� += +k +� +j=0 +δj,0 +� +h, regz→0(eaj+1 · · · eak) +� +· +(by a1 = z) += ⟨h, regz→0(ea1 · · · eak)⟩ += ⟨h, dist ◦ regz→0(ea1 · · · eak)⟩ +(Proposition 12) +(6.6) +If ea1 · · · eak ∈ B′′′ then +k +� +j=0 +� +h, λ(ea1 · · · eaj) +� +· +� +h, regz→0(eaj+1 · · · eak) +� +· += +k +� +j=0 +� +h, λ(ea1 · · · eaj) +� +· +� +h, ek−j +0 +� +(aj+1, . . . , ak ∈ {0, −z2}) += +k +� +j=0 +� +h, λ(ea1 · · · eaj) +� +· δk−j +(the coefficient of e0 in h is 0) += ⟨h, λ(ea1 · · · eak). += ⟨h, ̺(ǫ(ea1 · · · eak))⟩ += ⟨h, ℘(ǫ(ea1 · · · eak))⟩ +(Lemma (3.2)) +(6.7) +Thus, for u ∈ B′′ and v ∈ B′′, we have +� +exp(αt12)T (h1,2,34)δ(h)2,3,4, ψ(g(u +� v)) +� += +� +exp(αt12)T (h1,2,34)δ(h)2,3,4, ψ(g(u)) +� +· +� +exp(αt12)T (h1,2,34)δ(h)2,3,4, ψ(g(v)) +� += ⟨h, dist(regz→0(u))⟩ · ⟨h, ℘(ǫ(v))⟩ +(by (6.6) and (6.7)) += ⟨h, dist(regz→0(u)) +� ℘(ǫ(v))⟩ += ⟨h, regz→0(u +� v)⟩ . +(6.8) +Now, the lemma follows from (6.5) and (6.8). +□ +Lemma 30. For u ∈ B, we have +� +exp(−α(t13 + t23 ++ + t23 +− ))(h12,3,4)−1(h1,2,34)−1δ(h)2,3,4 exp(−α(t23 ++ + t24 ++ + t34 ++ )), ψ(g(u)) +� += ⟨h, ϕ(u)⟩. +Proof. We have +exp(−α(t13 + t23 ++ + t23 +− ))(h12,3,4)−1(h1,2,34)−1δ(h)2,3,4 exp(−α(t23 ++ + t24 ++ + t34 ++ )) += T (h12,3,4) exp(αt34 ++ )(h1,2,34)−1δ(h)2,3,4 exp(−α(t23 ++ + t24 ++ + t34 ++ )) +(Broadhurst duality) += T (h12,3,4)(h1,2,34)−1 exp(−α(t23 ++ + t24 ++ ))δ(h)2,3,4 +([t34 ++ , (h1,2,34)−1] = [δ(h)2,3,4, t23 ++ + t24 ++ + t34 ++ ]) += T (h12,3,4) exp(αt12)T (h1,2,34)δ(h)2,3,4 +(Broadhurst duality), +and thus +� +exp(−α(t13 + t23 ++ + t23 +− ))(h12,3,4)−1(h1,2,34)−1δ(h)2,3,4 exp(−α(t23 ++ + t24 ++ + t34 ++ )), ψ(g(u)) +� += +� +T (h12,3,4) exp(αt12)T (h1,2,34)δ(h)2,3,4, ψ(g(u)) +� +. +(6.9) +We have +T (h12,3,4) = h(t34 ++ , t13 + t23 ++ + t23 +− , −t13 − t23 ++ − t23 +− − t34 ++ − t34 +− ) + +THE CYCLOTOMIC GROTHENDIECK-TEICHMÜLLER GROUP AND THE MOTIVIC GALOIS GROUP +16 +and +∂(t34 ++ )g(u) = g(∂0(u)), +∂(t13+t23 ++ +t23 +− )g(u) = g(∂1(u)), +∂(−t13−t23 ++ −t23 +− −t34 ++ −t34 +− )g(u) = g(∂−1(u)). +Thus we have +� +T (h12,3,4) exp(αt12)T (h1,2,34)δ(h)2,3,4, ψ(g(u)) +� += +∞ +� +l=0 +� +c1,...,cl∈{0,1,−1} +⟨h, ec1 · · · ecl⟩ · +� +exp(αt12)T (h1,2,34)δ(h)2,3,4, g(∂c1 · · · ∂clu) +� += +∞ +� +l=0 +� +c1,...,cl∈{0,1,−1} +⟨h, ec1 · · · ecl⟩ · ⟨h, regz→0(∂c1 · · · ∂cl(u))⟩ +(Lemma 29) += +∞ +� +l=0 +� +c1,...,cl∈{0,1,−1} +⟨h, reg +�(ec1 · · · ecl)⟩ · ⟨h, regz→0(∂c1 · · · ∂clu)⟩ += ⟨h, regz→0(∂c1 · · · ∂clu) +� reg +�(ec1 · · · ecl)⟩ += ⟨h, ϕ(u)⟩ . +(6.10) +Hence the lemma follows from (6.9) and (6.10). +□ +Now, we can prove the main theorem of this paper. +Proof of Theorem 5. Recall that ICF is defined by +ICF := {u |z→1 −ϕ(u) | u ∈ B}. +For h ∈ PENT(2, k) and u ∈ B, by (6.2) and Lemmas 27 and 30, we have +⟨h, u |z→1 −ϕ(u)⟩ = 0. +Thus Theorem 5 is proved. Furthermore, by Theorem 14, we have +⟨h, u⟩ = 0 +(h ∈ PENT(2, k), u ∈ ker Lm), +and thus +⟨h, u⟩ = 0 +(h ∈ ϕ ∈ GRTM(¯1,1)(2, k), u ∈ ker La). +As explained in the introduction, this implies Theorem 3. +□ +Acknowledgements. The author would like to thank Hidekazu Furusho and Kenji Sakugawa for useful +comments and advices. This work was supported by JSPS KAKENHI Grant Numbers JP18K13392 and +JP22K03244. +References +[1] F. Brown, ‘Mixed Tate motives over Z’, Ann. Math., 175 (2012), 949-976. +[2] P. Deligne, ‘Le groupe fondamental unipotent motivique de Gm − µN, pour N = 2, 3, 4, 6 ou 8’, Publ. Math. Inst. Hautes +Etudes Sci. (2010), 101–141. +[3] P. Deligne and A. B. Goncharov, ‘Groupes fondamentaux motiviques de Tate mixte’, Annales scientifiques de l’École +Normale Supérieure 38.1 (2005): 1-56. +[4] V. G. Drinfeld, ‘On quasitriangular quasi-Hopf algebras and a group closely connected with Gal(¯Q/Q)’, Leningrad Math. +J. 2 (1991), no. 4, 829–860. +[5] B. Enriquez, ‘Quasi-reflection algebras and cyclotomic associators’, Selecta Math. (N.S.) 13 (2007), 391–463. +[6] B. Enriquez and H. Furusho, ‘Mixed pentagon, octagon and Broadhurst duality equations’, J. Pure Appl. Algebra 216 +(2012), 982-995. +[7] H. Furusho, ‘Pentagon and hexagon equations’, Ann. of Math. 171 (2010), 545-556. +[8] H. Furusho, ‘Double shuffle relation for associators’, Ann. of Math. 174 (2011), 341-360. +[9] A. Grothendieck, ‘Esquisse d’un programme,’ in “Geometric Galois actions, 1”, London Math. Soc. Lecture Note Ser., 242, +(1997) 5–48. +[10] M. Hirose and N. Sato, ‘Algebraic differential formulas for the shuffle, stuffle and duality relations of iterated integrals,’ J. +Algebra 556 (2020), 363-384. + +THE CYCLOTOMIC GROTHENDIECK-TEICHMÜLLER GROUP AND THE MOTIVIC GALOIS GROUP +17 +[11] M. Hirose and N. Sato, ‘The motivic Galois group of mixed Tate motives over Z[1/2] and its action on the fundamental +group of P1 \ {0, ±1, ∞}’, arXiv:2007.04288v2 [math.NT]. +[12] Y. Ihara, Some arithmetic aspects of Galois actions in the pro-p fundamental group of P1 − {0, 1, ∞}. In: Arithmetic +fundamental groups and noncommutative algebra (Berkeley, CA, 1999), Proc. Sympos. Pure Math., 70, Amer. Math. Soc., +Providence, RI, 2002, 247–273 +(Minoru Hirose) Institute for Advanced Research, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, 464- +8602, Japan +Email address: minoru.hirose@math.nagoya-u.ac.jp + diff --git a/S9E2T4oBgHgl3EQfswjv/content/tmp_files/load_file.txt b/S9E2T4oBgHgl3EQfswjv/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e3e928e769c84b8251a7d17fce5e91d25a55bc4b --- /dev/null +++ b/S9E2T4oBgHgl3EQfswjv/content/tmp_files/load_file.txt @@ -0,0 +1,712 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfswjv/content/2301.04064v1.pdf,len=711 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfswjv/content/2301.04064v1.pdf'} +page_content='04064v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfswjv/content/2301.04064v1.pdf'} +page_content='QA] 10 Jan 2023 THE CYCLOTOMIC GROTHENDIECK-TEICHMÜLLER GROUP AND THE MOTIVIC GALOIS GROUP MINORU HIROSE Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfswjv/content/2301.04064v1.pdf'} +page_content=' We show that the level 2 case of the cyclotomic Grothendieck-Teichmüller groups introduced by Enriquez coincides with the motivic Galois group of mixed Tate motives over Z[1/2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfswjv/content/2301.04064v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfswjv/content/2301.04064v1.pdf'} +page_content=' Introduction The purpose of this paper is to show the coincidence of the cyclotomic Grothendieck-Teichmüller group for level N = 2 and the motivic Galois group of mixed Tate motives over Z[1/2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfswjv/content/2301.04064v1.pdf'} +page_content=' This work concerns with the cyclotomic cases of the Grothendieck-Teichmüller theory, which are originated from the level one case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfswjv/content/2301.04064v1.pdf'} +page_content=' So let us start from the brief introduction for the original case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfswjv/content/2301.04064v1.pdf'} +page_content=' Let MT M(Z) be the Tannakian category of mixed Tate motives over Z, and G1 its motivic Galois group with respect to the canonical fiber functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfswjv/content/2301.04064v1.pdf'} +page_content=' Then G1 is decomposed as G1 = Gm ⋉ U1 where U1 is the prounipotent part of G1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfswjv/content/2301.04064v1.pdf'} +page_content=' For p, q ∈ {0, 1}, let πmot 1 (P1 \\ {0, 1, ∞}, p, q) be the the motivic fundamental torsor of path from p to q on P1 \\ {0, ∞, 1}, and qΠp its realization with respect to the canonical fiber functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfswjv/content/2301.04064v1.pdf'} +page_content=' We will write exp t0 3,1(k) for the set of group-like power series in k⟨⟨e0, e1⟩⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfswjv/content/2301.04064v1.pdf'} +page_content='1 Then qΠp(k) is canonically isomorphic to exp t0 3,1(k) for any p, q ∈ {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfswjv/content/2301.04064v1.pdf'} +page_content='2 For a group-like power series f ∈ exp t0 3,1(k), we write the corresponding element in qΠp(k) as qfp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfswjv/content/2301.04064v1.pdf'} +page_content=' Then G1 acts on qΠp for p, q ∈ {0, 1}, and these actions can be recovered only from the data of the map λ1 : U1 σ�→σ(110) −−−−−−→ 1Π0 ≃ exp t0 3,1 (see [3, Section 5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfswjv/content/2301.04064v1.pdf'} +page_content=' It was proved by Brown [1] that λ1 is injective, or equivalently the motivic Galois action G1 ↷ 1Π0 is faithful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfswjv/content/2301.04064v1.pdf'} +page_content=' On the other hand, Drinfeld [4] introduced an intermediate closed subscheme im(λ1) ⊂ GRT1 ⊂ exp t0 3,1 called (the prounipotent version of) the Grothendieck-Teichmüller group, which is closely related to Grothendieck’s approach in [9] to the description of the action of the absolute Galois group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfswjv/content/2301.04064v1.pdf'} +page_content=' The following is a fundamental conjecture in this area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfswjv/content/2301.04064v1.pdf'} +page_content=' Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfswjv/content/2301.04064v1.pdf'} +page_content=' im(λ1) = GRT1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfswjv/content/2301.04064v1.pdf'} +page_content=' Now let us consider the cyclotomic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfswjv/content/2301.04064v1.pdf'} +page_content=' Let N be a positive integer and µN the set of N-th roots of unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfswjv/content/2301.04064v1.pdf'} +page_content=' Let MT M(Z[µN, 1/N]) be the Tannakian category of mixed Tate motives over Z[µN, 1/N], and GN its motivic Galois group with respect to the canonical fiber functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfswjv/content/2301.04064v1.pdf'} +page_content=' Then GN is decomposed as GN = Gm ⋉ UN where UN is the prounipotent part of GN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfswjv/content/2301.04064v1.pdf'} +page_content=' For p, q ∈ {0} ∪ µN, let πmot 1 (P1 \\ {0, ∞} ∪ µN, p, q) be the the motivic fundamental torsor of path from p to q on P1 \\ {0, ∞} ∪ µN, and qΠ(N) p its realization with respect to the canonical fiber functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfswjv/content/2301.04064v1.pdf'} +page_content=' We will write exp t0 3,N(k) for the set of group-like power series in k⟨⟨ea | a ∈ {0} ∪ µN⟩⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfswjv/content/2301.04064v1.pdf'} +page_content=' Then qΠ(N) p (k) is canonically isomorphic to exp t0 3,N(k) for any p, q ∈ {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfswjv/content/2301.04064v1.pdf'} +page_content=' For f ∈ exp t0 3,N(k), we write the corresponding element in qΠ(N) p (k) as qfp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfswjv/content/2301.04064v1.pdf'} +page_content=' Then GN acts on qΠp for all p, q ∈ {0} ∪ µN, and these actions can be recovered only from the data of the map λN : UN σ�→σ(100) −−−−−−→ 1Π(N) 0 ≃ exp t0 3,N (see [3, Section 5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfswjv/content/2301.04064v1.pdf'} +page_content=' For general N > 1, λN is not necessary injective, but it is proved by Deligne [2] that λN is injective for N ∈ {2, 3, 4, 8}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfswjv/content/2301.04064v1.pdf'} +page_content=' Furthermore, Enriquez [5] generalize GRT1 to an intermediate closed 2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfswjv/content/2301.04064v1.pdf'} +page_content=' 11M32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfswjv/content/2301.04064v1.pdf'} +page_content=' 1Later, we will define a Lie algebra tn,N and its sub-Lie algebra t0 n,N for general n ≥ 2 and N ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E2T4oBgHgl3EQfswjv/content/2301.04064v1.pdf'} +page_content=' 2In this paper, we will use a convention such that the path γ ∈ π1(P1 \\ {0, 1, ∞}, p, q) corresponds to a power series whose coefficients of eak · · · ea1 is � 0 0; see Lemma 3.1. Combining these two results, +we finish the proof by a simple case distinction. This is done in Section 5. +We remark that our proof does no explicitly construct ε-separated sets with +arbitrarily large cardinality. We only establish their existence by an appli- +cation of Tur´an’s theorem. We believe that an explicit construction of such +sets would be worthwhile but probably very difficult. +1.2. Acknowledgements. The first named author is indebted to Urs Lang, +Alexander Lytchak, and Stephan Stadler for useful discussions about convex +hulls. + +6 +GIULIANO BASSO AND YANNICK KRIFKA +2. Preliminaries +2.1. Basic metric notions. We use N = {1, 2, . . . } to denote the set of +positive integers. A non-negative function ϱ: X × X → R is called semi- +metric if it is symmetric, satisfies the triangle inequality and ϱ(x, x) = 0 +for all x ∈ X. In other words, all axioms of a metric are satisfied except +(possibly) the positivity axiom, that is, there might exist distinct x, y ∈ X +such that ϱ(x, y) = 0. In the literature, such a function is sometimes also +called a pseudometric (see, for example, [8]). However, in the present article +we will only use the term semi-metric. Let X = (X, d) be a metric space. +We use X to denote the metric completion of X. If readability demands it +we will sometimes tacitly identify X with its canonical isometric copy in X. +A metric space is said to be totally bounded if for every ε > 0 there exists a +finite subset A ⊂ X such that for every x ∈ X there exists a ∈ A such that +d(x, a) < ε. We recall that X is totally bounded if and only if X is compact. +2.2. Graph theory. We use standard notation from graph theory as found +in [7, 12]. Let G = (V, E) be a graph, that is, V is a (possibly infinite) set +and E ⊂ {e ⊂ V : |e| = 2}. If {x, y} ∈ E then we often write x ∼ y. We +let G denote the complement graph of G. That is, G has vertex set V and +x ∼ y in G if and only if x ̸= y and x, y are not adjacent in G. We will also +need to consider graph powers of G. Let m ≥ 1 be an integer. We let Gm +denote the m-th power of G. By definition, Gm is a graph with vertex set V +and distinct vertices x, y ∈ V are adjacent if and only if there exists a path +in G of length at most m that connects x to y. We use the convention that +G0 denotes the empty graph (V, ∅). Given an integer r ≥ 1, we let Kr+1 +denote the complete graph on (r + 1)-vertices. The following theorem by +Tur´an is a foundational result in extremal graph theory. +Theorem 2.1 (Tur´an’s theorem). Let G = (V, E) be a finite graph and +r ≥ 1 an integer. If G does not contain Kr+1 as a subgraph, then +|E| ≤ +� +1 − 1 +r +� +· |V |2 +2 . +We will apply this theorem to graphs of the form Gm to obtain m- +separated sets in G with respect to the shortest-path metric dG; see Lemma 4.1 +and Corollary 4.2. Recall that the shortest-path metric dG : V × V → R is +defined by +dG(x, y) = min +� +k : (x0, . . . , xk) is a path in G form x to y +� +(2.1) +for all x, y ∈ V . +2.3. Bicombings. In the following we introduce bicombings and the various +properties one can impose on them. We decided to be a little more detailed +than would be strictly necessary for the main body of this article; see in +particular Theorem 2.2. All definitions appearing below are essentially due +to Descombes and Lang (see [11]). + +A NON-COMPACT CONVEX HULL IN NON-POSITIVE CURVATURE +7 +Let X be a metric space. +We say that σ: [0, 1] → X is a geodesic if +d(σ(s), σ(t)) = |s − t| · d(σ(0), σ(1)) for all s, t ∈ [0, 1]. A map +σ: X × X × [0, 1] → X +is called (geodesic) bicombing if for all x, y ∈ X, the path σxy(·): [0, 1] → X +defined by σxy(t) = σ(x, y, t) is a geodesic connecting x to y. We remark +that, in contrast, a map σ: X × [0, 1] → X is called combing with basepoint +p ∈ X if for all x ∈ X, the path σ(x, ·) is a geodesic connecting p to x. +However, we will not make use of this definition. Bicombings are also called +system of good geodesics; see [17, 19, 34]. Clearly, every geodesic metric +spaces admits a bicombing. We often consider bicombings in metric spaces +that have non-unique geodesics such as, for example, Rn equipped with the +p-norm for p ̸= 2. Therefore, it is useful to formalize some of the natural +properties of the bicombing on a uniquely geodesic metric space. We say +that σ is reversible if σxy(t) = σyx(1 − t) for all x, y ∈ X and all t ∈ [0, 1]. +In [5, Proposition 1.3] it is shown that any complete metric space with a +conical bicombings also admits a conical reversible bicombing (see also [10] +for an earlier result). Furthermore, we say that a bicombing σ is consistent +if it is reversible and σ(x, y, st) = σ(x, σxy(t), s) for all x, y ∈ X and all s, +t ∈ [0, 1]. Consistent bicombings are used in [18, 23], and a variant of the +definition that allows for a bounded error is studied in [14, Definition 2.6]. +We do not know if every space with a bicombing also admits a consistent +bicombing. This seemingly straightforward question does not seem to be so +easy to answer on closer inspection. For proper metric spaces admitting a +conical bicombing, it turns out to be true (see [3, Theorem 1.4]). +Descombes and Lang [11] introduced the following two non-positive cur- +vature conditions for a bicombing σ: +(1) if (1.1) holds, then σ is said to be conical. +(2) if for all x, y, x′, y′ ∈ X, the map t �→ d(σxy(t), σx′y′(t)) is convex +on [0, 1], then σ is called convex. +There are many examples of conical bicombings that are not convex (see +[11, Example 2.2] and [3, Example 3.6]). However, any consistent conical +bicombings is convex. One may wonder if any convex bicombing is automat- +ically consistent. This turns out to be not to be the case, as is demonstrated +in [5, Theorem 1.1]. To the authors’ knowledge, a relatively simple example +of a convex non-consistent bicombings seems to be missing. +The following theorem is a ’state of the art’ collection of general facts +about spaces that admit a conical bicombing. All of these properties are +usually associated with ’non-positive curvature’. +Theorem 2.2. Let X be a complete metric space admitting a conical bi- +combing. Then the following holds true. +(1) X is contractible, +(2) X admits barycenter map in the sense of Sturm [40], +(3) all Lipschitz homotopy groups πLip +k +(X) are trivial, + +8 +GIULIANO BASSO AND YANNICK KRIFKA +(4) X admits an isoperimetric inequality of Euclidean type for Ik(X). +Moreover, if X is proper then +(5) X is an absolute retract, +(6) X admits a visual boundary which is a Z-boundary in the sense of +Bestvina [6], +(7) any subgroup of the isometry group of X with bounded orbits has a +non-empty fixed-point set. +Proof. We prove each item separately. Fix o ∈ X. Clearly, H : X × [0, 1] → +X defined by H(x, t) = σ(x, o, t) is a homotopy between the identity map +on X and the constant map with value o. This shows (1). A proof of (2) +can be found in [3, Theorem 2.6]. We proceed by showing (3). A metric +space X is called Lipschitz k-connected with constant c if for every ℓ ∈ +{0, . . . , k}, every L-Lipschitz map f : Sℓ → X has a cL-Lipschitz extension +¯f : Bℓ+1 → X. Here, Sℓ, Bℓ+1 ⊂ Rℓ+1 denote the Euclidean unit sphere +and closed Euclidean unit ball, respectively. To prove that πLip +k +(X) is trivial +it suffices to show that X is Lipschitz k-connected for some constant c. +Therefore, the statement follows, since in [39, Proposition 6.2.2] it is proved +that X is Lipschitz k-connected with constant 3. +For a proof of (4) we +refer to Corollary 1.4 in [41]. Next, we prove (5). Using that X admits a +conical bicombing, it is not difficult to show that X is strictly equiconnected. +Therefore, it follows from a result by Himmelberg [22, Theorem 4] that +X is an absolute retract. The next statement, (6), follows directly from +Theorem 1.5 in [3]. +To finish the proof, we establish (7). Let Γ be a subgroup of the isometry +group of X with bounded orbits. Fix x0 ∈ X and consider the orbit A = +{f(x0) : f ∈ Γ}. In the following we combine results from [3] and [4] to +show that the fixed-point set of Γ is non-empty. In view of [4, Theorem 1.2] +it suffices to show that X admits a Γ-equivariant conical bicombing. We +now use the proof strategy of [3, Lemma 4.5] to show that such a bicombing +exists. Let CB(X) be the set of all conical bicombings on X and for every +x ∈ X let the metric Dx on CB(X) be given as in [3, Section 4]. We define +˜D = supx∈A Dx. Clearly, ˜D defines a metric on CB(X) and by considering +the proof of [3, Lemma 4.2] it is straightforward to show that (CB(X), ˜D) is +a compact metric space. Let f ∈ Γ and let F : CB(X) → CB(X) be defined +by F(σ)(x, y, t) = f−1(σ(f(x), f(y), t)). Since f(A) = A, it follows that F +is distance-preserving if CB(X) is equipped with ˜D. Now, one can argue +exactly as in the proof of [3, Lemma 4.5] to conclude that there exists some +σ∗ ∈ CB(X) such that F(σ∗) = σ∗ for all f ∈ Γ. In other words, σ∗ is a Γ- +equivariant concial bicombing, as desired. We remark that additional fixed- +point results for spaces with a conical bicombing can be found in [26, 27]. +□ +2.4. Conical midpoint maps. In this section we introduce conical mid- +point maps and derive some of their basic properties. We are mainly inter- +ested in this notion since it can be seen as a discrete analogue of conical + +A NON-COMPACT CONVEX HULL IN NON-POSITIVE CURVATURE +9 +bicombings. Indeed, any conical midpoint map on a metric space X induces +a conical bicombing on X. This is discussed at the end of this section. +Definition 2.3. We say that m: X × X → X is a conical midpoint map if +for all x, y, z ∈ X, the following holds: +(1) m(x, x) = x, +(2) m(x, y) = m(y, x), +(3) d(m(x, y), m(x, z)) ≤ 1 +2d(y, z). +We remark that for midpoints in Euclidean space, the inequality in (3) +becomes in fact an equality. It is easy to see that if m is as in Definition 2.3, +then z = m(x1, x2) is a midpoint of x1 and x2. Indeed, +d(z, xi) = d(z, m(xi, xi)) ≤ 1 +2d(x1, x2) +and thus using the triangle inequality, we find that d(z, xi) = 1 +2d(x1, x2). +Hence, a conical midpoint map is a midpoint map in the usual sense. +Furthermore, (3) can be upgraded to a more general inequality involving +four points. For all x, y, x′, y′ ∈ X, one has +d(m(x, y), m(x′, y′)) ≤ 1 +2d(x, x′) + 1 +2d(y, y′). +(2.2) +This can be seen as follows. Using (2) and the triangle inequality, we get +d(m(x, y), m(x′, y′)) ≤ d(m(x, y), m(x, y′)) + d(m(y′, x), m(y′, x′)) +and thus by virtue of (3) we obtain (2.2). Next, we show that conical mid- +point maps induce conical bicombings in a natural way. The used recursive +construction is well-known and goes back to Menger (see [31, Section 6]). +Let m be a concial midpoint map on X and x, y ∈ X. +Further, let +Gn = (2−n · Z) ∩ [0, 1], where n ≥ 0, be the 2−n-grid in [0, 1]. We define +σxy : � Gn → X recursively as follows. We put σxy(0) = x, σxy(1) = y and +if t ∈ Gn \ Gn−1, then we set +σxy(t) = m(σxy(r), σxy(s)), +where r, s ∈ Gn−1 are the unique points such that t = 1 +2r + 1 +2s and |r − s| = +2−(n−1). +Lemma 2.4. The map σxy extends uniquely to a geodesic σxy : [0, 1] → X. +Moreover, +d(σxy(t), σx′y′(t)) ≤ (1 − t)d(x, x′) + td(y, y′) +(2.3) +for all x, y, x′, y′ ∈ X. +Proof. To begin, we show that σxy|Gn is an isometric embedding for all n ≥ 0. +We proceed by induction. Clearly, σxy|G0 is an isometric embedding. Now, +fix ti ∈ Gn, i = 1, 2 and let ri, si ∈ Gn−1 with si ≤ ri be points such that +ti = 1 +2si + 1 +2ri and σxy(ti) = m(σxy(si), σxy(ri)). By construction of σxy + +10 +GIULIANO BASSO AND YANNICK KRIFKA +such points clearly exist. Without loss of generality, we may suppose that +t1 ≤ t2. Using the triangle inequality, we get +d(σxy(t1), σxy(t2)) ≤ d(σxy(t1), σxy(r1)) + d(σxy(r1), σxy(s2)) ++ d(σxy(s2), σxy(t2)), +and so, by the induction hypothesis and because m is a midpoint map, +d(σxy(t1), σxy(t2)) ≤ +�r1 − s1 +2 ++ |s2 − r1| + r2 − s2 +2 +� +d(x, y). +But, since t1 ≤ t2, it holds r1 ≤ s2. Hence, by the above, d(σxy(t1), σxy(t2)) ≤ +|t1 − t2|d(x, y). As a result, +d(x, y) ≤ d(x, σxy(t1)) + d(σxy(t1), σxy(t2)) + d(σxy(t2), y) +≤ +� +t1 + |t1 − t2| + |t2 − 1| +� +d(x, y). +This implies that d(σxy(t1), σxy(t2)) = |t1 − t2|d(x, y), and so σxy|Gn is an +isometric embedding. It follows by induction that σxy|Gn is an isometric +embedding for every n ≥ 0, as claimed. Now, since � Gn is a dense subset of +[0, 1], it follows that σxy can be uniquely extended to an isometric embedding +σxy : [0, 1] → X. Next, we show (2.3). Clearly, +d(σxy(1/2), σx′y′(1/2)) ≤ 1 +2d(x, x′) + 1 +2d(y, y′), +as σxy(1/2) = m(x, y), σx′y′(1/2) = m(x′, y′) and m is conical midpoint map +and thus satisfies (2.2). We now proceed by induction and show that if (2.3) +is valid for all t ∈ Gn−1, then it is also valid for all t ∈ Gn. Fix t ∈ Gn and +let s, r ∈ Gn−1 be the unique points with s ≤ r such that t = 1 +2s + 1 +2t. We +compute +d(σxy(t), σx′y′(t)) ≤ 1 +2d(σxy(s), σx′y′(s)) + 1 +2d(σxy(r), σx′y′(r)) +≤ +�1 − s +2 ++ 1 − r +2 +� +d(x, x′) + +�s +2 + r +2 +� +d(y, y′); +hence, (2.3) holds for all t ∈ Gn. Since � Gn is a dense subset of [0, 1] and +σxy and σx′y′ are geodesics, (2.3) is valid for all t ∈ [0, 1]. +□ +Thus, we have constructed a map σ: X × X × [0, 1] → X such that (1.1) +holds for all geodesics σxy and σx′y′. Now, given x, y ∈ X, we set +σxy(t) = lim +n→∞ σxnyn(t) +where xn, yn ∈ X are points such that xn → x and yn → y as n → ∞, +respectively. It follows that σ is a well-defined conical bicombing on X. We +call σ the conical bicombing induced by m. We point out that m is defined +on an arbitrary metric space X but σ is always a bicombing on X. +We conclude this section by giving a description of σ-convex hulls in terms +of m. Indeed, as with conical bicombings, conical midpoint maps give rise to + +A NON-COMPACT CONVEX HULL IN NON-POSITIVE CURVATURE +11 +’convex hulls’. For any A ⊂ X, we let m –conv(A) ⊂ X denote the closure +of the set +� +n∈N +Mn(A), +where M1(A) = +� +m(a, a′) : a, a′ ∈ A +� +and Mn(A) = M1(Mn−1(A)) for all +n ≥ 2. +Lemma 2.5. Let m be a conical midpoint map on a metric space X and +suppose σ denotes the conical bicombing on X induced by m. Then +σ–conv(A) = m –conv(A) +for all A ⊂ X. +Proof. Clearly, m –conv(A) ⊂ σ–conv(A). +Thus, it suffices to show that +the closed set m –conv(A) is σ-convex. To this end, let n ≥ 1 and let x, +y ∈ Mn(A). By construction of σ, it follows that σxy(Gm) ⊂ Mn+m(A) +for all m ∈ N. +Hence, σxy([0, 1]) ⊂ m –conv(A). +Now, suppose that x, +y ∈ m –conv(A). There exist points xk, yk ∈ Mnk(A) such that xk → x and +yk → y as k → ∞, respectively. Moreover, σxkyk → σxy uniformly. This +implies that σxy([0, 1]) ⊂ m –conv(A), and so m –conv(A) is σ-convex. +□ +3. Appending midpoints +Throughout this section we fix n0 ∈ N. This n0 will correspond to the +parameter from Theorem 1.3. We follow the proof strategy outlined in Sec- +tion 1.1 to construct the metric space X0. To begin, we construct recursively +a sequence of graphs Gn = (Vn, En). The whole construction is quite formal. +The basic idea is that Vn is obtained from Vn−1 by appending ’midpoints’ +and two midpoints in Vn are adjacent if and only if they are part of a cone +whose base is an edge of Gn−1. +We let G0 denote the null graph and G1 the complete graph on n0 vertices +with vertex set V1 = {1, . . . , n0}. For n ≥ 2 we set +Vn = Vn−1 ∪ +� +{x, y} : x, y ∈ Vn−1, x ̸= y +� +. +To formalize the notion of ’midpoint’ we use the following notation +m(a, b) = +� +{a, b} +if a ̸= b, +a +otherwise. +(3.1) +Notice that Vn = m(Vn−1 × Vn−1). +Moreover, we remark that we have +constructed an infinite nested sequence +V0 ⊂ V1 ⊂ V2 ⊂ · · · +Now, the edge set En is uniquely determined by {x, y} ∈ En if and only +if there exist v ∈ Vn−1 and {u, w} ∈ En−1 such that x = m(v, u) and +y = m(v, w). Loosely speaking, x and y are adjacent in Gn if and only if +x, y are the midpoints parallel to the base of a cone with vertex v ∈ Vn−1 + +12 +GIULIANO BASSO AND YANNICK KRIFKA +and base u ∼ w in Gn−1. See Figure 1.1 for an illustration. For example, if +n0 = 2, one has +V2 = +� +0, 1, {0, 1} +� +and +E2 = +� +{0, {0, 1}}, {{0, 1}, 1} +� +. +The graphs Gn for n0 = 2 and n = 1, 2, 3, 4 are depicted in Figure 3.1. +Figure 3.1. The graphs Gn for small n with n0 = 2. +To begin, we collect some basic facts about the cardinalities of Vn and En +that will be used later on. +Lemma 3.1. One has |V0| = 0, |V1| = n0, and for all n ≥ 2, +|Vn| = 1 +2 · +� +|Vn−1| + |Vn−2| +� +· +� +|Vn−1| − |Vn−2| + 1 +� +(3.2) +Moreover, for every ε > 0, +lim +n→∞ +|En| +|Vn|1+ε = 0. +(3.3) +Proof. By construction, Vn−2 ⊂ Vn−1. Thus, letting Wn−1 = Vn−1 \ Vn−2 +and using that m is symmetric, we find +Vn = m(Vn−1×Vn−1) = m(Vn−2×Vn−2)∪m(Vn−2×Wn−1)∪m(Wn−1×Wn−1). +Therefore, as these sets are pairwise disjoint, +|Vn| = |Vn−1| + |Vn−2| · |Wn−1| + |Wn−1| · +� +|Wn−1| − 1 +� +2 +. +Since |Wn−1| = |Vn−1| − |Vn−2|, this yields (3.2). To finish the proof, we +establish (3.3). Clearly, this is valid if n0 = 1. Thus, in the following, we + +A NON-COMPACT CONVEX HULL IN NON-POSITIVE CURVATURE +13 +may suppose that n0 ≥ 2. Notice that |E0| = 0, |E1| = 1 +2(n0 − 1)n0 and +|En| ≤ |Vn−1| · |En−1| for all n ≥ 2. Consequently, +|En| ≤ C · +n−1 +� +i=1 +|Vi|, +(3.4) +where C = 1 +2(n0 − 1)n0. We claim that +|Vn−1|2 +|Vn| +≤ 3 +(3.5) +for all n ∈ N. For n = 1, 2 this can be seen by a direct verification. Let now +n ≥ 3. Letting αn = |Vn−2|/|Vn−1| and using (3.2), we find that +|Vn−1|2 +|Vn| +≤ +2 +1 − α2n +. +In particular, if αn ≤ 1/ +√ +3, then (3.5) follows. Now, suppose that n0 ≥ 3. +It follows that α3 ≤ 1/ +√ +3 and hence (3.5) is valid for n = 3. Clearly, if +(3.5) holds for n − 1, then αn ≤ 3/|Vn−2|. Thus, as |V2| ≥ 6 ≥ 3 +√ +3, the +desired inequality (3.5) follows by induction. This establishes (3.5) when +n0 ≥ 3. We now treat the special case when n0 = 2. We have |V2| = 3, +|V3| = 5, |V4| = 12, and |V5| = 68. Hence, (3.5) holds true if n = 3, 4, 5. +The general case now follows as before by noting that |V4| = 12 ≥ 3 +√ +3, and +so (3.5) can be established by induction. This completes the proof of (3.5). +By combining (3.4) with (3.5), we arrive at +|En| +|Vn|1+ε ≤ C · +3n +|Vn|ε . +We claim that |Vn| ≥ |Vn−2|2 for all n ≥ 6. Letting β = |Vn−1|/|Vn−2|, we +obtain +|Vn| +|Vn−2|2 ≥ 1 +2(β + 1)(β − 1). +(3.6) +Since |V4| = 12 if n0 = 2, it follows that |V4| ≥ 12 for every n0 ≥ 2. +Therefore, |Vn−2| +3 +≥ +√ +3 for all n ≥ 6, and thus by virtue of |Vn−1| ≥ 1 +3|Vn−2|, +we obtain β2 ≥ 3. This is equivalent to 1 +2(β + 1)(β − 1) ≥ 1. By the use of +(3.6), we can conclude that |Vn| ≥ |Vn−2|2 for all n ≥ 6, as desired. Now, by +repeated use of this inequality and using that |V3| ≥ |V2|, we get +|Vn| ≥ |Vn−2|2 ≥ · · · ≥ +� +|V2| +�2 +n−3 +2 +for all n ≥ 6. Thus, letting c = +ε +2 +√ +2 and using that |V2| ≥ 3, we obtain +lim +n→∞ +|En| +|Vn|1+ε ≤ C · lim +n→∞ +3n +|Vn|ε ≤ C · lim +n→∞ |V2|n−c( +√ +2)n = 0. +□ + +14 +GIULIANO BASSO AND YANNICK KRIFKA +Let dn : Vn × Vn → R denote the shortest-path metric on Gn. The defi- +nition of the shortest-path metric of a graph is recalled in (2.1). Our next +result shows that any two distinct points in V1 ⊂ Vn realize the diameter of +Vn with respect to dn. +Lemma 3.2. For all distinct x, y ∈ V1, +dn(x, y) = diam Vn = 2n−1. +Proof. To begin, we show that +dn(x, y) ≤ 2dn−1(x, y) +(3.7) +for all n ≥ 2 and all x, y ∈ Vn−1. +Let (x0, x1, . . . , xk) be a shortest- +path in Gn−1 connecting x to y. +We set x′ +i = m(xi−1, xi) for all i = +1, . . . , k. +Clearly, xi−1 ∼ x′ +i and x′ +i ∼ xi in Gn for all i = 1, . . . , k. +Hence, (x0, x′ +1, x1, x′ +2, . . . , x′ +k, xk) is a path in Gn connecting x to y, and +so dn(x, y) ≤ 2k = 2dn−1(x, y), as desired. +By construction, diam V1 = 1. Hence, it follows from (3.7) that diam Vn ≤ +2n−1. To finish the proof we thus need to show that dn(x, y) ≥ 2n−1 for all +distinct x, y ∈ V1. +For this we will use the following construction. +We +define the functions δn : Vn → ∆n0−1 ∩ 2−(n−1) · Zn0 recursively as follows. +We may suppose that V1 = {1, . . . , n0} and we set δ1(i) = ei for each +i = 1, . . . , n0. Here, ei ∈ Rn0 is the vector with a one at the ith position and +zeros everywhere else. Suppose now n ≥ 2 and x ∈ Vn. We set +δn(x) = 1 +2 +� +δn−1(a) + δn−1(b) +� +if x = m(a, b) with a ̸= b, and δn(x) = δn−1(x) otherwise. It follows by +induction that if {x, y} ∈ En, then +|δn(x) − δn(y)|∞ = +1 +2n−1 , +(3.8) +where |·|∞ denotes the supremum norm on Rn0. Clearly, δn(i) = ei for all n ∈ +N and all i = 1, . . . , n0. Now, let x, y ∈ V1 be distinct and (x0, x1, . . . , xk) a +path in Gn connecting x to y. By the above, it follows that +1 = |δn(x) − δn(y)|∞ ≤ +k−1 +� +i=0 +|δn(xi) − δn(xi+1)|∞ = +k +2n−1 . +Hence, dn(x, y) ≥ 2n−1, as was to be shown. +□ +Our next lemma relates shortest-paths in Gn to shortest-paths in Gn−1. +The proof follows easily from the definition of dn and the recursive construc- +tion of En. +Lemma 3.3. For all x1, x2, y1, y2 ∈ Vn−1, +dn(m(x1, x2), m(y1, y2)) ≤ dn−1(x1, y1) + dn−1(x2, y2). +(3.9) + +A NON-COMPACT CONVEX HULL IN NON-POSITIVE CURVATURE +15 +Proof. Let (p0, . . . , pk) and (q0, . . . , qℓ) be shortest-paths in Gn−1 connecting +x1 to y1, and x2 to y2, respectively. We construct a path (r0, . . . , rk+ℓ) in +Gn as follows. We set ri = m(p0, qi) for all i = 0, . . . , ℓ and rℓ+j = m(pj, qℓ) +for all j = 1, . . . , k. By construction, ri−1 ∼ ri in Gn for all i = 1, . . . , k + ℓ. +Hence, (r0, . . . , rk+ℓ) is a path in Gn connecting r0 = m(x1, x2) to rk+ℓ = +m(y1, y2), and so it follows that +dn(m(x1, x2), m(y1, y2)) ≤ k + ℓ. +But dn−1(x1, y1) = k and dn−1(x2, y2) = ℓ. This finishes the proof of (3.9). +□ +We remark that (3.9) should be thought of as a discrete analogue of the +conical inequality (1.1). +Indeed, by considering the scaled metrics ϱn = +(diam Vn)−1 · dn and using that diam Vn = 2n−1 by Lemma 3.2, we obtain +that +ϱn(m(x1, x2), m(y1, y2)) ≤ 1 +2ϱn−1(x1, y1) + 1 +2ϱn−1(x2, y2) +(3.10) +for all x1, x2, y1, y2 ∈ Vn−1. In particular, if x, y ∈ Vn−1, then ϱn(x, y) ≤ +ϱn−1(x, y). In view of these inequalities, letting +V = +� +n≥1 +Vn +we find that the map ϱ: V × V → R defined by +ϱ(x, y) = lim +n→∞ ϱn(x, y) +is a semi-metric on V (see Section 2.1 for the definition). More formally, V +could also be constructed as the direct limit of the sequence of metric spaces +(Vn, ϱn) with morphisms Vn → Vm, for n ≤ m, induced by the identity. +By the above, the semi-metric space (V, ϱ) is naturally equipped with a +’conical midpoint map’. Indeed, because of (3.10), m: V × V → V defined +by (x, y) �→ m(x, y) satisfies +ϱ(m(x, y), m(x, z)) ≤ 1 +2ϱ(y, z) +(3.11) +for all x, y, z ∈ V . +It is now not difficult to upgrade m to a concial +midpoint map on a metric space X. Indeed, let us denote by (X, d) the +metric space induced by (V, ϱ). +By definition, X = V/ ∼ with x ∼ y if +and only if ϱ(x, y) = 0 and d is the quotient metric on X. We recall that +d([x], [y]) = ϱ(x, y) for all x, y ∈ V . +Lemma 3.4. The map m: X × X → X defined by m([x], [y]) = [m(x, y)] +for all [x], [y] ∈ X is a concial midpoint map on X. Moreover, +X = +� +n∈N +Mn(A), +where A = [V1] ⊂ X. + +16 +GIULIANO BASSO AND YANNICK KRIFKA +Proof. By applying (3.11), we get +ϱ(m(x, y), m(x′, y′)) ≤ ϱ(m(x, y), m(x, y′)) + ϱ(m(x, y′), m(x′, y′)) +≤ 1 +2ϱ(y, y′) + 1 +2ϱ(x, x′). +Hence, if ϱ(x, x′) = 0 and ϱ(y, y′) = 0, then ϱ(m(x, y), m(x′, y′)) = 0. This +shows that m: X × X → X defined by m([x], [y]) = [m(x, y)] for all [x], +[y] ∈ X is well-defined. Moreover, it follows directly from the inequality +above that m is a concial midpoint map on X. By construction, m(Vn−1 × +Vn−1) = Vn for all n ≥ 2, and so V ⊂ m –conv(V1). This implies the desired +equality. +□ +In summary, we have shown that the map m: V × V → V descends to +conical midpoint map on X, where X denotes the metric space associated +to V . For simplicity this map is also denoted by m. Due to the results in +Section 2.4, m now induces a conical bicombing on X. We set X0 = X. +In Section 5 we show that X0 is non-compact. We achieve this by showing +that X is not totally bounded. In order to work effectively with X, it seems +natural to determine how much the semi-metric ϱ (and hence d) differs from +the metric ϱn on Vn. The following lemma shows that ϱ does not collapse +the distances too much. +Lemma 3.5. For all n ≥ 2, +ϱn(x, y) − 8 +2n ≤ ϱ(x, y) ≤ ϱn(x, y). +(3.12) +for all x, y ∈ Vn. +Notice that due to Lemma 3.5, if x, y ∈ Vn satisfy dn(x, y) ≥ 5, then +d(x, y) > 0. In particular, ε-separated sets in (Vn, ϱn) induce ε′-separated +sets in X. See Lemma 4.1 for the exact statement. +Proof of Lemma 3.5. The desired upper bound of ϱ(x, y) follows directly +from (3.7). In what follows we show the lower bound. To begin, we claim +that +2dn−1(x, y) ≤ dn(x, y) + 4 +(3.13) +for all x, y ∈ Vn. Fix distinct points x, y ∈ Vn−1 and let {x, x′} and {y, y′} +be edges in En−1. Since Gn−1 is connected such edges surely exists. Because +of (3.8), it follows that p := m(x, x′), q := m(y, y′) ∈ Vn \ Vn−1. Moreover, +since x ∼ p and y ∼ q in Gn, by the triangle inequality, +|dn(x, y) − dn(p, q)| ≤ 2. +We claim that +dn(p, q) = min +� +dn−1(x, y) + dn−1(x′, y′), dn−1(x, y′) + dn−1(x′, y) +� +. (3.14) +Indeed, let (x0, . . . , xℓ) be a shortest-path in Gn connecting p to q. For each +i = 1, . . . , ℓ there is vi ∈ Vn−1 and {ui, wi} ∈ En−1 such that +xi−1 = m(vi, ui) +and +xi = m(vi, wi). + +A NON-COMPACT CONVEX HULL IN NON-POSITIVE CURVATURE +17 +x +x′ +p +x1 +x2 +x3 +x4 +x5 +y +y′ +q +γ +η +Figure 3.2. The construction from Lemma 3.5. +We define a0, . . . , aℓ ∈ Vn−1 and b0, . . . , bℓ+1 ∈ Vn−1 by induction as follows. +We put a0 = v1 and b0 = u1, b1 = w1. Now, for every i = 1, . . . , ℓ − 1, we +set +� +ai = wi+1 and bi+1 = bi +if ui+1 = ai−1, +ai = ai−1 and bi+1 = wi+1 +if ui+1 = bi. +By construction, m(a0, b0) = x0 = p and m(aℓ−1, bℓ) = xℓ = q. Moreover, af- +ter deleting repeated entries, γ = (a0, . . . , aℓ−1) and η = (b0, . . . , bℓ) are (pos- +sibly degenerate) shortest-paths in Gn−1 such that length(γ) + length(η) = +ℓ = dn(p, q). See Figure 3.2. Hence, +dn−1(a0, aℓ−1) + dn−1(b0, bℓ) ≤ dn(p, q). +Because of p, q /∈ Vn−1, without loss of generality we have a0 = x, aℓ−1 = y, +b0 = x′ and bℓ = y′, and so the desired equality (3.14) now follows due to +Lemma 3.3. +Having (3.14) at hand, (3.13) now follows easily. +Indeed, using that +dn−1(x, x′) = dn−1(y, y′) = 1, we have +dn−1(x, y) − 1 ≤ dn−1(x, y′) +dn−1(x, y) − 1 ≤ dn−1(x′, y) +and +dn−1(x, y) − 2 ≤ dn−1(x′, y′), +and so using (3.14), we deduce that +dn(x, y) ≥ dn(p, q) − 2 ≥ 2dn−1(x, y) − 4. +This shows (3.13). Now, by dividing (3.13) by 2n−1, we obtain +ϱn−1(x, y) ≤ ϱn(x, y) + 8 +2n . + +18 +GIULIANO BASSO AND YANNICK KRIFKA +In particular, for every k ∈ N, +ϱn(x, y) ≤ ϱn+k(x, y) + 8 +2n +k +� +i=1 +1 +2i +and the left inequality of (3.12) follows by taking the limit k → ∞. +□ +We remark that in (3.13) at least an additive error of 2 must occur. This +is discussed further in the following example. +x +xʹ +v +b +a +Figure 3.3. Illustration of the construction in +Example 3.6. +Example 3.6. Let n0 = 2 and consider the graph G4 depicted in Figure 3.3. +In particular, v = m(0, 1), a = m(0, v), b = m(v, 1) and +x = m(a, 1) +and +x′ = m(v, b). +Clearly, d4(x, x′) = 2. We claim that d5(x, x′) = 2 as well. Since x ∼ b in +G4, the points x0 := m(v, b) and x1 := m(v, x) are adjacent in G5. Thus, as +m(x, v) ∼ m(x, x) in G5, it follows that (x0, x1, x2) is a path in G5 connecting +x′ to x. Hence, d5(x, x′) ≤ 2. On the other hand, it is not difficult to see that +δ5(x) = δ5(x′) and thus due to (3.8), it follows that x and x′ are not adjacent +in G5. This shows that d5(x, x′) = 2. Hence, 2d4(x, x′) − d5(x, x′) = 2, and +so the additive error in (3.13) must be at least 2. +4. Gm +n has few edges +In this section, we find a sufficient condition that X is not totally bounded +in terms of the number of edges of Gm +n . The basic graph theory notation +that is needed in the sequel can be found in Section 2.2. +Lemma 4.1. Let n, r ≥ 1 and m ≥ 6 be integers. If Gm +n has an (r + 1)- +clique, then there exist r + 1 points x1, . . . , xr+1 ∈ X such that +d(xi, xj) ≥ m +2n +for all distinct i, j = 1, . . . , r + 1. +Proof. If v1, . . . , vr+1 ∈ Vn are the vertices of an (r + 1)-clique in Gm +n , then +by definition one has +dn(vi, vj) ≥ m + 1 + +A NON-COMPACT CONVEX HULL IN NON-POSITIVE CURVATURE +19 +for all distinct i, j = 1, . . . , r + 1. Hence, by dividing by 2n−1 on both sides +and using Lemma 3.5, we obtain +d(vi, vj) ≥ 2m − 6 +2n +≥ m +2n , +as desired. +□ +Fix an integer k ≥ 1 sufficiently large to be determined later. We ab- +breviate m(n) = 2n−k. +Using Tur´an’s theorem we obtain the following +non-compactness criterion for X. +Corollary 4.2. Let n0 ≥ 2 and let X be constructed as in Section 3. If +lim inf +n→∞ +|E +� +Gm(n) +n +� +| +|Vn|2 += 0, +(4.1) +then X is not totally bounded. +Proof. We prove the contrapositive. Suppose that X is totally bounded. +There exists r ≥ 1 such that X does not contain a +1 +2k -net of cardinality +r + 1. Hence, by Lemma 4.1, for n ≥ 1 sufficiently large, the complement +graph of the m(n)-th power Gn does not contain an (r + 1)-clique. Thus, +Tur´an’s theorem, see Theorem 2.1, tells us that +|E +� +Gm(n) +n +� +| ≤ +� +1 − 1 +r +� +· |Vn|2 +2 +for all n sufficiently large. Therefore, +|Vn| · (|Vn| − 1) +2 +− |E +� +Gm(n) +n +� +| ≤ +� +1 − 1 +r +� +· |Vn|2 +2 +and it follows that +lim inf +n→∞ +|E +� +Gm(n) +n +� +| +|Vn|2 +≥ 1 +2r > 0, +as desired. We remark that to show the lower bound on the liminf we have +used that |Vn| is an unbounded sequence, which is only valid if n0 ≥ 2. +□ +Thus, to prove that X not totally bounded, it suffices to establish (4.1). +To this end, in the next subsection we derive some upper bounds for |E +� +Gm(n) +n +� +|. +4.1. Upper bounds. The following estimate is not sharp in general, but +is sufficient for our purposes. It is the crucial building block for inequality +(4.3), which is our key tool in the proof of Theorem 1.2. +Lemma 4.3. Let n, m ∈ N. Then there exist non-negative integers a, b +such that a + b = m and +|E(Gm +n )| ≤ 2m |E(Ga +n−1)| · |E(Gb +n−1)|. +We recall that we use the convention that |E(G0)| = |V | for any finite +graph G = (V, E). + +20 +GIULIANO BASSO AND YANNICK KRIFKA +Proof of Lemma 4.3. Suppose that x is adjacent to y in Gm +n . By definition, +there exist a shortest-path (x0, . . . , xℓ) in Gn of length ≤ m connecting x to +y. For each i = 1, . . . , ℓ there is vi ∈ Vn−1 and {ui, wi} ∈ En−1 such that +xi−1 = m(vi, ui) +and +xi = m(vi, wi). +As in the proof of Lemma 3.5, we define a0, . . . , aℓ ∈ Vn−1 and b0, . . . , bℓ+1 ∈ +Vn−1 by induction as follows. We put a0 = v1 and b0 = u1, b1 = w1. Now, +for every i = 1, . . . , ℓ − 1, we set +� +ai = wi+1 and bi+1 = bi +if ui+1 = ai−1, +ai = ai−1 and bi+1 = wi+1 +if ui+1 = bi. +By construction, m(a0, b0) = x0 = x and m(aℓ−1, bℓ) = xℓ = y. +More- +over, after deleting repeated entries, γ = (a0, . . . , aℓ−1) and η = (b0, . . . , bℓ) +are (possibly degenerate) shortest-paths in Gn−1 such that length(γ) + +length(η) = ℓ = dn(x, y). See Figure 3.2. Moreover, any two non-degenerate +shortest-paths γ and η induce at most two edges in Gm +n in this way. Conse- +quently, +|E(Gm +n )| ≤ |Vn−1| · |E(Gm +n−1)| + 2 +m−1 +� +i=1 +|E(Gi +n−1)| · |E(Gm−i +n−1)|. +We put +M = max +� +|E(Gi +n−1)| · |E(Gm−i +n−1)| : i = 0, . . . , m +� +. +By the above, it follows that |E(Gm +n )| ≤ M + 2(m − 1)C ≤ 2mM. +□ +Recall that we have fixed an integer k ≥ 1 which is sufficiently large to +be determined later, and we use the notation +m(n) = 2n−k +and +¯n = n − k. +Using Lemma 4.3, it is possible to obtain an upper bound on the number of +edges of Gm(n) +n +in terms of a product with factors |E(Gmi +k )| and |Vn−i|ki. +Lemma 4.4. Let n ≥ 1 be sufficiently large. Then there exist an integer +K ∈ {1, . . . , m(n)}, positive integers m1, . . . , mK such that m1 +· · ·+mK = +m(n), and integers ki ∈ {0, . . . , 2i − 1} for i = 1, . . . , ¯n satisfying +¯n +� +i=1 +ki · 2¯n−i = m(n) − K, +(4.2) +such that +|E(Gm(n) +n +)| ≤ 32m(n)� K +� +i=1 +|E(Gmi +k )| +�� +¯n +� +i=1 +|Vn−i|ki +� +. +(4.3) +Proof. We consider the following replacement rule: +|E(Gm +n )| → +� +2m |E(Ga +n−1)| · |E(Gb +n−1)| +if m > 0, where a, b are as in Lemma 4.3 +|Vn| +if m = 0. + +A NON-COMPACT CONVEX HULL IN NON-POSITIVE CURVATURE +21 +By using this rule and Lemma 4.3 sufficiently many times, we obtain integers +ℓi ∈ {1, . . . , 2i}, for i = 0, . . . , ¯n − 1, such that +|E(Gm(n) +n +)| ≤ +� +¯n +� +i=1 +2ℓi−1 · Ai · |Vn−i|ki +� +· +� K +� +i=1 +|E(Gmi +k )| +� +, +(4.4) +where +Ai := +ℓi−1 +� +j=1 +αi,j +for some positive integers αi,j > 0 satisfying αi,1 +· · ·+αi,ℓi = m(n). Notice +that in particular ℓ0 = 1. Using the inequality of arithmetic and geometric +means, we get +ℓi +� +j=1 +αi,j ≤ +�m(n) +ℓi +�ℓi = 2(¯n−log2 ℓi)·2log2 ℓi. +The function f(x) := (¯n − x) · 2x is increasing on [0, ¯n − 2], and +max +x∈[0,¯n] f(x) = +2¯n +e log 2 ≤ 2¯n. +Hence, using that ℓi ∈ {1, . . . , 2i}, we have A¯n ≤ 2¯n and for all i = 1, . . . , ¯n− +2, +Ai ≤ 2(¯n−(i−1))·2i−1. +Thus, since +¯n−2 +� +j=0 +(¯n − j)2j ≤ 2¯n−1 + +¯n−1 +� +j=1 +2j ≤ 2¯n−1 + 2¯n, +we find that +¯n +� +i=1 +Ai ≤ 22¯n+2¯n−1+2¯n ≤ 162¯n. +Moreover, +¯n +� +i=1 +2ℓi−1 ≤ +¯n−1 +� +i=0 +22i ≤ 22¯n, +and thus (4.3) follows from (4.4). +□ +We remark that if Lemma 4.3 were true for a = b = m +2 , by exactly the +same reasoning as in the proof of Lemma 4.4, we would get the following +slightly more elegant upper bound in (4.3), +8m(n) · |E(Gk)|m(n), +but we do not know how to prove this. + +22 +GIULIANO BASSO AND YANNICK KRIFKA +5. Proof of main results +In this section we prove the main results from the introduction. Theo- +rem 1.2 is an immediate consequence of the following result. +Theorem 5.1. Let n0 ∈ N and let X0 be the complete metric space con- +structed in Section 3. +Then X0 admits a conical bicombing σ and there +is a finite subset A ⊂ X0 such that σ–conv(A) = X0. +Moreover, X0 is +non-compact for every n0 ≥ 2. +Proof. In the following, we retain the notation of Section 3. Recall that +X0 = X, where (X, d) is the metric space associated to the semi-metric +space (V, ϱ). We set A = V1 ⊂ X0. Lemma 3.4 tells us that m: X × X → X +defines a conical midpoint map on X and +X = +� +n∈N +Mn(A). +Let σ be the conical bicombing on X0 induced by m. For the construction of +σ we refer to Section 2.4. Because of Lemma 2.5, it follows that σ–conv(A) = +X0. +Let now n0 ≥ 2. To finish the proof we show that X0 is not compact. +This is achieved by showing that X is not totally bounded, which in turn is +established via Corollary 4.2. Fix ε ∈ (0, 2−4) and choose k ≥ 1 sufficiently +large such that +max +� 1 +|Vk|, |E(Gk)| +|Vk|(1+ε) +� +≤ +1 +(2α) +1 +ε +, +(5.1) +for some large constant α > 0 to be determined later. The existence of k +is guaranteed by Lemma 3.1. As in Section 4, we set m(n) = 2n−k and +¯n = n − k. We claim that +|E(Gm(n) +n +)| +|Vn|2 +≤ +�1 +2 +�m(n) +(5.2) +for all n ≥ 1 sufficently large. By Lemma 4.4, there exists an integer K ∈ +{1, . . . , m(n)}, positive integers m1, . . . , mK such that m1+· · ·+mK = m(n), +and ki ∈ {0, . . . , 2i − 1} for i = 1, . . . , ¯n, such that (4.2) holds and +|E(Gm(n) +n +)| ≤ 32m(n)� K +� +i=1 +|E(Gmi +k )| +�� +¯n +� +i=1 +|Vn−i|ki +� +. +(5.3) +In the following, we derive an upper bound for 1/|Vn|2. Due to (3.5), we +have +|Vn−1|2 +|Vn| +≤ 3, +(5.4) +and so we find that +1 +|Vn|2 ≤ +32 +|Vn−1|4 = +3b0 +|Vn−1|k1 · +1 +|Vn−1|b1 , + +A NON-COMPACT CONVEX HULL IN NON-POSITIVE CURVATURE +23 +where b0 = 2 and b1 = 2b0 −k1. We define the integers b0, . . . , b¯n recursively +as follows. We set b0 = 2, and bi = 2bi−1 − ki for all i = 1, . . . , ¯n. Hence, by +using (5.4) repeatedly, we arrive at +1 +|Vn|2 ≤ +� +¯n +� +i=1 +3bi−1 +|Vn−i|ki +� +· +1 +|Vk|b¯n +(5.5) +Via a straightforward computation, we find +¯n−1 +� +i=0 +bi ≤ +¯n−1 +� +i=0 +2i+1 ≤ 2¯n+1, +b¯n = 2 · m(n) − +¯n−1 +� +i=0 +k¯n−i · 2i. +Hence, because of (4.2), it follows that b¯n = m(n) + K. By combining (5.3) +with (5.5), we obtain +|E(Gm(n) +n +)| +|Vn|2 +≤ αm(n) · +�K +i=1|E(Gmi +k )| +|Vk|2K +· +1 +|Vk|m(n)−K , +(5.6) +where α = 32 · 9. In the following, we consider the cases K ≤ (1 − ε)m(n) +and K > (1 − ε)m(n) separately. First, we suppose that K ≤ (1 − ε)m(n). +From (5.6), we find that +|E(Gm(n) +n +)| +|Vn|2 +≤ αm(n) · +1 +|Vk|m(n)−K . +Since ε · m(n) ≤ m(n) − K, it follows from our assumption (5.1) on k that +|E(Gm(n) +n +)| +|Vn|2 +≤ αm(n) · +� +1 +(2α) +1 +ε +�ε·m(n) +≤ +�1 +2 +�m(n) +. +Second, suppose that K > (1 − ε)m(n). Since mi ≥ 1 and m1 + . . . + mK = +m(n), it follows that mj ≥ 2 for at most 2ε·m(n) many indices j. To ease the +notation, we may suppose m1 = · · · = mL = 1, where L = ⌈K − 2εm(n)⌉. +Hence, using (5.6) once again, we find that +|E(Gm(n) +n +)| +|Vn|2 +≤ αm(n) · +� |E(Gk)| +|Vk|(1+ε) +�L +· +1 +|Vk|(1−2ε)m(n)−εL +≤ αm(n) · +� 1 +2α +�(1−2ε)m(n)+(1−ε)L +, +where in the last inequality we used (5.1), our assumption on k. By con- +struction, L ≥ (1 − 3ε)m(n), and so we get +(1 − 2ε)m(n) + (1 − ε)L ≥ (1 − ε)(2 − 5ε)m(n) ≥ m(n), +where in the last step we used that ε ∈ (0, 2−4). Therefore, it follows from +the above that +|E(Gm(n) +n +)| +|Vn|2 +≤ +�1 +2 +�m(n) +. + +24 +GIULIANO BASSO AND YANNICK KRIFKA +This concludes the case distinction and establishes (5.2). Finally, having +(5.2) at hand we find that +lim inf +n→∞ +|E(Gm(n) +n +)| +|Vn|2 += 0, +since m(n) → ∞ as n → ∞. So Corollary 4.2 tells us that X is not totally +bounded. Hence, X0 is not compact. +□ +A metric space Y is called injective if whenever A ⊂ B are metric spaces +and f : A → Y a 1-Lipschitz map, then there exists a 1-Lipschitz extension +¯f : B → Y of f. More formally, Y is an injective object in the category of +metric spaces with 1-Lipschitz maps as morphisms. Injective metric spaces +have been introduced by Aronszajn and Panitchpakdi in [2] and are some- +times also called hyperconvex metric spaces by some authors. We refer to +[16, 28] for an introduction to injective metric spaces. As observed by Lang in +[28, Proposition 3.8], every injective metric spaces admits a conical bicomb- +ing. Indeed, given an injective metric space Y , by applying Kuratowski’s +embedding theorem, we may suppose that Y ⊂ Cb(Y ), and so because Y is +injective, there is a 1-Lipschitz retraction r: Cb(Y ) → Y and thus +σ(x, y, t) = r((1 − t)x + ty) +defines a conical bicombing on Y . Using an extension result of [3], we find +that Theorem 1.2 is also valid for an injective metric space. +Theorem 5.2. There exists an injective metric space Y with a conical bi- +combing σ such that there is a finite subset of Y whose closed σ-convex hull +is not compact. +Proof. Let n0 ≥ 2 and let X0 be constructed as in Section 3. We recall that +by definition X0 = X and X is naturally equipped with a conical midpoint +map m. Let σ denote the conical bicombing on X0 induced by m. As m is +symmetric, it is not difficult to see that σxy(t) = σyx(1−t) for all x, y ∈ X0. +This shows that σ is a reversible conical bicombing. Hence, by virtue of [3, +Theorem 1.2], there exists an injective metric space Y containing X0, and +a conical bicombing ˜σ on Y such that ˜σxy = σxy for all x, y ∈ X0. As +X0 is complete, it follows that ˜σ–conv(A) = σ–conv(A) for any A ⊂ X0. +Therefore, due to Theorem 5.1, Y admits a finite subset whose closed ˜σ- +convex hull is not compact. +□ +We finish this section by proving the following more general version of +Theorem 1.3. +Theorem 5.3. Let n0 ∈ N. Then there exists a complete metric space X0 +with a conical bicombing such that whenever A ⊂ Y is an n0-point subset of +some complete metric space Y with a conical midpoint map m, then there +exists a Lipschitz map Φ: X0 → Y with A ⊂ Φ(X0) and furthermore Φ(X0) +is σ-convex with respect to the conical bicombing σ induced by m. + +A NON-COMPACT CONVEX HULL IN NON-POSITIVE CURVATURE +25 +Proof. Let X0 = (X0, d) be the metric space constructed in Section 3. We +set A0 = V1 ⊂ X0. By Lemma 3.2, it follows that +d(x, y) = d1(x, y) = 1 +(5.7) +for all distinct x, y ∈ A0. In particular, A0 ⊂ X0 is an n0-point subset. Now, +let A be as in the statement of the theorem. Since A and A0 are both n0- +point sets, there is a surjective map ϕ: A0 → A. Clearly, ϕ is L-Lipschitz for +some L ≥ 1. We define L-Lipschitz maps ϕn : (Vn, ϱn) → Mn(A) recursively +as follows. +Because of (5.7), it follows that ϕ1 = ϕ is L-Lipschitz with +respect to ϱ1. Given n ≥ 2 and x ∈ Vn, we set +ϕn(x) = m +� +ϕn−1(a), ϕn−1(b) +� +if x = m(a, b) with a, b ∈ Vn−1. Let x, y ∈ Vn be such that x ∼ y in +Gn. Hence, by definition, there is v ∈ Vn−1 and u ∼ w in Gn−1 such that +x = m(v, u) and y = m(v, w), and so +d(ϕn(x), ϕn(y)) = d +� +m(ϕn−1(v), ϕn−1(u)), m(ϕn−1(v), ϕn−1(w)) +� +≤ 1 +2d(ϕn−1(u), ϕn−1(w)) ≤ L · +1 +2n−1 , +where in the last step we have used that ϕn−1 is L-Lipschitz with respect to +ϱn−1. Since ϱn = 2−(n−1) · dGn, it now follows directly from the above and +the definition of the shortest-path metric dGn that ϕn is L-Lipschitz with +respect to ϱn. By construction, ϕn(x) = ϕm(x) for all x ∈ Vn and m ≥ n. +Hence, as Y is complete these maps naturally give rise to a L-Lipschitz map +Φ: X0 → Y . +To finish the proof we show that Φ(X0) is σ-convex. For simplicity, in +the following we will denote the bicombings on X0 and Y both by σ. By +construction of Φ and since σ is induced by a conical midpoint map, it +follows that Φ(σ(x, y, t)) = σ(Φ(x), Φ(y), t) for all x, y ∈ Mn(A0) and +all t ∈ [0, 1]. +Let now x, y ∈ X0 be arbitrary. +Then there exists xk, +yk ∈ Mnk(A0) such that xk → x and yk → y as k → ∞, respectively. +Moreover, σxkyk → σxy uniformly. Hence, as Φ is Lipschitz continuous, we +have Φ(σ(x, y, t)) = σ(Φ(x), Φ(y), t) for all t ∈ [0, 1]. This shows that Φ(X0) +is σ-convex. +□ +6. Does X0 admits a consistent conical bicombing? +In practice, it is often desirable to impose stronger properties on a bi- +combing than (1.1). By asserting that a conical bicombing is consistent, see +Section 2.3 for the definition, one obtains an interesting class of bicombings +which seem to be quite rigid. +Following Haettel, we call a metric space +a CUB-space if it admits a unique consistent conical bicombing (see [21]). +The class of CUB-space is already quite rich and still growing. For example, +in [5] it is shown that any convex body in a dual Banach space is CUB. +Moreover, proper, finite-dimensional injective metric space are CUB and +Deligne complexes of certain Artin groups are CUB if they are re-metrized + +26 +GIULIANO BASSO AND YANNICK KRIFKA +by considering the length metric induced by the ℓ∞-metric on each cell (see +[11, 21]). +However, using a non-affine isometry first introduced by Schechtman [38], +one can construct a complete metric space with two distinct consistent coni- +cal bicombings (see [5, Example 4.4]). On the other hand, up to the author’s +knowledge, there is no example of a metric space with a conical bicombing +that does not also admit a consistent conical bicombing. In other words, the +following question of Descombes and Lang [11] is still open. +Question 6.1 (Descombes–Lang). Let X be a complete metric space. Is it +true that X admits a conical bicombing if and only if it admits a consistent +conical bicombing. +This question also appears in the problem list [35, p. 385]. A partial +result that indicates a positive answer when X is proper has been obtained +in [3, Theorem 1.4]. One difficulty in finding a negative answer to Ques- +tion 6.1 lies in the fact that many know examples of metric spaces with a +conical bicombing have locally a nice structure. In this situation one can +then employ a generalized version of the Cartan-Hadamard theorem due to +Miesch [32] to construct a consistent conical bicombing. 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Krifka (krifka@mpim-bonn.mpg.de) +Max Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany + diff --git a/SdE2T4oBgHgl3EQfWgfj/content/tmp_files/load_file.txt b/SdE2T4oBgHgl3EQfWgfj/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3f0ba6d1162d1eaddff5c4b6a27629ddbc0d117d --- /dev/null +++ b/SdE2T4oBgHgl3EQfWgfj/content/tmp_files/load_file.txt @@ -0,0 +1,1268 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf,len=1267 +page_content='A NON-COMPACT CONVEX HULL IN GENERAL NON-POSITIVE CURVATURE GIULIANO BASSO AND YANNICK KRIFKA Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' In this article, we are interested in metric spaces that sat- isfy a weak non-positive curvature condition in the sense that they admit a conical bicombing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Recently, these spaces have begun to be studied in more detail, and a rich theory is beginning to emerge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' In this paper, we contribute to this study by constructing a complete metric space X with a conical bicombing σ such that there is a finite subset of X whose closed σ-convex hull is non-compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' In CAT(0)-geometry, the analogous statement is an open question, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' it is not known whether closed convex hulls of finite subsets of complete CAT(0) space are com- pact or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' This question goes back to Gromov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Our result shows that to obtain a positive answer to Gromov’s question, more than just the convexity properties of the metric must be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' The constructed space X has the additional property that there is an integer n such that it is an initial object in the category of convex hulls of n-point sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Thus, roughly speaking, X can be thought of as the largest possible convex hull of n-points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Introduction A family σ = (σxy)x,y∈X of geodesics σxy : [0, 1] → X of a metric space X with the property that σxy(0) = x and σxy(1) = y for all x, y ∈ X is called (geodesic) bicombing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' The terms combing and bicombing have been coined by Thurston [15, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' 84] and variants of it have originally been studied in the context of geometric group theory (see [1, 20, 25]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' In the present article, we are mainly concerned with metric spaces that admit bicombings whose geodesics share properties with geodesics in non-positively curved spaces such as CAT(0) spaces or, more generally, Busemann spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Following Descombes and Lang [11], we say that a bicombing σ is conical if d(σxy(t), σx′y′(t)) ≤ (1 − t)d(x, x′) + td(y, y′) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='1) for all x, x′, y, y′ ∈ X and all t ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' We remark that in CAT(0) spaces the function t �→ d(γ(t), η(t)) is convex on [0, 1] for all linearly reparametrized geodesics γ, η: [0, 1] → X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' In particular, the unique geodesics of a CAT(0) space form a conical bicombing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Recently, conical bicombing have gained some interest and have begun to be studied in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' This is partly Date: January 11, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' 53C23, 51F99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' non-positive curvature, convex hulls, conical bicombing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='03835v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='MG] 10 Jan 2023 2 GIULIANO BASSO AND YANNICK KRIFKA due to some applications in the context of Helly groups (see [9, 24]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' More- over, they also naturally occur as target spaces in the context of Lipschitz extension problems (see [30, 33]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Indeed, a metric space with a conical bicombing has many more properties that are usually associated to ’non- positive curvature’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' See Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='2 below for a collection of some of those results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' In this article, we study convex hulls in metric spaces with a conical bicombing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' The starting point of our considerations is the following intrigu- ing question regarding convex hulls in CAT(0) spaces due to Gromov (see [20, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='B1(f)]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Question 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='1 (Gromov).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Let X be a complete CAT(0) space and K ⊂ X a compact subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Is it true that the closed convex hull of K is compact?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Gromov’s question has been popularized by Petrunin (see [36] and also [37, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' 77]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Since the closed convex hull of K has the same diameter as K, it is not difficult to see that Question 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='1 has a positive answer if X is proper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' However, for non-proper spaces it seems to be very difficult to answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' In fact, already for three-point subsets the question is completely open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' We remark that using standard techniques from CAT(0)-geometry one can show that Question 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='1 has a positive answer if and only if it has a positive answer for finite subsets;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' see [13, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' However, also for finite subsets the closure of the convex hull needs to be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Indeed, already the convex hull of three points is not closed if the points do not lie on a geodesic and are contained in a generic complete Riemannian manifold of dimension ≥ 3 (see [29, Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Clearly, Question 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='1 can also be stated for spaces with a conical bicomb- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Let σ be a conical bicombing on a complete metric space X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' We say that A ⊂ X is σ-convex if for all x, y ∈ A, the geodesic σxy is contained in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' We consider the closed σ-convex hull of A, σ–conv(A) = � C, where the intersection is taken over all closed σ-convex subsets C ⊂ X containing A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Our main result shows that in the setting of spaces with conical bicombings the analogue of Gromov’s question has a negative answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='2 (Non-compact convex hull).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' There exists a complete metric space X with a conical bicombing σ such that there is a finite subset of X whose closed σ-convex hull is not compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Thus, to obtain a positive answer to Gromov’s question, more than just the convexity properties of the metric must be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' We remark that there is a metric space X as in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='2 which is additionally an injective metric space, see Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='2 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Injective metric spaces are prime ex- amples of metric spaces with a conical bicombing (see [28, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Descombes and Lang [11] showed that injective metric spaces of finite com- binatorial dimension admit a unique bicombing which satisfies a stronger convexity property than (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' More precisely, such spaces admit a unique A NON-COMPACT CONVEX HULL IN NON-POSITIVE CURVATURE 3 convex bicombing which is furthermore consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' The exact definitions are recalled in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' We do not know whether Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='2 holds also for such bicombings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' The construction of the metric space X in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='2 is discrete in nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Indeed, X is the metric completion of the direct limit V of a sequence of finite graphs Gn = (Vn, En).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' The morphisms in question are injective maps Vn → Vm, which are 1-Lipschitz with respect to an appropriate scaling of the shortest-path metric on Gn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' The conical bicombing σ on X is then constructed using a midpoint map m: V × V → V which satisfies a discrete version of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' More details about the construction of X can be found in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' The original idea behind this construction was to ensure the existence of the initial object X0 in the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Indeed, the metric space X in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='2 can be taken to be X0 for any n0 ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='3 (Initial object).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Let n0 ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Then there exists a complete metric space X0 with a conical bicombing such that whenever A ⊂ Y is an n0-point subset of some complete CAT(0) space Y , then there exists a Lipschitz map Φ: X0 → Y such that Φ(X0) is convex and contains A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' We actually prove a stronger statement than Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Instead of complete CAT(0) spaces Y , more general non-positively curved target spaces such as Busemann spaces can be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' See Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='3 below for the exact statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' We remark that, by construction, conv(A) ⊂ closure(Φ(X0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Therefore, if Φ(X0) is precompact, then the closed convex hull of A is com- pact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Given this relation, it seems reasonable to suspect that X0 is not compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' As it turns out, this is indeed the case for every n0 ≥ 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' see Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' In addition, it also follows immediately from the construction of X0 that there is some finite subset A ⊂ X0 such that σ–conv(A) = X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Hence, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='2 is a direct consequence of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' One may of course wonder whether there also exists such a space X0 as above, which is in addition a complete CAT(0) space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' The existence of such spaces would reduce Gromov’s question to the problem of deciding whether these spaces X0 are compact or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' If they are all compact, then Question 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='1 would have a positive answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' On the other hand, the non- compactness of X0 for some n0 ∈ N would give a negative answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' However, our proof does not seem to be directly amenable for generating CAT(0) spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Strategy of proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' In the following, we give a brief overview of how the metric space X0 in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='3 is constructed as a direct limit of a sequence of graphs Gn = (Vn, En).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' We fix n0 ∈ N and we let G0 denote the null graph and G1 the complete graph on n0 vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' The basic idea is that we have an increasing sequence of vertex sets V0 ⊂ V1 ⊂ · · · 4 GIULIANO BASSO AND YANNICK KRIFKA such that the vertex set Vn is obtained from Vn−1 by appending all possible midpoints, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=', Vn = Vn−1 ∪ midpoints(Vn−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' The formal definition of the midpoint construction m(a, b) can be found in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Any connected graph can naturally be viewed as a metric space by equipping it with the shortest-path metric (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='1) for the definition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' The edge set En is now defined such that the shortest-path metric of Gn satisfies a discrete version of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='1) for x = x′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Loosely speaking, En is obtained by considering cones in Gn−1, and then the ’cone midpoints’ in Gn are adjacent, and indeed every edge in Gn arises in this way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' More concretely, we have x ∼ y in Gn if and only if there there exists a vertex v ∈ Vn−1 (the cone point) and an edge u ∼ w in Gn−1 (the base) such that x = midpoint(v, u) and y = midpoint(v, w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Hence, midpoint(v, u) ∼ midpoint(v, w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' whenever v ∈ Vn−1 and u ∼ w in Gn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' This is illustrated in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' u w v x=m(v,u) y=m(v,w) Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Cone midpoints are adjacent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' For n0 = 2 and n = 1, 2, 3, 4, the graphs Gn = (Vn, En) obtained by applying this rule are shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' The graph G5 has already 68 vertices and 184 edges and quite an intricate structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Letting dGn denote the shortest-path metric of Gn, we find by definition of En that dGn � midpoint(x, y), midpoint(x, z) � ≤ dGn−1(y, z) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='2) for all x, y, z ∈ Vn−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' see Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' We interpret this as a discrete version of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='1) with x = x′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' In particular, as x = midpoint(x, x), the inclusion (Vn−1, dGn−1) �→ (Vn, dGn) is 1-Lipschitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Hence, letting V = � n Vn, it follows that ϱ: V × V → R given by ϱ(x, y) = lim n→∞ (diam Vn)−1· dGn(x, y), A NON-COMPACT CONVEX HULL IN NON-POSITIVE CURVATURE 5 defines a semi-metric on V (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='1 for the definition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' We remark that V is the direct limit of the sequence (Vn) with morphisms Vn → Vm, for n ≤ m, induced by the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' By construction, V is equipped with a midpoint map m: V × V → V defined by m(x, y) = midpoint(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Since diam Vn = 2n−1, it follows because of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='2) that ϱ(m(x, y), m(x, z)) ≤ 1 2ϱ(y, z) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='3) for all x, y, z ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Now, X0 = (X0, d) is defined as the metric completion of the metric space (X, d) associated to (V, ϱ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' We prove in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='4 that m extends to a map m: X × X → X such that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='3) still holds true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' It is not difficult to show that such a map m induces a conical bicombing σ on X0 such that X0 = σ–conv(V1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' see Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='4 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' We finish this overview with the main ideas that go into the proof of the non-compactness of X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' As a first reduction, it is clearly sufficient to show that X is not totally bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Now an important observation is that to prove that X has an (m · 2−n)-separated set of cardinality r + 1, it suffices to show that the graph Gm n has an (r + 1)-clique;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' see Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Here, Gm n denotes the m-th power of Gn and Gm n its complement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' This is standard terminology from graph theory, which is recalled in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Thus, by the above, the problem has been completely reduced to the existence of cliques in graph powers of Gn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' This opens the field for applications of techniques from extremal graph theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Indeed, using Tur´an’s theorem, see Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='1, it is not difficult to show that if for a certain sequence of integers m(n) one has that lim inf n→∞ |E � Gm(n) n � | |Vn|2 = 0, then X is not totally bounded;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' see Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Hence, to conclude the proof we need to show that Gm n does not contain too many edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' This is achieved by exploiting the explicit construction of En.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' In particular, the number of edges of Gm n is related to the edge counts of Ga n−1 and Gb n−1 with a + b = m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' see Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Moreover, one has that |En| |Vn|1+ε → 0 as n → ∞ for every ε > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' see Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Combining these two results, we finish the proof by a simple case distinction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' This is done in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' We remark that our proof does no explicitly construct ε-separated sets with arbitrarily large cardinality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' We only establish their existence by an appli- cation of Tur´an’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' We believe that an explicit construction of such sets would be worthwhile but probably very difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' The first named author is indebted to Urs Lang, Alexander Lytchak, and Stephan Stadler for useful discussions about convex hulls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' 6 GIULIANO BASSO AND YANNICK KRIFKA 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Preliminaries 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Basic metric notions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' We use N = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' } to denote the set of positive integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' A non-negative function ϱ: X × X → R is called semi- metric if it is symmetric, satisfies the triangle inequality and ϱ(x, x) = 0 for all x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' In other words, all axioms of a metric are satisfied except (possibly) the positivity axiom, that is, there might exist distinct x, y ∈ X such that ϱ(x, y) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' In the literature, such a function is sometimes also called a pseudometric (see, for example, [8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' However, in the present article we will only use the term semi-metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Let X = (X, d) be a metric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' We use X to denote the metric completion of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' If readability demands it we will sometimes tacitly identify X with its canonical isometric copy in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' A metric space is said to be totally bounded if for every ε > 0 there exists a finite subset A ⊂ X such that for every x ∈ X there exists a ∈ A such that d(x, a) < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' We recall that X is totally bounded if and only if X is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Graph theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' We use standard notation from graph theory as found in [7, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Let G = (V, E) be a graph, that is, V is a (possibly infinite) set and E ⊂ {e ⊂ V : |e| = 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' If {x, y} ∈ E then we often write x ∼ y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' We let G denote the complement graph of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' That is, G has vertex set V and x ∼ y in G if and only if x ̸= y and x, y are not adjacent in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' We will also need to consider graph powers of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Let m ≥ 1 be an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' We let Gm denote the m-th power of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' By definition, Gm is a graph with vertex set V and distinct vertices x, y ∈ V are adjacent if and only if there exists a path in G of length at most m that connects x to y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' We use the convention that G0 denotes the empty graph (V, ∅).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Given an integer r ≥ 1, we let Kr+1 denote the complete graph on (r + 1)-vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' The following theorem by Tur´an is a foundational result in extremal graph theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='1 (Tur´an’s theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Let G = (V, E) be a finite graph and r ≥ 1 an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' If G does not contain Kr+1 as a subgraph, then |E| ≤ � 1 − 1 r � |V |2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' We will apply this theorem to graphs of the form Gm to obtain m- separated sets in G with respect to the shortest-path metric dG;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' see Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='1 and Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Recall that the shortest-path metric dG : V × V → R is defined by dG(x, y) = min � k : (x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' , xk) is a path in G form x to y � (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='1) for all x, y ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Bicombings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' In the following we introduce bicombings and the various properties one can impose on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' We decided to be a little more detailed than would be strictly necessary for the main body of this article;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' see in particular Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' All definitions appearing below are essentially due to Descombes and Lang (see [11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' A NON-COMPACT CONVEX HULL IN NON-POSITIVE CURVATURE 7 Let X be a metric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' We say that σ: [0, 1] → X is a geodesic if d(σ(s), σ(t)) = |s − t| · d(σ(0), σ(1)) for all s, t ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' A map σ: X × X × [0, 1] → X is called (geodesic) bicombing if for all x, y ∈ X, the path σxy(·): [0, 1] → X defined by σxy(t) = σ(x, y, t) is a geodesic connecting x to y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' We remark that, in contrast, a map σ: X × [0, 1] → X is called combing with basepoint p ∈ X if for all x ∈ X, the path σ(x, ·) is a geodesic connecting p to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' However, we will not make use of this definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Bicombings are also called system of good geodesics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' see [17, 19, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Clearly, every geodesic metric spaces admits a bicombing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' We often consider bicombings in metric spaces that have non-unique geodesics such as, for example, Rn equipped with the p-norm for p ̸= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Therefore, it is useful to formalize some of the natural properties of the bicombing on a uniquely geodesic metric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' We say that σ is reversible if σxy(t) = σyx(1 − t) for all x, y ∈ X and all t ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' In [5, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='3] it is shown that any complete metric space with a conical bicombings also admits a conical reversible bicombing (see also [10] for an earlier result).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Furthermore, we say that a bicombing σ is consistent if it is reversible and σ(x, y, st) = σ(x, σxy(t), s) for all x, y ∈ X and all s, t ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Consistent bicombings are used in [18, 23], and a variant of the definition that allows for a bounded error is studied in [14, Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' We do not know if every space with a bicombing also admits a consistent bicombing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' This seemingly straightforward question does not seem to be so easy to answer on closer inspection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' For proper metric spaces admitting a conical bicombing, it turns out to be true (see [3, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Descombes and Lang [11] introduced the following two non-positive cur- vature conditions for a bicombing σ: (1) if (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='1) holds, then σ is said to be conical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' (2) if for all x, y, x′, y′ ∈ X, the map t �→ d(σxy(t), σx′y′(t)) is convex on [0, 1], then σ is called convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' There are many examples of conical bicombings that are not convex (see [11, Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='2] and [3, Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' However, any consistent conical bicombings is convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' One may wonder if any convex bicombing is automat- ically consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' This turns out to be not to be the case, as is demonstrated in [5, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' To the authors’ knowledge, a relatively simple example of a convex non-consistent bicombings seems to be missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' The following theorem is a ’state of the art’ collection of general facts about spaces that admit a conical bicombing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' All of these properties are usually associated with ’non-positive curvature’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Let X be a complete metric space admitting a conical bi- combing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Then the following holds true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' (1) X is contractible, (2) X admits barycenter map in the sense of Sturm [40], (3) all Lipschitz homotopy groups πLip k (X) are trivial, 8 GIULIANO BASSO AND YANNICK KRIFKA (4) X admits an isoperimetric inequality of Euclidean type for Ik(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Moreover, if X is proper then (5) X is an absolute retract, (6) X admits a visual boundary which is a Z-boundary in the sense of Bestvina [6], (7) any subgroup of the isometry group of X with bounded orbits has a non-empty fixed-point set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' We prove each item separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Fix o ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Clearly, H : X × [0, 1] → X defined by H(x, t) = σ(x, o, t) is a homotopy between the identity map on X and the constant map with value o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' This shows (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' A proof of (2) can be found in [3, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' We proceed by showing (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' A metric space X is called Lipschitz k-connected with constant c if for every ℓ ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' , k}, every L-Lipschitz map f : Sℓ → X has a cL-Lipschitz extension ¯f : Bℓ+1 → X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Here, Sℓ, Bℓ+1 ⊂ Rℓ+1 denote the Euclidean unit sphere and closed Euclidean unit ball, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' To prove that πLip k (X) is trivial it suffices to show that X is Lipschitz k-connected for some constant c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Therefore, the statement follows, since in [39, Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='2] it is proved that X is Lipschitz k-connected with constant 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' For a proof of (4) we refer to Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='4 in [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Next, we prove (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Using that X admits a conical bicombing, it is not difficult to show that X is strictly equiconnected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Therefore, it follows from a result by Himmelberg [22, Theorem 4] that X is an absolute retract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' The next statement, (6), follows directly from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='5 in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' To finish the proof, we establish (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Let Γ be a subgroup of the isometry group of X with bounded orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Fix x0 ∈ X and consider the orbit A = {f(x0) : f ∈ Γ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' In the following we combine results from [3] and [4] to show that the fixed-point set of Γ is non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' In view of [4, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='2] it suffices to show that X admits a Γ-equivariant conical bicombing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' We now use the proof strategy of [3, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='5] to show that such a bicombing exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Let CB(X) be the set of all conical bicombings on X and for every x ∈ X let the metric Dx on CB(X) be given as in [3, Section 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' We define ˜D = supx∈A Dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Clearly, ˜D defines a metric on CB(X) and by considering the proof of [3, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='2] it is straightforward to show that (CB(X), ˜D) is a compact metric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Let f ∈ Γ and let F : CB(X) → CB(X) be defined by F(σ)(x, y, t) = f−1(σ(f(x), f(y), t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Since f(A) = A, it follows that F is distance-preserving if CB(X) is equipped with ˜D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Now, one can argue exactly as in the proof of [3, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='5] to conclude that there exists some σ∗ ∈ CB(X) such that F(σ∗) = σ∗ for all f ∈ Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' In other words, σ∗ is a Γ- equivariant concial bicombing, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' We remark that additional fixed- point results for spaces with a conical bicombing can be found in [26, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Conical midpoint maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' In this section we introduce conical mid- point maps and derive some of their basic properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' We are mainly inter- ested in this notion since it can be seen as a discrete analogue of conical A NON-COMPACT CONVEX HULL IN NON-POSITIVE CURVATURE 9 bicombings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Indeed, any conical midpoint map on a metric space X induces a conical bicombing on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' This is discussed at the end of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' We say that m: X × X → X is a conical midpoint map if for all x, y, z ∈ X, the following holds: (1) m(x, x) = x, (2) m(x, y) = m(y, x), (3) d(m(x, y), m(x, z)) ≤ 1 2d(y, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' We remark that for midpoints in Euclidean space, the inequality in (3) becomes in fact an equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' It is easy to see that if m is as in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='3, then z = m(x1, x2) is a midpoint of x1 and x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Indeed, d(z, xi) = d(z, m(xi, xi)) ≤ 1 2d(x1, x2) and thus using the triangle inequality, we find that d(z, xi) = 1 2d(x1, x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Hence, a conical midpoint map is a midpoint map in the usual sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Furthermore, (3) can be upgraded to a more general inequality involving four points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' For all x, y, x′, y′ ∈ X, one has d(m(x, y), m(x′, y′)) ≤ 1 2d(x, x′) + 1 2d(y, y′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='2) This can be seen as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Using (2) and the triangle inequality, we get d(m(x, y), m(x′, y′)) ≤ d(m(x, y), m(x, y′)) + d(m(y′, x), m(y′, x′)) and thus by virtue of (3) we obtain (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Next, we show that conical mid- point maps induce conical bicombings in a natural way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' The used recursive construction is well-known and goes back to Menger (see [31, Section 6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Let m be a concial midpoint map on X and x, y ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Further, let Gn = (2−n · Z) ∩ [0, 1], where n ≥ 0, be the 2−n-grid in [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' We define σxy : � Gn → X recursively as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' We put σxy(0) = x, σxy(1) = y and if t ∈ Gn \\ Gn−1, then we set σxy(t) = m(σxy(r), σxy(s)), where r, s ∈ Gn−1 are the unique points such that t = 1 2r + 1 2s and |r − s| = 2−(n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' The map σxy extends uniquely to a geodesic σxy : [0, 1] → X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Moreover, d(σxy(t), σx′y′(t)) ≤ (1 − t)d(x, x′) + td(y, y′) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='3) for all x, y, x′, y′ ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' To begin, we show that σxy|Gn is an isometric embedding for all n ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' We proceed by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Clearly, σxy|G0 is an isometric embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Now, fix ti ∈ Gn, i = 1, 2 and let ri, si ∈ Gn−1 with si ≤ ri be points such that ti = 1 2si + 1 2ri and σxy(ti) = m(σxy(si), σxy(ri)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' By construction of σxy 10 GIULIANO BASSO AND YANNICK KRIFKA such points clearly exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Without loss of generality, we may suppose that t1 ≤ t2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Using the triangle inequality, we get d(σxy(t1), σxy(t2)) ≤ d(σxy(t1), σxy(r1)) + d(σxy(r1), σxy(s2)) + d(σxy(s2), σxy(t2)), and so, by the induction hypothesis and because m is a midpoint map, d(σxy(t1), σxy(t2)) ≤ �r1 − s1 2 + |s2 − r1| + r2 − s2 2 � d(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' But, since t1 ≤ t2, it holds r1 ≤ s2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Hence, by the above, d(σxy(t1), σxy(t2)) ≤ |t1 − t2|d(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' As a result, d(x, y) ≤ d(x, σxy(t1)) + d(σxy(t1), σxy(t2)) + d(σxy(t2), y) ≤ � t1 + |t1 − t2| + |t2 − 1| � d(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' This implies that d(σxy(t1), σxy(t2)) = |t1 − t2|d(x, y), and so σxy|Gn is an isometric embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' It follows by induction that σxy|Gn is an isometric embedding for every n ≥ 0, as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Now, since � Gn is a dense subset of [0, 1], it follows that σxy can be uniquely extended to an isometric embedding σxy : [0, 1] → X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Next, we show (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Clearly, d(σxy(1/2), σx′y′(1/2)) ≤ 1 2d(x, x′) + 1 2d(y, y′), as σxy(1/2) = m(x, y), σx′y′(1/2) = m(x′, y′) and m is conical midpoint map and thus satisfies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' We now proceed by induction and show that if (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='3) is valid for all t ∈ Gn−1, then it is also valid for all t ∈ Gn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Fix t ∈ Gn and let s, r ∈ Gn−1 be the unique points with s ≤ r such that t = 1 2s + 1 2t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' We compute d(σxy(t), σx′y′(t)) ≤ 1 2d(σxy(s), σx′y′(s)) + 1 2d(σxy(r), σx′y′(r)) ≤ �1 − s 2 + 1 − r 2 � d(x, x′) + �s 2 + r 2 � d(y, y′);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' hence, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='3) holds for all t ∈ Gn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Since � Gn is a dense subset of [0, 1] and σxy and σx′y′ are geodesics, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='3) is valid for all t ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' □ Thus, we have constructed a map σ: X × X × [0, 1] → X such that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='1) holds for all geodesics σxy and σx′y′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Now, given x, y ∈ X, we set σxy(t) = lim n→∞ σxnyn(t) where xn, yn ∈ X are points such that xn → x and yn → y as n → ∞, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' It follows that σ is a well-defined conical bicombing on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' We call σ the conical bicombing induced by m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' We point out that m is defined on an arbitrary metric space X but σ is always a bicombing on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' We conclude this section by giving a description of σ-convex hulls in terms of m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Indeed, as with conical bicombings, conical midpoint maps give rise to A NON-COMPACT CONVEX HULL IN NON-POSITIVE CURVATURE 11 ’convex hulls’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' For any A ⊂ X, we let m –conv(A) ⊂ X denote the closure of the set � n∈N Mn(A), where M1(A) = � m(a, a′) : a, a′ ∈ A � and Mn(A) = M1(Mn−1(A)) for all n ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Let m be a conical midpoint map on a metric space X and suppose σ denotes the conical bicombing on X induced by m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Then σ–conv(A) = m –conv(A) for all A ⊂ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Clearly, m –conv(A) ⊂ σ–conv(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Thus, it suffices to show that the closed set m –conv(A) is σ-convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' To this end, let n ≥ 1 and let x, y ∈ Mn(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' By construction of σ, it follows that σxy(Gm) ⊂ Mn+m(A) for all m ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Hence, σxy([0, 1]) ⊂ m –conv(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Now, suppose that x, y ∈ m –conv(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' There exist points xk, yk ∈ Mnk(A) such that xk → x and yk → y as k → ∞, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Moreover, σxkyk → σxy uniformly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' This implies that σxy([0, 1]) ⊂ m –conv(A), and so m –conv(A) is σ-convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Appending midpoints Throughout this section we fix n0 ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' This n0 will correspond to the parameter from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' We follow the proof strategy outlined in Sec- tion 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='1 to construct the metric space X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' To begin, we construct recursively a sequence of graphs Gn = (Vn, En).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' The whole construction is quite formal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' The basic idea is that Vn is obtained from Vn−1 by appending ’midpoints’ and two midpoints in Vn are adjacent if and only if they are part of a cone whose base is an edge of Gn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' We let G0 denote the null graph and G1 the complete graph on n0 vertices with vertex set V1 = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' , n0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' For n ≥ 2 we set Vn = Vn−1 ∪ � {x, y} : x, y ∈ Vn−1, x ̸= y � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' To formalize the notion of ’midpoint’ we use the following notation m(a, b) = � {a, b} if a ̸= b, a otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='1) Notice that Vn = m(Vn−1 × Vn−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Moreover, we remark that we have constructed an infinite nested sequence V0 ⊂ V1 ⊂ V2 ⊂ · · · Now, the edge set En is uniquely determined by {x, y} ∈ En if and only if there exist v ∈ Vn−1 and {u, w} ∈ En−1 such that x = m(v, u) and y = m(v, w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Loosely speaking, x and y are adjacent in Gn if and only if x, y are the midpoints parallel to the base of a cone with vertex v ∈ Vn−1 12 GIULIANO BASSO AND YANNICK KRIFKA and base u ∼ w in Gn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' See Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='1 for an illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' For example, if n0 = 2, one has V2 = � 0, 1, {0, 1} � and E2 = � {0, {0, 1}}, {{0, 1}, 1} � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' The graphs Gn for n0 = 2 and n = 1, 2, 3, 4 are depicted in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' The graphs Gn for small n with n0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' To begin, we collect some basic facts about the cardinalities of Vn and En that will be used later on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' One has |V0| = 0, |V1| = n0, and for all n ≥ 2, |Vn| = 1 2 · � |Vn−1| + |Vn−2| � � |Vn−1| − |Vn−2| + 1 � (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='2) Moreover, for every ε > 0, lim n→∞ |En| |Vn|1+ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='3) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' By construction, Vn−2 ⊂ Vn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Thus, letting Wn−1 = Vn−1 \\ Vn−2 and using that m is symmetric, we find Vn = m(Vn−1×Vn−1) = m(Vn−2×Vn−2)∪m(Vn−2×Wn−1)∪m(Wn−1×Wn−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Therefore, as these sets are pairwise disjoint, |Vn| = |Vn−1| + |Vn−2| · |Wn−1| + |Wn−1| · � |Wn−1| − 1 � 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Since |Wn−1| = |Vn−1| − |Vn−2|, this yields (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' To finish the proof, we establish (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Clearly, this is valid if n0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Thus, in the following, we A NON-COMPACT CONVEX HULL IN NON-POSITIVE CURVATURE 13 may suppose that n0 ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Notice that |E0| = 0, |E1| = 1 2(n0 − 1)n0 and |En| ≤ |Vn−1| · |En−1| for all n ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Consequently, |En| ≤ C · n−1 � i=1 |Vi|, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='4) where C = 1 2(n0 − 1)n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' We claim that |Vn−1|2 |Vn| ≤ 3 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='5) for all n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' For n = 1, 2 this can be seen by a direct verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Let now n ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Letting αn = |Vn−2|/|Vn−1| and using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='2), we find that |Vn−1|2 |Vn| ≤ 2 1 − α2n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' In particular, if αn ≤ 1/ √ 3, then (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='5) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Now, suppose that n0 ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' It follows that α3 ≤ 1/ √ 3 and hence (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='5) is valid for n = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Clearly, if (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='5) holds for n − 1, then αn ≤ 3/|Vn−2|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Thus, as |V2| ≥ 6 ≥ 3 √ 3, the desired inequality (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='5) follows by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' This establishes (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='5) when n0 ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' We now treat the special case when n0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' We have |V2| = 3, |V3| = 5, |V4| = 12, and |V5| = 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Hence, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='5) holds true if n = 3, 4, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' The general case now follows as before by noting that |V4| = 12 ≥ 3 √ 3, and so (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='5) can be established by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' This completes the proof of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' By combining (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='4) with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='5), we arrive at |En| |Vn|1+ε ≤ C · 3n |Vn|ε .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' We claim that |Vn| ≥ |Vn−2|2 for all n ≥ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Letting β = |Vn−1|/|Vn−2|, we obtain |Vn| |Vn−2|2 ≥ 1 2(β + 1)(β − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='6) Since |V4| = 12 if n0 = 2, it follows that |V4| ≥ 12 for every n0 ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Therefore, |Vn−2| 3 ≥ √ 3 for all n ≥ 6, and thus by virtue of |Vn−1| ≥ 1 3|Vn−2|, we obtain β2 ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' This is equivalent to 1 2(β + 1)(β − 1) ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' By the use of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='6), we can conclude that |Vn| ≥ |Vn−2|2 for all n ≥ 6, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Now, by repeated use of this inequality and using that |V3| ≥ |V2|, we get |Vn| ≥ |Vn−2|2 ≥ · · · ≥ � |V2| �2 n−3 2 for all n ≥ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Thus, letting c = ε 2 √ 2 and using that |V2| ≥ 3, we obtain lim n→∞ |En| |Vn|1+ε ≤ C · lim n→∞ 3n |Vn|ε ≤ C · lim n→∞ |V2|n−c( √ 2)n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' □ 14 GIULIANO BASSO AND YANNICK KRIFKA Let dn : Vn × Vn → R denote the shortest-path metric on Gn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' The defi- nition of the shortest-path metric of a graph is recalled in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Our next result shows that any two distinct points in V1 ⊂ Vn realize the diameter of Vn with respect to dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' For all distinct x, y ∈ V1, dn(x, y) = diam Vn = 2n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' To begin, we show that dn(x, y) ≤ 2dn−1(x, y) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='7) for all n ≥ 2 and all x, y ∈ Vn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Let (x0, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' , xk) be a shortest- path in Gn−1 connecting x to y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' We set x′ i = m(xi−1, xi) for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' , k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Clearly, xi−1 ∼ x′ i and x′ i ∼ xi in Gn for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' , k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Hence, (x0, x′ 1, x1, x′ 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' , x′ k, xk) is a path in Gn connecting x to y, and so dn(x, y) ≤ 2k = 2dn−1(x, y), as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' By construction, diam V1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Hence, it follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='7) that diam Vn ≤ 2n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' To finish the proof we thus need to show that dn(x, y) ≥ 2n−1 for all distinct x, y ∈ V1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' For this we will use the following construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' We define the functions δn : Vn → ∆n0−1 ∩ 2−(n−1) · Zn0 recursively as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' We may suppose that V1 = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' , n0} and we set δ1(i) = ei for each i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' , n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Here, ei ∈ Rn0 is the vector with a one at the ith position and zeros everywhere else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Suppose now n ≥ 2 and x ∈ Vn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' We set δn(x) = 1 2 � δn−1(a) + δn−1(b) � if x = m(a, b) with a ̸= b, and δn(x) = δn−1(x) otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' It follows by induction that if {x, y} ∈ En, then |δn(x) − δn(y)|∞ = 1 2n−1 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='8) where |·|∞ denotes the supremum norm on Rn0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Clearly, δn(i) = ei for all n ∈ N and all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' , n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Now, let x, y ∈ V1 be distinct and (x0, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' , xk) a path in Gn connecting x to y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' By the above, it follows that 1 = |δn(x) − δn(y)|∞ ≤ k−1 � i=0 |δn(xi) − δn(xi+1)|∞ = k 2n−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Hence, dn(x, y) ≥ 2n−1, as was to be shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' □ Our next lemma relates shortest-paths in Gn to shortest-paths in Gn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' The proof follows easily from the definition of dn and the recursive construc- tion of En.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' For all x1, x2, y1, y2 ∈ Vn−1, dn(m(x1, x2), m(y1, y2)) ≤ dn−1(x1, y1) + dn−1(x2, y2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='9) A NON-COMPACT CONVEX HULL IN NON-POSITIVE CURVATURE 15 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Let (p0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' , pk) and (q0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' , qℓ) be shortest-paths in Gn−1 connecting x1 to y1, and x2 to y2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' We construct a path (r0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' , rk+ℓ) in Gn as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' We set ri = m(p0, qi) for all i = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' , ℓ and rℓ+j = m(pj, qℓ) for all j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' , k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' By construction, ri−1 ∼ ri in Gn for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' , k + ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Hence, (r0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' , rk+ℓ) is a path in Gn connecting r0 = m(x1, x2) to rk+ℓ = m(y1, y2), and so it follows that dn(m(x1, x2), m(y1, y2)) ≤ k + ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' But dn−1(x1, y1) = k and dn−1(x2, y2) = ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' This finishes the proof of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' □ We remark that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='9) should be thought of as a discrete analogue of the conical inequality (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Indeed, by considering the scaled metrics ϱn = (diam Vn)−1 · dn and using that diam Vn = 2n−1 by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='2, we obtain that ϱn(m(x1, x2), m(y1, y2)) ≤ 1 2ϱn−1(x1, y1) + 1 2ϱn−1(x2, y2) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='10) for all x1, x2, y1, y2 ∈ Vn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' In particular, if x, y ∈ Vn−1, then ϱn(x, y) ≤ ϱn−1(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' In view of these inequalities, letting V = � n≥1 Vn we find that the map ϱ: V × V → R defined by ϱ(x, y) = lim n→∞ ϱn(x, y) is a semi-metric on V (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='1 for the definition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' More formally, V could also be constructed as the direct limit of the sequence of metric spaces (Vn, ϱn) with morphisms Vn → Vm, for n ≤ m, induced by the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' By the above, the semi-metric space (V, ϱ) is naturally equipped with a ’conical midpoint map’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Indeed, because of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='10), m: V × V → V defined by (x, y) �→ m(x, y) satisfies ϱ(m(x, y), m(x, z)) ≤ 1 2ϱ(y, z) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='11) for all x, y, z ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' It is now not difficult to upgrade m to a concial midpoint map on a metric space X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Indeed, let us denote by (X, d) the metric space induced by (V, ϱ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' By definition, X = V/ ∼ with x ∼ y if and only if ϱ(x, y) = 0 and d is the quotient metric on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' We recall that d([x], [y]) = ϱ(x, y) for all x, y ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' The map m: X × X → X defined by m([x], [y]) = [m(x, y)] for all [x], [y] ∈ X is a concial midpoint map on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Moreover, X = � n∈N Mn(A), where A = [V1] ⊂ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' 16 GIULIANO BASSO AND YANNICK KRIFKA Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' By applying (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='11), we get ϱ(m(x, y), m(x′, y′)) ≤ ϱ(m(x, y), m(x, y′)) + ϱ(m(x, y′), m(x′, y′)) ≤ 1 2ϱ(y, y′) + 1 2ϱ(x, x′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Hence, if ϱ(x, x′) = 0 and ϱ(y, y′) = 0, then ϱ(m(x, y), m(x′, y′)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' This shows that m: X × X → X defined by m([x], [y]) = [m(x, y)] for all [x], [y] ∈ X is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Moreover, it follows directly from the inequality above that m is a concial midpoint map on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' By construction, m(Vn−1 × Vn−1) = Vn for all n ≥ 2, and so V ⊂ m –conv(V1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' This implies the desired equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' □ In summary, we have shown that the map m: V × V → V descends to conical midpoint map on X, where X denotes the metric space associated to V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' For simplicity this map is also denoted by m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Due to the results in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='4, m now induces a conical bicombing on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' We set X0 = X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' In Section 5 we show that X0 is non-compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' We achieve this by showing that X is not totally bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' In order to work effectively with X, it seems natural to determine how much the semi-metric ϱ (and hence d) differs from the metric ϱn on Vn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' The following lemma shows that ϱ does not collapse the distances too much.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' For all n ≥ 2, ϱn(x, y) − 8 2n ≤ ϱ(x, y) ≤ ϱn(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='12) for all x, y ∈ Vn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Notice that due to Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='5, if x, y ∈ Vn satisfy dn(x, y) ≥ 5, then d(x, y) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' In particular, ε-separated sets in (Vn, ϱn) induce ε′-separated sets in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' See Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='1 for the exact statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' The desired upper bound of ϱ(x, y) follows directly from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' In what follows we show the lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' To begin, we claim that 2dn−1(x, y) ≤ dn(x, y) + 4 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='13) for all x, y ∈ Vn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Fix distinct points x, y ∈ Vn−1 and let {x, x′} and {y, y′} be edges in En−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Since Gn−1 is connected such edges surely exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Because of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='8), it follows that p := m(x, x′), q := m(y, y′) ∈ Vn \\ Vn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Moreover, since x ∼ p and y ∼ q in Gn, by the triangle inequality, |dn(x, y) − dn(p, q)| ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' We claim that dn(p, q) = min � dn−1(x, y) + dn−1(x′, y′), dn−1(x, y′) + dn−1(x′, y) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='14) Indeed, let (x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' , xℓ) be a shortest-path in Gn connecting p to q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' For each i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' , ℓ there is vi ∈ Vn−1 and {ui, wi} ∈ En−1 such that xi−1 = m(vi, ui) and xi = m(vi, wi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' A NON-COMPACT CONVEX HULL IN NON-POSITIVE CURVATURE 17 x x′ p x1 x2 x3 x4 x5 y y′ q γ η Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' The construction from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' We define a0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' , aℓ ∈ Vn−1 and b0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' , bℓ+1 ∈ Vn−1 by induction as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' We put a0 = v1 and b0 = u1, b1 = w1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Now, for every i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' , ℓ − 1, we set � ai = wi+1 and bi+1 = bi if ui+1 = ai−1, ai = ai−1 and bi+1 = wi+1 if ui+1 = bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' By construction, m(a0, b0) = x0 = p and m(aℓ−1, bℓ) = xℓ = q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Moreover, af- ter deleting repeated entries, γ = (a0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' , aℓ−1) and η = (b0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' , bℓ) are (pos- sibly degenerate) shortest-paths in Gn−1 such that length(γ) + length(η) = ℓ = dn(p, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' See Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Hence, dn−1(a0, aℓ−1) + dn−1(b0, bℓ) ≤ dn(p, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Because of p, q /∈ Vn−1, without loss of generality we have a0 = x, aℓ−1 = y, b0 = x′ and bℓ = y′, and so the desired equality (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='14) now follows due to Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Having (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='14) at hand, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='13) now follows easily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Indeed, using that dn−1(x, x′) = dn−1(y, y′) = 1, we have dn−1(x, y) − 1 ≤ dn−1(x, y′) dn−1(x, y) − 1 ≤ dn−1(x′, y) and dn−1(x, y) − 2 ≤ dn−1(x′, y′), and so using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='14), we deduce that dn(x, y) ≥ dn(p, q) − 2 ≥ 2dn−1(x, y) − 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' This shows (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Now, by dividing (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='13) by 2n−1, we obtain ϱn−1(x, y) ≤ ϱn(x, y) + 8 2n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' 18 GIULIANO BASSO AND YANNICK KRIFKA In particular, for every k ∈ N, ϱn(x, y) ≤ ϱn+k(x, y) + 8 2n k � i=1 1 2i and the left inequality of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='12) follows by taking the limit k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' □ We remark that in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='13) at least an additive error of 2 must occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' This is discussed further in the following example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' x xʹ v b a Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Illustration of the construction in Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Let n0 = 2 and consider the graph G4 depicted in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' In particular, v = m(0, 1), a = m(0, v), b = m(v, 1) and x = m(a, 1) and x′ = m(v, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Clearly, d4(x, x′) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' We claim that d5(x, x′) = 2 as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Since x ∼ b in G4, the points x0 := m(v, b) and x1 := m(v, x) are adjacent in G5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Thus, as m(x, v) ∼ m(x, x) in G5, it follows that (x0, x1, x2) is a path in G5 connecting x′ to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Hence, d5(x, x′) ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' On the other hand, it is not difficult to see that δ5(x) = δ5(x′) and thus due to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='8), it follows that x and x′ are not adjacent in G5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' This shows that d5(x, x′) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Hence, 2d4(x, x′) − d5(x, x′) = 2, and so the additive error in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='13) must be at least 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Gm n has few edges In this section, we find a sufficient condition that X is not totally bounded in terms of the number of edges of Gm n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' The basic graph theory notation that is needed in the sequel can be found in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Let n, r ≥ 1 and m ≥ 6 be integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' If Gm n has an (r + 1)- clique, then there exist r + 1 points x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' , xr+1 ∈ X such that d(xi, xj) ≥ m 2n for all distinct i, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' , r + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' If v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' , vr+1 ∈ Vn are the vertices of an (r + 1)-clique in Gm n , then by definition one has dn(vi, vj) ≥ m + 1 A NON-COMPACT CONVEX HULL IN NON-POSITIVE CURVATURE 19 for all distinct i, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' , r + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Hence, by dividing by 2n−1 on both sides and using Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='5, we obtain d(vi, vj) ≥ 2m − 6 2n ≥ m 2n , as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' □ Fix an integer k ≥ 1 sufficiently large to be determined later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' We ab- breviate m(n) = 2n−k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Using Tur´an’s theorem we obtain the following non-compactness criterion for X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Let n0 ≥ 2 and let X be constructed as in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' If lim inf n→∞ |E � Gm(n) n � | |Vn|2 = 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='1) then X is not totally bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' We prove the contrapositive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Suppose that X is totally bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' There exists r ≥ 1 such that X does not contain a 1 2k -net of cardinality r + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Hence, by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='1, for n ≥ 1 sufficiently large, the complement graph of the m(n)-th power Gn does not contain an (r + 1)-clique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Thus, Tur´an’s theorem, see Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='1, tells us that |E � Gm(n) n � | ≤ � 1 − 1 r � |Vn|2 2 for all n sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Therefore, |Vn| · (|Vn| − 1) 2 − |E � Gm(n) n � | ≤ � 1 − 1 r � |Vn|2 2 and it follows that lim inf n→∞ |E � Gm(n) n � | |Vn|2 ≥ 1 2r > 0, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' We remark that to show the lower bound on the liminf we have used that |Vn| is an unbounded sequence, which is only valid if n0 ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' □ Thus, to prove that X not totally bounded, it suffices to establish (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' To this end, in the next subsection we derive some upper bounds for |E � Gm(n) n � |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Upper bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' The following estimate is not sharp in general, but is sufficient for our purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' It is the crucial building block for inequality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='3), which is our key tool in the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Let n, m ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Then there exist non-negative integers a, b such that a + b = m and |E(Gm n )| ≤ 2m |E(Ga n−1)| · |E(Gb n−1)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' We recall that we use the convention that |E(G0)| = |V | for any finite graph G = (V, E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' 20 GIULIANO BASSO AND YANNICK KRIFKA Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Suppose that x is adjacent to y in Gm n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' By definition, there exist a shortest-path (x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' , xℓ) in Gn of length ≤ m connecting x to y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' For each i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' , ℓ there is vi ∈ Vn−1 and {ui, wi} ∈ En−1 such that xi−1 = m(vi, ui) and xi = m(vi, wi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' As in the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='5, we define a0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' , aℓ ∈ Vn−1 and b0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' , bℓ+1 ∈ Vn−1 by induction as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' We put a0 = v1 and b0 = u1, b1 = w1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Now, for every i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' , ℓ − 1, we set � ai = wi+1 and bi+1 = bi if ui+1 = ai−1, ai = ai−1 and bi+1 = wi+1 if ui+1 = bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' By construction, m(a0, b0) = x0 = x and m(aℓ−1, bℓ) = xℓ = y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' More- over, after deleting repeated entries, γ = (a0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' , aℓ−1) and η = (b0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' , bℓ) are (possibly degenerate) shortest-paths in Gn−1 such that length(γ) + length(η) = ℓ = dn(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' See Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Moreover, any two non-degenerate shortest-paths γ and η induce at most two edges in Gm n in this way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Conse- quently, |E(Gm n )| ≤ |Vn−1| · |E(Gm n−1)| + 2 m−1 � i=1 |E(Gi n−1)| · |E(Gm−i n−1)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' We put M = max � |E(Gi n−1)| · |E(Gm−i n−1)| : i = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' , m � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' By the above, it follows that |E(Gm n )| ≤ M + 2(m − 1)C ≤ 2mM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' □ Recall that we have fixed an integer k ≥ 1 which is sufficiently large to be determined later, and we use the notation m(n) = 2n−k and ¯n = n − k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Using Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='3, it is possible to obtain an upper bound on the number of edges of Gm(n) n in terms of a product with factors |E(Gmi k )| and |Vn−i|ki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Let n ≥ 1 be sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Then there exist an integer K ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' , m(n)}, positive integers m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' , mK such that m1 +· · ·+mK = m(n), and integers ki ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' , 2i − 1} for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' , ¯n satisfying ¯n � i=1 ki · 2¯n−i = m(n) − K, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='2) such that |E(Gm(n) n )| ≤ 32m(n)� K � i=1 |E(Gmi k )| �� ¯n � i=1 |Vn−i|ki � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='3) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' We consider the following replacement rule: |E(Gm n )| → � 2m |E(Ga n−1)| · |E(Gb n−1)| if m > 0, where a, b are as in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='3 |Vn| if m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' A NON-COMPACT CONVEX HULL IN NON-POSITIVE CURVATURE 21 By using this rule and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='3 sufficiently many times, we obtain integers ℓi ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' , 2i}, for i = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' , ¯n − 1, such that |E(Gm(n) n )| ≤ � ¯n � i=1 2ℓi−1 · Ai · |Vn−i|ki � � K � i=1 |E(Gmi k )| � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='4) where Ai := ℓi−1 � j=1 αi,j for some positive integers αi,j > 0 satisfying αi,1 +· · ·+αi,ℓi = m(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Notice that in particular ℓ0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Using the inequality of arithmetic and geometric means, we get ℓi � j=1 αi,j ≤ �m(n) ℓi �ℓi = 2(¯n−log2 ℓi)·2log2 ℓi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' The function f(x) := (¯n − x) · 2x is increasing on [0, ¯n − 2], and max x∈[0,¯n] f(x) = 2¯n e log 2 ≤ 2¯n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Hence, using that ℓi ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' , 2i}, we have A¯n ≤ 2¯n and for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' , ¯n− 2, Ai ≤ 2(¯n−(i−1))·2i−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Thus, since ¯n−2 � j=0 (¯n − j)2j ≤ 2¯n−1 + ¯n−1 � j=1 2j ≤ 2¯n−1 + 2¯n, we find that ¯n � i=1 Ai ≤ 22¯n+2¯n−1+2¯n ≤ 162¯n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Moreover, ¯n � i=1 2ℓi−1 ≤ ¯n−1 � i=0 22i ≤ 22¯n, and thus (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='3) follows from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' □ We remark that if Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='3 were true for a = b = m 2 , by exactly the same reasoning as in the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='4, we would get the following slightly more elegant upper bound in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='3), 8m(n) · |E(Gk)|m(n), but we do not know how to prove this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' 22 GIULIANO BASSO AND YANNICK KRIFKA 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Proof of main results In this section we prove the main results from the introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Theo- rem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='2 is an immediate consequence of the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Let n0 ∈ N and let X0 be the complete metric space con- structed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Then X0 admits a conical bicombing σ and there is a finite subset A ⊂ X0 such that σ–conv(A) = X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Moreover, X0 is non-compact for every n0 ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' In the following, we retain the notation of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Recall that X0 = X, where (X, d) is the metric space associated to the semi-metric space (V, ϱ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' We set A = V1 ⊂ X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='4 tells us that m: X × X → X defines a conical midpoint map on X and X = � n∈N Mn(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Let σ be the conical bicombing on X0 induced by m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' For the construction of σ we refer to Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Because of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='5, it follows that σ–conv(A) = X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Let now n0 ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' To finish the proof we show that X0 is not compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' This is achieved by showing that X is not totally bounded, which in turn is established via Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Fix ε ∈ (0, 2−4) and choose k ≥ 1 sufficiently large such that max � 1 |Vk|, |E(Gk)| |Vk|(1+ε) � ≤ 1 (2α) 1 ε , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='1) for some large constant α > 0 to be determined later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' The existence of k is guaranteed by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' As in Section 4, we set m(n) = 2n−k and ¯n = n − k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' We claim that |E(Gm(n) n )| |Vn|2 ≤ �1 2 �m(n) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='2) for all n ≥ 1 sufficently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='4, there exists an integer K ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' , m(n)}, positive integers m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' , mK such that m1+· · ·+mK = m(n), and ki ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' , 2i − 1} for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' , ¯n, such that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='2) holds and |E(Gm(n) n )| ≤ 32m(n)� K � i=1 |E(Gmi k )| �� ¯n � i=1 |Vn−i|ki � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='3) In the following, we derive an upper bound for 1/|Vn|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Due to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='5), we have |Vn−1|2 |Vn| ≤ 3, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='4) and so we find that 1 |Vn|2 ≤ 32 |Vn−1|4 = 3b0 |Vn−1|k1 · 1 |Vn−1|b1 , A NON-COMPACT CONVEX HULL IN NON-POSITIVE CURVATURE 23 where b0 = 2 and b1 = 2b0 −k1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' We define the integers b0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' , b¯n recursively as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' We set b0 = 2, and bi = 2bi−1 − ki for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' , ¯n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Hence, by using (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='4) repeatedly, we arrive at 1 |Vn|2 ≤ � ¯n � i=1 3bi−1 |Vn−i|ki � 1 |Vk|b¯n (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='5) Via a straightforward computation, we find ¯n−1 � i=0 bi ≤ ¯n−1 � i=0 2i+1 ≤ 2¯n+1, b¯n = 2 · m(n) − ¯n−1 � i=0 k¯n−i · 2i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Hence, because of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='2), it follows that b¯n = m(n) + K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' By combining (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='3) with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='5), we obtain |E(Gm(n) n )| |Vn|2 ≤ αm(n) · �K i=1|E(Gmi k )| |Vk|2K 1 |Vk|m(n)−K , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='6) where α = 32 · 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' In the following, we consider the cases K ≤ (1 − ε)m(n) and K > (1 − ε)m(n) separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' First, we suppose that K ≤ (1 − ε)m(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' From (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='6), we find that |E(Gm(n) n )| |Vn|2 ≤ αm(n) · 1 |Vk|m(n)−K .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Since ε · m(n) ≤ m(n) − K, it follows from our assumption (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='1) on k that |E(Gm(n) n )| |Vn|2 ≤ αm(n) · � 1 (2α) 1 ε �ε·m(n) ≤ �1 2 �m(n) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Second, suppose that K > (1 − ε)m(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Since mi ≥ 1 and m1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' + mK = m(n), it follows that mj ≥ 2 for at most 2ε·m(n) many indices j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' To ease the notation, we may suppose m1 = · · · = mL = 1, where L = ⌈K − 2εm(n)⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Hence, using (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='6) once again, we find that |E(Gm(n) n )| |Vn|2 ≤ αm(n) · � |E(Gk)| |Vk|(1+ε) �L 1 |Vk|(1−2ε)m(n)−εL ≤ αm(n) · � 1 2α �(1−2ε)m(n)+(1−ε)L , where in the last inequality we used (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='1), our assumption on k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' By con- struction, L ≥ (1 − 3ε)m(n), and so we get (1 − 2ε)m(n) + (1 − ε)L ≥ (1 − ε)(2 − 5ε)m(n) ≥ m(n), where in the last step we used that ε ∈ (0, 2−4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Therefore, it follows from the above that |E(Gm(n) n )| |Vn|2 ≤ �1 2 �m(n) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' 24 GIULIANO BASSO AND YANNICK KRIFKA This concludes the case distinction and establishes (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Finally, having (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='2) at hand we find that lim inf n→∞ |E(Gm(n) n )| |Vn|2 = 0, since m(n) → ∞ as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' So Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='2 tells us that X is not totally bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Hence, X0 is not compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' □ A metric space Y is called injective if whenever A ⊂ B are metric spaces and f : A → Y a 1-Lipschitz map, then there exists a 1-Lipschitz extension ¯f : B → Y of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' More formally, Y is an injective object in the category of metric spaces with 1-Lipschitz maps as morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Injective metric spaces have been introduced by Aronszajn and Panitchpakdi in [2] and are some- times also called hyperconvex metric spaces by some authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' We refer to [16, 28] for an introduction to injective metric spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' As observed by Lang in [28, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='8], every injective metric spaces admits a conical bicomb- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Indeed, given an injective metric space Y , by applying Kuratowski’s embedding theorem, we may suppose that Y ⊂ Cb(Y ), and so because Y is injective, there is a 1-Lipschitz retraction r: Cb(Y ) → Y and thus σ(x, y, t) = r((1 − t)x + ty) defines a conical bicombing on Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Using an extension result of [3], we find that Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='2 is also valid for an injective metric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' There exists an injective metric space Y with a conical bi- combing σ such that there is a finite subset of Y whose closed σ-convex hull is not compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Let n0 ≥ 2 and let X0 be constructed as in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' We recall that by definition X0 = X and X is naturally equipped with a conical midpoint map m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Let σ denote the conical bicombing on X0 induced by m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' As m is symmetric, it is not difficult to see that σxy(t) = σyx(1−t) for all x, y ∈ X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' This shows that σ is a reversible conical bicombing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Hence, by virtue of [3, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='2], there exists an injective metric space Y containing X0, and a conical bicombing ˜σ on Y such that ˜σxy = σxy for all x, y ∈ X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' As X0 is complete, it follows that ˜σ–conv(A) = σ–conv(A) for any A ⊂ X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Therefore, due to Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='1, Y admits a finite subset whose closed ˜σ- convex hull is not compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' □ We finish this section by proving the following more general version of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Let n0 ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Then there exists a complete metric space X0 with a conical bicombing such that whenever A ⊂ Y is an n0-point subset of some complete metric space Y with a conical midpoint map m, then there exists a Lipschitz map Φ: X0 → Y with A ⊂ Φ(X0) and furthermore Φ(X0) is σ-convex with respect to the conical bicombing σ induced by m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' A NON-COMPACT CONVEX HULL IN NON-POSITIVE CURVATURE 25 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Let X0 = (X0, d) be the metric space constructed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' We set A0 = V1 ⊂ X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='2, it follows that d(x, y) = d1(x, y) = 1 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='7) for all distinct x, y ∈ A0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' In particular, A0 ⊂ X0 is an n0-point subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Now, let A be as in the statement of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Since A and A0 are both n0- point sets, there is a surjective map ϕ: A0 → A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Clearly, ϕ is L-Lipschitz for some L ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' We define L-Lipschitz maps ϕn : (Vn, ϱn) → Mn(A) recursively as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Because of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='7), it follows that ϕ1 = ϕ is L-Lipschitz with respect to ϱ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Given n ≥ 2 and x ∈ Vn, we set ϕn(x) = m � ϕn−1(a), ϕn−1(b) � if x = m(a, b) with a, b ∈ Vn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Let x, y ∈ Vn be such that x ∼ y in Gn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Hence, by definition, there is v ∈ Vn−1 and u ∼ w in Gn−1 such that x = m(v, u) and y = m(v, w), and so d(ϕn(x), ϕn(y)) = d � m(ϕn−1(v), ϕn−1(u)), m(ϕn−1(v), ϕn−1(w)) � ≤ 1 2d(ϕn−1(u), ϕn−1(w)) ≤ L · 1 2n−1 , where in the last step we have used that ϕn−1 is L-Lipschitz with respect to ϱn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Since ϱn = 2−(n−1) · dGn, it now follows directly from the above and the definition of the shortest-path metric dGn that ϕn is L-Lipschitz with respect to ϱn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' By construction, ϕn(x) = ϕm(x) for all x ∈ Vn and m ≥ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Hence, as Y is complete these maps naturally give rise to a L-Lipschitz map Φ: X0 → Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' To finish the proof we show that Φ(X0) is σ-convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' For simplicity, in the following we will denote the bicombings on X0 and Y both by σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' By construction of Φ and since σ is induced by a conical midpoint map, it follows that Φ(σ(x, y, t)) = σ(Φ(x), Φ(y), t) for all x, y ∈ Mn(A0) and all t ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Let now x, y ∈ X0 be arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Then there exists xk, yk ∈ Mnk(A0) such that xk → x and yk → y as k → ∞, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Moreover, σxkyk → σxy uniformly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Hence, as Φ is Lipschitz continuous, we have Φ(σ(x, y, t)) = σ(Φ(x), Φ(y), t) for all t ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' This shows that Φ(X0) is σ-convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Does X0 admits a consistent conical bicombing?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' In practice, it is often desirable to impose stronger properties on a bi- combing than (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' By asserting that a conical bicombing is consistent, see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='3 for the definition, one obtains an interesting class of bicombings which seem to be quite rigid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Following Haettel, we call a metric space a CUB-space if it admits a unique consistent conical bicombing (see [21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' The class of CUB-space is already quite rich and still growing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' For example, in [5] it is shown that any convex body in a dual Banach space is CUB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Moreover, proper, finite-dimensional injective metric space are CUB and Deligne complexes of certain Artin groups are CUB if they are re-metrized 26 GIULIANO BASSO AND YANNICK KRIFKA by considering the length metric induced by the ℓ∞-metric on each cell (see [11, 21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' However, using a non-affine isometry first introduced by Schechtman [38], one can construct a complete metric space with two distinct consistent coni- cal bicombings (see [5, Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' On the other hand, up to the author’s knowledge, there is no example of a metric space with a conical bicombing that does not also admit a consistent conical bicombing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' In other words, the following question of Descombes and Lang [11] is still open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Question 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='1 (Descombes–Lang).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Let X be a complete metric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Is it true that X admits a conical bicombing if and only if it admits a consistent conical bicombing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' This question also appears in the problem list [35, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' 385].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' A partial result that indicates a positive answer when X is proper has been obtained in [3, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' One difficulty in finding a negative answer to Ques- tion 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='1 lies in the fact that many know examples of metric spaces with a conical bicombing have locally a nice structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' In this situation one can then employ a generalized version of the Cartan-Hadamard theorem due to Miesch [32] to construct a consistent conical bicombing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' The metric space X0 is locally not ’nice’ as it is fractal-like in nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' So we believe that it could be a potential candidate for a counterexample to Question 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='1.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' 1, 383-417 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=', 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' [36] Anton Petrunin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Convex hull in cat(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' MathOverflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' 28 GIULIANO BASSO AND YANNICK KRIFKA [37] Anton Petrunin.' metadata={'source': 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MathOverflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' [39] Thilo Schlichenmaier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' A quasisymmetrically invariant notion of dimension and abso- lute Lipschitz retracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Doctoral thesis, ETH Zurich, Z¨urich, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' [40] Karl-Theodor Sturm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Probability measures on metric spaces of nonpositive curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' In Heat kernels and analysis on manifolds, graphs, and metric spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Lecture notes from a quarter program on heat kernels, random walks, and analysis on manifolds and graphs, April 16–July 13, 2002, Paris, France, pages 357–390.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Providence, RI: American Mathematical Society (AMS), 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' [41] Stefan Wenger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Isoperimetric inequalities of Euclidean type in metric spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Funct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=', 15(2):534–554, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Basso (basso@mpim-bonn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='mpg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='de) Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content=' Krifka (krifka@mpim-bonn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='mpg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} +page_content='de) Max Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE2T4oBgHgl3EQfWgfj/content/2301.03835v1.pdf'} diff --git a/StE0T4oBgHgl3EQfUwAJ/content/tmp_files/2301.02253v1.pdf.txt 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Received YYY; in original form ZZZ +ABSTRACT +The dynamical state and morphological features of galaxies and galaxy clusters, and their +high-redshift precursors, are tightly connected with their assembly history, encoding crucial +information about the formation and evolution of such cosmic structures. As a first step towards +finding an optimal indicator of the assembly state of observed structures, we use a cosmological +simulation of a moderate volume to critically examine the best definition of an indicator +that is able to discriminate dark matter haloes undergoing mergers and/or strong accretion +from haloes experimenting a relaxed evolution. Using a combination of centre offset, virial +ratio, mean radial velocity, sparsity and ellipticity of the dark matter halo, we study how the +thresholds on these parameters, as well as their relative weights, should evolve with redshift +to provide the best classification possible. This allows us to split a sample of haloes in a +totally relaxed, a marginally relaxed and an unrelaxed subsamples. The resulting classification +strongly correlates with the merging activity obtained from the analysis of complete merger +trees extracted from whole simulation data. The results on how the different indicators depend +on redshift and halo mass, and their optimal combination to better match the true assembly +history of haloes, could constitute relevant hints to find a suitable set of indicators applicable +to observational data. +Key words: large-scale structure of Universe – dark matter – galaxies: clusters: general — +methods: numerical +1 +INTRODUCTION +Deeply interwoven through a complex network of filaments and +sheets, dark matter (DM) haloes are bound, diffuse structures which +result from the gravitational collapse of the primordial density fluc- +tuations and a hierarchical merging history (Zel’dovich 1970; Press +& Schechter 1974; Gott & Rees 1975). DM haloes constitute the +fundamental building blocks of the large-scale structure (LSS) of +the Universe, and host their baryonic counterparts that we observe +over the electromagnetic spectrum (see, for instance, Planelles et al. +2015, for a review). At the galactic scale, the current theories of +galaxy formation typically assume DM haloes to be virialised (e.g., +White & Rees 1978), although this does not necessarily hold for +each galactic DM halo; while, at larger masses (at the galaxy cluster +scale), most DM haloes are still expected to be in the process of viri- +alisation, since they are the latest objects to have assembled (e.g., +Kravtsov & Borgani 2012, for a review on galaxy cluster formation). +However, the dynamical state of individual haloes is tightly +connected to their assembly history and, in particular, to the pres- +ence of mergers and the accretion rates in the last one or few dynam- +ical times. A merger or a period of intense accretion usually triggers +★ E-mail: david.valles-perez@uv.es +many morphological and dynamical disturbances in the halo (as- +phericity, higher velocity dispersions, abundance of substructures, +changes to the internal structure, etc.), which gradually fade away +once the assembly episode is over (see, for example, Poole et al. +2006 for a thorough analysis of the disturbances and the subsequent +relaxation after a merger event at cluster scales). +Since dynamically relaxed and disturbed structures often +present fundamentally different properties, a characterisation of the +dynamical state of the sample of cosmic structures is often a neces- +sary procedure in many analyses of very different natures, such as in +studies about the geometry of the cosmic web (Gouin et al. 2021), +statistical properties of the population of galaxy clusters (scaling +relations, mass functions, etc.; e.g., Chen et al. 2019; Seppi et al. +2021), hydrostatic mass bias (Nelson et al. 2014; Biffi et al. 2016; +Angelinelli et al. 2020), turbulence (Vazza et al. 2017; Valdarnini +2019; Vallés-Pérez et al. 2021a,b; Simonte et al. 2022), or galactic +environments (Kuchner et al. 2022), just to mention a few. +Even though we usually define haloes using the virial radius +prescription of Eke et al. (1996) and Bryan & Norman (1998), +based on the spherical collapse model, this does not imply that, in +general, three-dimensional haloes (non necessarily spherical, in a +full-cosmological, i.e., not isolated environment) defined this way +are necessarily in virial equilibrium. While in simulations one can +© 2023 The Authors +arXiv:2301.02253v1 [astro-ph.CO] 5 Jan 2023 + +2 +Vallés-Pérez et al. +access the whole temporal evolution of the objects, and thus recover +the assembly history of the halo under study in order to assess the +dynamical state, this is not possible in observations. Thus, for the +sake of a more direct comparison with observational works, simple +schemes for characterising the dynamical state using halo properties +at a given time are usually involved in many analyses. +For the time being, most works have relied on placing a thresh- +old on some halo property expected to correlate with the dynamical +state, in order to split the relaxed and unrelaxed subsamples. Per- +haps, the most direct of such indicators is the virial ratio, usually +defined as 𝜂 ≡ 2𝑇/|𝑊|, where 𝑇 is the intrinsic kinetic energy of +the halo and 𝑊 is its gravitational potential energy. 𝜂 would be +expected to be 1 for an isolated system in a steady state. However, +different works have found different thresholds to best suite their +particular classification (e.g., Shaw et al. 2006; Neto et al. 2007; +Knebe & Power 2008, see also the discussion in Cui et al. 2017). +Similarly, there is debate about the necessity of including a surface +tension term to account for the fact that haloes are not isolated +(Poole et al. 2006; Shaw et al. 2006; Knebe et al. 2011). Another +frequently used indicator, both in simulations and observations, is +the centre offset, which quantifies the departure from smoothness +and spherical symmetry of the matter distribution, and serves as an +indicator of substructure (Crone et al. 1996). In practice, however, +there are many possibilities regarding the choice of centres (see Cui +et al. 2016) and how to set the thresholds (cf. D’Onghia & Navarro +2007; Macciò et al. 2007). Additionally, in observations the centre +offset may depend crucially on the orientation, posing an additional +challenge. Last, other authors have used the fraction of mass in sub- +structures as a measure of the dynamical unrelaxedness of a DM +halo (e.g., Neto et al. 2007; cf. other recently suggested approaches, +e.g. Kimmig et al. 2022). While the election of this magnitude is +well-motivated, the mass contained in substructures in simulated +haloes depends critically on numerical resolution and the precise +definition of the substructure extent (see, e.g, the discussion in +Vallés-Pérez et al. 2022), making this criterion less comparable. +Since it is difficult that a single property can reflect the complex +picture of the dynamical state of a halo, many recent studies have +used combinations of these indicators, either by considering as +relaxed the haloes which simultaneously fulfil several relaxation +criteria (Neto et al. 2007; Biffi et al. 2016), or by defining some +combined indicator (Haggar et al. 2020; Zhang et al. 2021a; De +Luca et al. 2021). Finally, other metrics of the dynamical state are +based on the X-ray morphology, such as the centroid shift 𝜔 (Mohr +et al. 1993), or the power ratio, 𝑃3/𝑃0 (Buote & Tsai 1995, see also +the review of Rasia et al. 2013 on X-ray morphological estimators for +galaxy clusters); or more sophisticated ones such as those involving +Fourier analyses of the fluctuations in mass and X-ray maps (Cerini +et al. 2022), or the expansion of the Compton 𝑦-maps in Zernike +polynomials (Capalbo et al. 2021). +However, in most of the previous works, the parameters being +used and, especially, the thresholds imposed on them have been +tuned in a somewhat empirical way. This has lead to variations in +the criteria from work to work, even though the underlying idea is +kept. Furthermore, a possible redshift evolution of these thresholds +or of their very relevance has been devoted marginal attention, either +because the studies were focused on a particular cosmic epoch or +because it had been implicitly assumed that these criteria should +not evolve with redshift. +In this work, we intend to critically examine a set of possi- +ble indicators of the assembly state, all of which can be obtained +from the complete three-dimensional information in simulations, +and develop a criterion which accommodates redshift-dependent +thresholds and the possibility that different indicators have more or +less relevance at different cosmic epochs. We note the reader that, +while in the following we may refer to the dynamical state of haloes, +our main focus is oriented towards the dynamical disturbances asso- +ciated to the assembly history of haloes (i.e., the presence of merger +events or episodes of strong accretion; rather than a more general +sense of dynamical unrelaxedness which could include, e.g., the +presence of substructures even when they are not associated to a +merger episode, since they have an impact on properties such as the +hydrostatic equilibrium). +The rest of the manuscript is organised as follows. In Sec. +2, we introduce our simulation, halo sample and the methodology +that we employ for setting the thresholds and relative weights of +the different dynamical state indicators. Our resulting criterion is +presented in Sec. 3, including the analysis of the mass dependence +of our results and a validation of our method with a different sim- +ulation. Finally, we discuss the applicability of our results in Sec. +4. Appendix A contains the fitting formulae for the thresholds and +weights applicable for massive haloes. +2 +METHODS +The results reported in this paper have been extracted from the +analysis of a ΛCDM cosmological simulation tracking the coupled +evolution of baryons and DM. We describe the relevant details of the +simulation in Sec. 2.1, then cover the halo catalogues and merger +tree elaboration in Sec. 2.2, and discuss how do we compute the +dynamical state indicators in Sec. 2.3. Finally, we introduce our +classification strategy in Sec. 2.4. +2.1 +The simulation +The haloes we analyse in this paper are extracted from a numerical +simulation run with MASCLET (Quilis 2004; Quilis et al. 2020), a +(magneto-)hydrodynamics and 𝑁-Body code primarily designed for +cosmological applications. For evolving the DM component, which +is the primary focus of this work, MASCLET implements a multilevel +Particle-Mesh (PM) scheme (Hockney & Eastwood 1988), which +takes advantage of the adaptive-mesh refinement (AMR) strategy +(Berger & Colella 1989) to gain spatial, temporal and force resolu- +tion. +We have simulated a periodic, cubic (𝐿 = 100 ℎ−1 Mpc) do- +main, under the assumption of a flat, ΛCDM cosmology specified +by the matter density parameter Ω𝑚 = 0.31 (ΩΛ = 1 − Ω𝑚), +baryon density parameter Ω𝑏 = 0.048, and Hubble parameter +ℎ ≡ 𝐻0/(100 km s−1) = 0.678. The initial conditions stem from +a realisation of the primordial gaussian random field assuming a +spectral index 𝑛𝑠 = 0.96 and an amplitude yielding 𝜎8 = 0.82, +and are set up at redshift 𝑧ini = 100 using a CDM transfer function +(Eisenstein & Hu 1998). The values selected for the cosmological +parameters are consistent with the latest results reported by Planck +Collaboration et al. (2020). +A first simulation is run at low resolution, using a fix grid of +𝑁3𝑥 = 2563 cells and the same number of equal-mass particles. This +is used to identify the Lagrangian regions in the initial conditions +which will evolve into dense structures by 𝑧 = 0, and mapping +them with enhanced numerical resolution already at 𝑧ini. We use +this approach to establish three nested levels of initial conditions, +resulting in a best mass resolution of 1.48 × 107 𝑀⊙. +Using these high-resolution initial conditions, the simulation +MNRAS 000, 1–15 (2023) + +On the assembly state of DM haloes +3 +is evolved again using AMR based on gas/DM overdensities, con- +verging flows, and Jeans length criteria, achieving a peak resolution +of Δ𝑥8 = 2.3 kpc at the maximum (ℓ = 𝑛ℓ ≡ 8) level of refinement. +While the baryonic component is not the primary focus of this work, +the simulation includes gas cooling, but no other baryonic effect or +feedback mechanism. +2.2 +Halo catalogue and merging history +For each snapshot of the simulation, we have identified the DM +haloes using the public halo finder ASOHF (Planelles & Quilis 2010; +Knebe et al. 2011; Vallés-Pérez et al. 2022)1, which is based on +the spherical-overdensity definition and uses the virial radius (ac- +cording to the prescription of Bryan & Norman 1998) to delimit the +extent of the haloes that are not substructure. +After determining the halo catalogues, these are linked in be- +tween snapshots using the merger tree code presented by Vallés- +Pérez et al. (2022, their section 2.6.2), which identifies all the haloes +at a given code output which have contributed to an object in a fol- +lowing one, allowing to skip an arbitrary number of snapshots, if +necessary. Using it, we determine the main evolutionary line of each +halo, as well as the presence and characterisation of mergers. +Following Planelles & Quilis (2009), Chen et al. (2019), and +Vallés-Pérez et al. (2020), we have classified each merger event in +the sample as either a major merger (if the mass ratio, 𝑀min/𝑀max, +between the two haloes involved exceeds 1/3), or a minor merger +(1/3 > 𝑀min/𝑀max ≥ 1/10). Mergers below a mass ratio of 1/10 +are disregarded. The merger time is determined as the moment in +which the centre of the infalling (the least massive) halo crosses the +virial boundary of the host (the most massive) halo. +2.2.1 +Fiducial classification: assembly history of the haloes +In order to determine the optimal thresholds on the dynamical state +indicators (see below, Sec. 2.3 and therein), we compare with a +reference, or fiducial, classification of the dynamical state based +on the full assembly history of haloes (i.e., the presence of past or +ongoing mergers, as well as the accretion rates). +As a tentative classification of the unrelaxedness induced by +a merger event, we will assume that a typical halo remains in a +disturbed state for one dynamical time after a major merger, or half +a dynamical time after a minor merger, with the dynamical time +𝜏dyn being defined as +𝜏dyn(𝑧) ≡ +1 +√︁ +𝐺𝜌 += +1 +√︁ +𝐺𝜌𝐵(𝑧)Δvir(𝑧) +, +(1) +with 𝐺 the gravitational constant, 𝜌 the density of the halo, 𝜌𝐵(𝑧) +the background matter density and Δvir(𝑧) the virial overdensity +(Bryan & Norman 1998). +While the choice of the timespan is a crude approximation, it +responds to the fact that many works have shown that the disturbance +triggered by a minor merger is, in general terms, smaller than the +effect of a major merger, both for the dark and for the baryonic +components (Planelles & Quilis 2009; Yu et al. 2014; Vallés-Pérez +et al. 2020; Zhang et al. 2021b). In practical terms, since 𝜏dyn(𝑧) +varies strongly with redshift and reaches considerable fractions of +the age of the Universe, especially at low redshift, we choose to +1 https://github.com/dvallesp/ASOHF. +0 +1 +2 +3 +4 +5 +Redshift, z +1012 +1013 +1014 +1015 +Mass, Mvir (M +) +Median +Mean +(16% +84%) CI +(2.5% +97.5%) CI +Min & max +Figure 1. Evolution of the distribution of halo masses in our sample. The +solid line indicates the median mass of the sample, with the dark and light +shaded regions enclosing the 16% − 84% (dark blue) and 2.5% − 97.5% +(light blue) confidence intervals (CIs) around it, respectively. The dashed +line corresponds to the mean mass, while the dotted lines correspond to the +maximum and minimum masses. +define the number of dynamical times between two moments, 𝑡1 +and 𝑡2, as in Jiang & van den Bosch (2016) and Wang et al. (2020): +𝑁𝜏 (𝑡1, 𝑡2) = +∫ 𝑡2 +𝑡1 +d𝑡 +𝜏dyn(𝑧) +(2) +Additionally, it might be the case that a halo is accreting +strongly, but without undergoing any significant merger (either phys- +ically or due to the finite resolution of a simulation). Thus, we also +consider as unrelaxed, for the purpose of the fiducial classification, +any halo which has assembled more than 50% of their mass in the +last dynamical time. +For the analyses within this work, all the 28 snapshots of the +simulation since redshift 𝑧 = 5 are considered. We select the 1000 +most massive haloes at each epoch, and discard all those which +cannot be reliably traced back in time for at least one 𝜏dyn(𝑧). +We show, in Fig. 1, the redshift evolution of the median mass in +the sample (solid line), together with shaded regions enclosing the +confidence intervals corresponding to the 16% − 84% (dark blue) +and 2.5% − 97.5% (light blue) percentiles of the distribution of +masses. The dotted lines mark the maximum mass (upper line) +and the minimum mass, or mass limit (lower line) in the sample +at each time. Thus, the mass limit in our sample evolves from +∼ 1012𝑀⊙ at 𝑧 = 5 to ∼ 4.5 × 1012𝑀⊙ at 𝑧 = 0. The wide redshift +interval considered in this study includes from the cluster-, group- +and massive galaxy-sized haloes at 𝑧 ≃ 0, to the DM counterpart +of galaxies and the progenitors of low-redshift clusters at the high- +redshift end. +The results of the fiducial classification are summarised in Fig. +2, where we show the number of haloes which are finally considered +at each snapshot (blue line, referring to the axis on the left). Only +at high redshift (𝑧 ≳ 3), a large fraction (10% to 25%) of the pre- +liminary haloes get discarded because they cannot be tracked back +in time for at least one dynamical time. The fraction of unrelaxed +MNRAS 000, 1–15 (2023) + +4 +Vallés-Pérez et al. +0 +1 +2 +3 +4 +5 +Redshift, z +0 +200 +400 +600 +800 +1000 +Number of haloes +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Fraction of haloes +Unrelaxed + Merging + Accreting +Figure 2. Fiducial classification of dynamical states of the halo sample. The +blue line represents the number of haloes selected per snapshot, according to +the left axis. The dashed/dotted lines, according to the right axis, describe the +evolution with redshift of the fraction of unrelaxed haloes (green), which can +have been labelled as such due to mergers (purple) or strong accretion/mass +growth (orange). +haloes according to the fiducial classification (green, dashed line; +referring to the axis on the right) varies from ∼ 80% to ∼ 30% +through the considered redshift interval. Purple and orange, dotted +lines show the number of haloes, as a fraction of the total, which are +unrelaxed due to either the condition on recent mergers or the con- +dition on the accretion rate, respectively. Most of the low-redshift +haloes which are labelled unrelaxed are undergoing mergers, while +at high redshift the cause for unrelaxedness is more usually a high +level of smooth accretion. This may be due to several reasons, +amongst which we can mention the higher density in the vincinity +of haloes at high redshift, or resolution limitations of the simulation +(i.e., at high redshift, a halo may be accreting small, underresolved +structures, which are therefore not accounted as mergers). +2.3 +Indicators for the assembly state +Many possible proxies for the dynamical and assembly state of a DM +halo, or their corresponding baryonic structure (e.g., a galaxy or a +galaxy cluster) have been proposed in the literature (see, for instance, +Cole & Lacey 1996; Crone et al. 1996; Shaw et al. 2006; Haggar +et al. 2020; Zhang et al. 2021a, just to cite a few). While simulations +allow to access the complete three-dimensional picture, the lack of +the whole information in observations (due to, e.g., projection or +the inability to observe the dark component, or even the plasma out +to large radii) requires that, generally, different quantities are used +for assessing the dynamical state in simulations and in observations +(Rasia et al. 2013; De Luca et al. 2021; Yuan et al. 2022). While +comparison with observations is crucial and will be dealt with in +future work, here we shall focus on the dynamical state indicators +extracted from the complete, three-dimensional data in simulations +as a first step. Unless otherwise specified, all quantities below are +referred to the virial volume. +Centre offset. The centre offset is usually defined as the distance +between two different choices of centre, in units of some aperture +radius (typically, the virial radius of the halo, 𝑅vir). Many exam- +ples for the choices of centre pair exist in the literature, such as +centre of mass (CM) vs. density peak (Baldi et al. 2017) or CM +vs. potential minimum (Biffi et al. 2016), extracted from the three- +dimensional description in simulations, or the morphological offset +of the BCG location vs. X-ray surface brightness peak (Rossetti et al. +2016), amongst many others. We address the interested reader to +Cui et al. (2016), who compare many different choices of observable +for defining the centre of galaxy clusters. +In this work, we have tested the three possible combinations +between the minimum of gravitational potential (defined as the +location of the most-bound DM particle, as obtained by ASOHF and +described in detail in Vallés-Pérez et al. 2022), the DM density peak, +and the DM centre-of-mass. We find that the most robust results are +obtained for the Peak-CM pair. Therefore, we defined the centre +offset parametre as: +Δ𝑟 = +��𝒓peak,DM − 𝒓CM,DM +�� +𝑅vir +(3) +Virial ratio. For a gravitational system in steady state, the virial +theorem predicts 2𝑇 + 𝑊 − 𝐸𝑠 = 0, where 𝑇 is the kinetic energy, +𝑊 is the gravitational binding energy, and 𝐸𝑠 is the surface energy +term (Chandrasekhar 1961). Neglecting the surface term, the virial +ratio is usually defined as +𝜂 ≡ 2𝑇 +|𝑊| , +(4) +and it is expected that 𝜂 → 1 for isolated systems. However, haloes +are not generally isolated systems, and therefore there is not a good +a priori reason to drop the surface term in the virial theorem. Thus, +many works define the virial ratio as 𝜂′ = (2𝑇 − 𝐸𝑠)/|𝑊| (Shaw +et al. 2006), while others claim that the surface term overcorrects the +virial ratio (Power et al. 2012). As the latter, we find that correcting +the virial ratio by the surface term wipes out the correlation with +merging activity, and thus we shall use the definition in Eq. 4 in the +remainder of this work. +Mean radial velocity. In a relaxed object, we do not expect impor- +tant changes in the radial structure, while an unrelaxed system will +experience significant disturbances as it settles down to equilibrium. +This motivates the consideration of the mean radial velocity of DM +particles, +⟨𝑣𝑟⟩DM = +� +𝑖 𝑚𝑖𝑣𝑟,𝑖 +� +𝑖 𝑚𝑖 +, +(5) +being 𝑚𝑖 the mass of the 𝑖-th DM particle, and 𝑣𝑟,𝑖 its radial velocity +relative to the halo reference frame. In practical terms, we scale this +quantity by the circular velocity at the virial radius, 𝑉circ,vir ≡ +√︁ +𝐺𝑀vir/𝑅vir, and define the corresponding normalised indicator +as +⟨ � +𝑣𝑟⟩DM ≡ |⟨𝑣𝑟⟩DM| +𝑉circ,vir +. +(6) +Sparsity. Systems which have experienced recent significant merg- +ers tend to display shallower central density profiles due to the +MNRAS 000, 1–15 (2023) + +On the assembly state of DM haloes +5 +disturbance caused by the infalling halo, and thus are less concen- +trated. Many works (e.g., Neto et al. 2007; Wang et al. 2020) have +pointed out the relation between the time spanned since the last +major merger and halo concentration, 𝑐vir = 𝑅vir/𝑅s, being 𝑅s the +scale radius of the Navarro et al. (1997) profile (or the radius where +the logarithmic slope of the DM density profile equals −2). +More recently, sparsity has been suggested as a non-parametric +alternative to concentration, which reduces the scatter with halo +mass (Balmès et al. 2014; Corasaniti et al. 2018), and has also been +found to correlate with the timing since the last relevant merger +(Richardson & Corasaniti 2022). While sparsity is generally defined +as the quotient between the masses at different spherical overdensi- +ties, we find that the one maximising the correlation with merging +activity is +𝑠200𝑐,500𝑐 ≡ 𝑀200𝑐 +𝑀500𝑐 +. +(7) +Ellipticity. DM haloes are generally triaxial (Frenk et al. 1988; +Knebe & Wießner 2006), with significant scatter in halo shape at a +given mass and redshift. Many recent studies have pointed out at the +correlation between triaxiality and/or ellipticity of the halo shape +and the formation history of a halo, with relaxed haloes tending to +be rounder (Chen et al. 2019; Lau et al. 2021). +We define the overall shape of the DM halo by finding the +eigenvalues of the shape tensor, defined as +𝑆𝛼𝛽 = +∑︁ +𝑖 +𝑚𝑖 +𝑟𝑖,𝛼𝑟𝑖,𝛽 +𝑟2 +𝑖 +, +(8) +which are proportional to the semiaxes squared. The positions, 𝒓𝑖, +are relative to the cluster centre (defined as the location of the density +peak), and we choose to normalise them to be unit length to prevent +the shape to be dominated by the particles in the outskirts of the +halo. Note this corresponds to the E2 method introduced by Zemp +et al. (2011). If 𝑎, 𝑏 and 𝑐 are the semiaxes sorted in non-increasing +order, we define the ellipticity of the halo, 𝜖, as: +𝜖 = 1 − 𝑐 +𝑎 . +(9) +Other indicators not considered in this work. Amongst the most +widely used proxies for the dynamical state of DM haloes in the +literature, we have not included the fraction of substructures, 𝑓sub, +in this study (neither defined as the mass in substructures as a +fraction of the host mass, nor as the ratio between the mass of +the heaviest substructure and the host mass, as in Cialone et al. +2018). While 𝑓sub should naturally correlate with the assembly +state (especially, with the merging state), its interpretation is very +subtle due to several factors. First, there is not a unique way to +define the extent of a subhalo, and differences amongst halo finders +have a dramatic impact on the recovered masses of substructures +(see Vallés-Pérez et al. 2022, their figures 5 and 10). In second +place, the amount of substructure produced in simulations depends +strongly, not only on resolution, but also on the numerical scheme +employed to solve gravity. This introduces strong mass biases (while +the most massive haloes in our simulation may host well-resolved +substructure, haloes with less than a few ten thousands particles +are likely to be substructure-deficient. These obscure dependencies +with mass, resolution and numerical scheme limit our ability to +consistently incorporate this indicator in our work. Simulations with +enhanced resolution, capable of fully resolving rich substructure in +Table 1. Summary of the redshift binning considered for the subsequent +analyses. Each bin contains the haloes extracted from the 𝑁snaps available +with 𝑧 ∈ [𝑧min, 𝑧max]. The mean redshift of the 𝑁haloes haloes in the bin +is ¯𝑧, with a fraction 𝑓unrelaxed of them being unrelaxed (either merging or +experiencing intense accretion) according to the fiducial classification. Note +we report ¯𝑧, instead of the median, because 𝑧 is not continuously distributed +(at each redshift bin, there are only 𝑁snaps different values of 𝑧). +𝑧min +𝑧max +𝑁snaps +¯𝑧 +𝑁haloes +𝑓unrelaxed +0 +0.2 +4 +0.084 +3828 +0.309 +0.2 +0.5 +4 +0.381 +3830 +0.360 +0.5 +0.75 +3 +0.651 +2890 +0.406 +0.75 +1.0 +3 +0.897 +2871 +0.468 +1.0 +1.5 +4 +1.253 +3769 +0.512 +1.5 +2.0 +3 +1.808 +2788 +0.562 +2.0 +3.0 +3 +2.536 +2742 +0.600 +3.0 +4.0 +2 +3.350 +1791 +0.657 +4.0 +5.0 +2 +4.443 +1564 +0.751 +our wide range of masses could be able to overcome this limitation +of our work. +Regarding the indicators describing the shape of the mass dis- +tribution, while 𝜖 alone does not fully characterise the shape of an +ellipsoid, we have not considered any additional parameter, such as +triaxiality𝑇 ≡ 𝑎2−𝑏2 +𝑎2−𝑐2 (Franx et al. 1991). While ellipticity measures +directly the deviation from sphericity, which is expected during as- +sembly episodes, the same is not true for triaxiality. As a matter +of fact, triaxiality is undefined for spherical objects, and we do +not find a clear reason to have a preference towards either prolate- +ness/oblateness during mergers or strong accretion periods. +2.4 +Classification strategy +2.4.1 +Redshift binning +A total of 28 snapshots of the simulation, since 𝑧 = 5, are saved and +used in this analysis. To augment the statistics, we have grouped the +snapshots in several redshift bins, which are described in Table 1.2 +2.4.2 +Optimising the thresholds +In a first step, we place a threshold, 𝑋thr +𝑖 , for each of the dynamical +state indicators, 𝑋𝑖, described in the previous section (𝑖 = 1, . . . , 5, +for the five dynamical state indicators). This is performed indepen- +dently at each redshift bin. To do so, we vary 𝑋thr +𝑖 +from the minimum +to the maximum value of 𝑋𝑖 through the sample, and identify how +well does 𝑋thr +𝑖 +separate the relaxed and the unrelaxed samples of +the fiducial classification. +For each value of 𝑋thr +𝑖 , we compute two complementary met- +2 Not all bins contain the same number of snapshots (or haloes): higher +redshift bins comprise less snapshots. While this may increase the scatter +in our results at high redshift, grouping more snapshots together at high +redshift would increase the systematic uncertainty due to stacking objects +of more different epochs. +MNRAS 000, 1–15 (2023) + +6 +Vallés-Pérez et al. +rics of the goodness of the classification3, namely the efficiency in +discriminating the unrelaxed haloes, +𝜖unrelaxed(𝑋thr +𝑖 ) = # of unrelaxed haloes properly identified +# of unrelaxed haloes (fiducial) +(10) +and the efficiency in discriminating the relaxed haloes, +𝜖relaxed(𝑋thr +𝑖 ) = # of relaxed haloes properly identified +# of relaxed haloes (fiducial) +. +(11) +Out of all the possible values of 𝑋thr +𝑖 , we choose the one which +maximises the product of both metrics, that is to say: +ˆ𝑋thr +𝑖 += argmax𝑋thr +𝑖 +� +𝜖unrelaxed(𝑋thr +𝑖 ) · 𝜖relaxed(𝑋thr +𝑖 ) +� +. +(12) +Since 𝜖unrelaxed (𝜖relaxed) can be thought, in a frequentist ap- +proach, as the probability of correctly identifying an unrelaxed (re- +laxed) halo as such, our choice of threshold in Eq. 12 corresponds +to picking the one which enhances the likelihood of correctly clas- +sifying both an unrelaxed and a relaxed halo, and thus serves as a +compromise between too generous and too stringent thresholds. +2.4.3 +Totally relaxed, marginally relaxed and disturbed haloes +Once the final (redshift-dependent) thresholds, +� +𝑋thr +𝑖 +(𝑧) +�5 +𝑖=1, are +established, any halo will be regarded as totally relaxed if +𝑋𝑖 < 𝑋thr +𝑖 +(𝑧) +∀𝑖 = 1, . . . , 5, +(13) +that is, if it has a low value of all the dynamical state indicators (low +centre offset, mean radial velocity and ellipticity, virial ratio and +sparsity close to unity). This allows a very conservative definition +of the most relaxed haloes. +However, it may be the case that a halo has a high value of one +of the parametres, but is relaxed according to the rest. This might +be the case for a variety of reasons, ranging from physical (e.g., a +halo with high ellipticity due to a strong tidal field generated by the +surrounding large-scale structure; Chen et al. 2016) to numerical +(e.g., underresolved haloes with higher sparsities, misidentification +of the centre, etc.). Thus, we deal with all haloes not falling into +the totally relaxed category by defining a combined relaxedness +indicator, in the manner of Haggar et al. (2020) (see also Kuchner +et al. 2020; Zhang et al. 2021a; Gouin et al. 2022) but adding weights +which account for the fact that some dynamical state indicators can +be more insightful than others at any given particular epoch. +𝜒 = +������ +𝑤1 +� Δ𝑟 +Δthr +𝑟 +�2 ++ 𝑤2 +� 𝜂 − 1 +𝜂thr − 1 +�2 ++ 𝑤3 +� +⟨ � +𝑣𝑟⟩DM +⟨ � +𝑣𝑟⟩thr +DM +�2 ++ +𝑤4 +� +𝑠200𝑐,500𝑐 − 1 +𝑠thr +200𝑐,500𝑐 − 1 +�2 ++ 𝑤5 +� 𝜖 +𝜖thr +�2������ +−1/2 +(14) +The weights, {𝑤𝑖}5 +𝑖=1, are normalised so that �5 +𝑖=1 𝑤𝑖 = 1, and +3 Note that the metrics introduced in Eqns. 10 and 11 also correspond, +respectively, to the True Positive Rate (TPR) or sensitivity, and the True +Negative Rate (TNR) or specificity in the usual jargon of binary classifica- +tions (e.g., Fawcett 2006). However, we choose this notation here for better +readability. +are fixed at each redshift bin to be proportional to the performance +of their corresponding indicator in splitting the merging and non- +merging subsamples of the fiducial classifications. In particular, +we set 𝑤𝑖 ∝ 𝜖relaxed𝜖unrelaxed − 0.25 (the absolute values being set +by the closure relation �5 +𝑖=1 𝑤𝑖 = 1). If, at a given redshift bin, +𝜖relaxed𝜖unrelaxed ≤ 0.25, we consider that the particular indicator is +not meaningful and its weight is set to 𝑤𝑖 = 0. +A particular halo which does not belong to the totally relaxed +category will be classified as marginally relaxed if 𝜒 ≥ 1, and dis- +turbed whenever 𝜒 < 1. Additionally, this classification scheme can +naturally handle missing data. For instance, if 𝑠200𝑐,500𝑐 is missing +(e.g., due to a low resolution not enabling to resolve 𝑅500𝑐), one +can simply evaluate 𝜒 neglecting the sparsity term (and multiplying +𝜒 by a factor √1 − 𝑤4; or, alternatively, renormalising the weights +after setting 𝑤4 = 0). +2.4.4 +Redshift evolution of the thresholds and weights +With the procedure outlined in Sec. 2.4.2 and 2.4.3, we obtain a +threshold and a weight for each dynamical state indicator at each of +the redshift bins specified in Table 1. In order to obtain a continu- +ous trend for each of these parameters, we fit them to polynomial +functions of arbitrary degree. +First, we estimate the uncertainties in the thresholds (𝑋thr +𝑖 ) and +weights (𝑤𝑖) by computing the standard deviation of the distribution +of these parametres obtained in 1000 bootstrap resamplings (Efron +1979). Then, we fit the redshift evolution of the given parameter to +polynomial functions of increasing degree, until the 𝑝-value of the +highest degree coefficient falls above 𝑝 = 0.046 (low significance), +or the reduced chi-squared falls below 1 (indicating possible overfit- +ting of the model). Fits are performed using least squares weighted +to the inverse of the variance of each data point. +3 +RESULTS +Following the procedure described in Sec. 2.4.2 and 2.4.4 over the +whole sample, we have found the optimal thresholds for the dynam- +ical state indicators, and fitted them to the best possible polynomial +models. The results are shown in Fig. 3, from top to bottom, for the +centre offset, virial ratio, mean radial velocity, sparsity and elliptic- +ity thresholds. +Most of the thresholds on the assembly state indicators present +a clear redshift evolution. At earlier times, the thresholds on the +dynamical state indicators tend to take higher values, reflecting the +fact that haloes at earlier times were more irregular or exhibited +more disturbed features, even when not having experienced any +relevant merging activity or growth during the last dynamical time. +The evolution of the thresholds ranges from very mild or al- +most nonexistent (e.g., Δthr +𝑟 , 𝜀thr) to noticeable (and definitely worth +taking into account; e.g., 𝜂thr, 𝑠thr +200𝑐,500𝑐, ⟨ � +𝑣𝑟⟩thr). This unequivo- +cally evidences that fixed, set thresholds on certain parameters may +not be able to correctly discriminate relaxed from merging haloes +through the whole evolutionary history of the objects, especially +when delving into the realm of high-redshift haloes. +The thresholds can be fitted by the following equations (solid +lines in Fig. 3, whose uncertainties are represented by the shaded +regions), valid for 0 ≤ 𝑧 ≤ 5, where the figures in parentheses cor- +respond to the uncertainty in the two last digits of each coefficient: +Δthr +𝑟 (𝑧) = 0.0849(13) +(15) +MNRAS 000, 1–15 (2023) + +On the assembly state of DM haloes +7 +0.075 +0.080 +0.085 +0.090 +0.095 +0.100 +0.105 +thr +r (z) +1.4 +1.5 +1.6 +1.7 +thr(z) +0.07 +0.08 +0.09 +0.10 +0.11 +0.12 +vr thr(z) +1.50 +1.55 +1.60 +1.65 +1.70 +1.75 +1.80 +sthr +200c, 500c(z) +0 +1 +2 +3 +4 +5 +0.25 +0.26 +0.27 +0.28 +thr(z) +0 +1 +2 +3 +4 +5 +Redshift, z +Figure 3. Redshift evolution of the thresholds on the dynamical state indi- +cators. From top to bottom, the panels refer to the centre offset (Δthr +𝑟 ), virial +ratio (𝜂thr), mean radial velocity (⟨� +𝑣𝑟 ⟩thr +DM), sparsity (𝑠thr +200𝑐,500𝑐), and ellip- +ticity (𝜀thr) thresholds. Dots correspond to the optimal threshold obtained +within the redshift bin, with the error bars obtained by means of bootstrap +resampling. Solid lines correspond to the best polynomial fits, with their +(16-84)% confidence interval as the shaded region. +𝜂thr(𝑧) = 1.3383(56) + 0.197(11)𝑧 − 0.0276(32)𝑧2 +(16) +⟨ � +𝑣𝑟⟩thr +DM(𝑧) = 0.0718(22) + 0.0056(14)𝑧 +(17) +𝑠thr +200𝑐,500𝑐(𝑧) = 1.491(16)+0.064(37)𝑧−0.031(22)𝑧2+0.0060(35)𝑧3 +(18) +𝜀thr(𝑧) = 0.2696(27) +(19) +Based on the performance of each assembly state indicator in +matching the fiducial classification, we fix the weights of each indi- +cator in Eq. 14 as described in Sec. 2.4.3. The results are summarised +in Fig. 4, which is analogous to Fig. 3 but this time showing the +weights instead of the thresholds. Note that, if all indicators were +equivalently important, 𝑤𝑖 = 0.2∀𝑖. Thus, 𝑤𝑖 > 0.2 (𝑤𝑖 < 0.2) +implies above-average (below-average) performance for the given +dynamical state indicator at the given epoch. +Interestingly, the importance of each indicator in determining +the dynamical state of DM haloes varies strongly with redshift. For +example, one of the most widely used indicators, the centre offset +Δ𝑟, is exceedingly effective in discriminating the disturbed haloes at +high redshift, but its effectiveness declines steeply with decreasing +redshift and has slightly below-average performance at 𝑧 ≃ 0. As +an example of the opposite trend, the virial ratio, 𝜂, appears to be +irrelevant at high redshift (𝑧 ≳ 2), and is only useful at low redshifts +(≲ 1). This dissimilar behaviour between centre offset and virial +ratio is also reported by the analyses at high redshift of Davis et al. +(2011). +As a purely dynamical parameter, the mean radial velocity +⟨ � +𝑣𝑟⟩ is especially relevant at high redshift, likely due to the fact that +smooth (nearly radial) accretion could be more important at these +stages given the relatively higher density in the surroundings of the +halo. Sparsity, as well as ellipticity, are especially correlated with +the fiducial dynamical state classification at more recent redshifts, +although they cannot generally be neglected at any epoch. As a +matter of fact, at low redshift, 𝜀 is the most relevant indicator of the +dynamical state of haloes. +With the same procedure as above, we have fitted the weights +to polynomial functions capturing their evolution (solid lines in Fig. +4, whose uncertainties are represented by the shaded regions), valid +for 0 ≤ 𝑧 ≤ 5: +𝑤[Δ𝑟](𝑧) ∝ 0.1679(70) + 0.0423(50)𝑧 +(20) +𝑤[𝜂](𝑧) ∝ 0.1965(78) − 0.1037(60)𝑧 + 0.0134(11)𝑧2 +(21) +𝑤[⟨ � +𝑣𝑟⟩DM](𝑧) ∝ 0.1370(70) + 0.0364(48)𝑧 +(22) +𝑤[𝑠200𝑐,500𝑐](𝑧) ∝ 0.2327(97) +0.051(14)𝑧 −0.0153(38)𝑧2 (23) +𝑤[𝜀](𝑧) ∝ 0.2603(75) − 0.0181(51)𝑧 +(24) +We note that, while at any epoch the data points fulfilled +MNRAS 000, 1–15 (2023) + +8 +Vallés-Pérez et al. +0.15 +0.20 +0.25 +0.30 +0.35 +0.40 +w[ +r](z) +0.00 +0.05 +0.10 +0.15 +0.20 +w[ ](z) +0.15 +0.20 +0.25 +0.30 +0.35 +w[ vr ](z) +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +w[s200c, 500c](z) +0 +1 +2 +3 +4 +5 +0.10 +0.15 +0.20 +0.25 +0.30 +w[ ](z) +0 +1 +2 +3 +4 +5 +Redshift, z +Figure 4. Redshift evolution of the weights on the dynamical state indicators. +From top to bottom, the panels refer to the centre offset (𝑤 [Δ𝑟 ]), virial +ratio (𝑤 [𝜂]), mean radial velocity (𝑤 [⟨ ˜𝑣𝑟 ⟩]), sparsity (𝑤 [𝑠200𝑐,500𝑐 ]), +and ellipticity (𝑤 [𝜀]) weights. Dots correspond to the weights obtained +within the redshift bin, with the error bars obtained by means of bootstrap +resampling. Solid lines correspond to the best polynomial fits, with their +(16-84)% confidence interval as the shaded region. +0 +1 +2 +3 +4 +5 +Redshift, z +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +0.35 +0.40 +Relative weights +r +vr +s200c, 500c +Figure 5. Overall fitted redshift evolution of the weights of the dynamical +state indicators, using the complete sample. +�5 +𝑖=1 𝑤𝑖 = 1, this is not necessarily true for the fitting polynomials +evaluated at any arbitrary redshift (although it holds to a few per- +cents). Thus, they must be normalised by their sum before plugging +them into Eq. 14. +For better comparison of the relative importance of each of the +indicators, Fig. 5 presents the fits of the weights on each dynamical +state indicator, as a function of decreasing redshift. The behaviour +can be roughly summarised as: +• At high redshift (𝑧 ∈ [2, 5]), the centre offset provides the most +insightful information about the recent assembly activity. This can +be primarily complemented by the mean radial velocity (especially +at 𝑧 ≳ 3), or sparsity and ellipticity (at 𝑧 ≲ 3). The virial ratio +does not seem to provide any insight on the dynamical state at high +redshift. +• At intermediate redshifts, (𝑧 ∈ [1, 2]), sparsity, ellipticity and +centre offset provide similarly useful information about the dynam- +ical state. The relevance of the virial ratio is still limited at this +epoch. +• At low redshifts (𝑧 ≲ 1), the ellipticity of the DM halo corre- +lates exceptionally well with the dynamical state, as well as sparsity +does. While no dynamical state indicator is negligible at this stage, +centre offset and virial ratio also present reasonable performances, +while mean radial velocity is the least useful indicator at this time. +3.1 +Dependence on halo mass +The previous analyses have considered all haloes on an equal +footing, despite their broad distribution in masses. Invoking self- +similarity (see, e.g., Navarro et al. 2010 for a thorough analysis on +the level of self-similarity of haloes), it may be argued that the same +thresholds and weights could be used for all halo masses. How- +ever, the fact that many halo properties (related to the dynamical +state indicators in our work) scale with mass demands to explicitly +check how our results depend on the mass scale of the haloes being +considered. +We have split the complete sample, introduced in Sec. 2.4.3, +in two subsamples, namely a low-mass and a high-mass subsample. +The high-mass subsample contains, at each redshift, the 20% most +massive haloes in the complete sample. This is chosen so as to +MNRAS 000, 1–15 (2023) + +On the assembly state of DM haloes +9 +0 +1 +2 +3 +4 +5 +Redshift, z +1012 +1013 +1014 +1015 +Mvir (M +) +Low mass subsample +High mass subsample +Figure 6. Definition of the mass subsamples in terms of mass, as a function +of redshift. The gray (salmon) shaded regions correspond to the low-mass +(high-mass) subsamples. The dotted lines mark the mass limits. +contain, at redshift 0, all the haloes associated to massive groups +and clusters (𝑀DM > 3 × 1013𝑀⊙). Fig. 6 shows the evolution of +the mass limits on each subsample. Note that, therefore, our mass +groups do not correspond to fixed-mass ranges, but rather to two +sub-populations with a redshift-dependent mass threshold. While +it would definetely be interesting to explore the dependence of our +thresholds and weights with actual mass ranges, our limited statistics +prevent us from this goal and we may defer this for future work. +Repeating the analyses above separately for each mass sub- +sample, we can infer the mass dependence of the thresholds on the +dynamical state indicators, and that of their corresponding weights +in Eq. 14. The redshift evolution of the thresholds for each mass +subsample is presented in Fig. 7, which is analogous to Fig. 3 but +displaying only the fits for clarity. The same colour coding as in Fig. +6 is used here. +For some indicators, such as ellipticity, there is no hint for +any significant trend of the evolution of the threshold with mass, at +least within the statistical uncertainties given by our sample size. +That is to say, at least within the mass range considered in this +work (roughly, [1012 − 1015] 𝑀⊙), the relaxation criteria based on +this indicator can be used regardless of the scale of the objects (as +customarily done with many indicators, e.g. Power et al. 2012). +However, the rest of indicators of the dynamical state do present +significant dependence on halo mass. In particular, the threshold on +Δ𝑟 remains constant with redshift for the low-mass subsample, while +it increases linearly with increasing redshift for group and cluster- +sized DM haloes. This would suggest that imposing a constant +threshold on the centre offset may be too conservative and could, +for massive haloes at high redshift, artificially increase in excess the +number of disturbed haloes. +Regarding virial ratio, there is a minor trend with mass at +intermediate and low redshifts (𝑧 ≲ 3), with higher-mass haloes +preferring a slightly more stringent threshold to separate dynami- +cally relaxed and unrelaxed haloes, but this difference corresponds +to a small variation on the value of the parameter (Δ𝜂thr ∼ 0.05). +The most striking difference appears at high redshift, but is not rel- +evant since we have found that 𝜂 itself is not meaningful at high +redshift (see the second panel in Fig. 4, as well as Fig. 8 below). +0.09 +0.10 +0.11 +0.12 +0.13 +thr +r (z) +1.4 +1.5 +1.6 +1.7 +1.8 +thr(z) +0.07 +0.08 +0.09 +0.10 +vr thr(z) +1.50 +1.55 +1.60 +1.65 +1.70 +sthr +200c, 500c(z) +0 +1 +2 +3 +4 +5 +0.2625 +0.2650 +0.2675 +0.2700 +0.2725 +0.2750 +thr(z) +0 +1 +2 +3 +4 +5 +Redshift, z +Figure 7. Mass dependence of the redshift evolution of the thresholds on the +dynamical state indicators. The figure is analogous to Fig. 3, with each line +corresponding to the fit performed with a mass subsample (the same colour +coding as in Fig. 6 is used: gray [salmon] lines correspond to the low-mass +[high-mass] subsamples), and the shaded regions enclosing 1𝜎 confidence +intervals. +MNRAS 000, 1–15 (2023) + +10 +Vallés-Pérez et al. +The mass dependence of ⟨ � +𝑣𝑟⟩ is moderate, with massive haloes +preferring a constant threshold around ⟨ � +𝑣𝑟⟩ ≈ 0.085 and low-mass +systems displaying a decreasing trend with decreasing redshift. +Last, the two mass subsamples present a different behaviours +with respect to the threshold on halo sparsity, 𝑠thr +200𝑐,500𝑐. In this case, +lower mass haloes require consistently larger thresholds on sparsity +to discriminate relaxed and merging objects. While smaller haloes +tend to be more concentrated (see, for instance, Dutton & Macciò +2014), the mass-dependence of sparsity is much more contained +(Corasaniti et al. 2018; Corasaniti & Rasera 2019). It might be the +case that, both for physical (e.g., stronger influence of the environ- +ment) and numerical (e.g., less mass resolution elements leading to +more unresolved central regions) reasons, low-mass haloes present +a broader distribution in sparsities (see, e.g., figure 10 in Balmès +et al. 2014) and thus a larger sparsity threshold is required at the +low mass end. +Finally, we show in Fig. 8 the evolution of the weights on each +of the dynamical state indicators (as they appear in Eq. 14), for each +of the mass subsamples. Besides the general trends already analysed +for the whole sample when Fig. 4 was presented, several differences +emerge between the high-mass and the low-mass subsamples, espe- +cially at intermediate and high redshifts. At low redshift, however, +the weights are essentially compatible amongst the two subsam- +ples, with only a small hint of virial ratio and centre offset being +–comparatively– more effective in higher-mass haloes. +At intermediate redshifts, 𝑧 ∼ 2-3, high-mass haloes find el- +lipticity and mean radial velocity to be better indicators of their +dynamical state, and centre offset and sparsity comparatively worse +ones, when confronted to the low-mass sample. At high redshifts, +𝑧 ∼ 5, the performance of sparsity gets more penalised for lower- +mass haloes and, in these objects, centre offset can be relatively +more important than in higher-mass haloes. This further highlights +that, besides not being able to put fix criteria (in the sense of them +not evolving with redshift) for assessing the dynamical state of dark +matter haloes, they have to be carefully chosen depending on the +scale of the object being studied. +We provide fits for the thresholds and weights for the high-mass +(group and cluster-sized) subsample in Appendix A. +3.2 +Classification assessment +The aim of this section is to validate to which extent the dynam- +ical state classification introduced in this work, which only uses +information at a given timestep, is capable of predicting the merg- +ing state of the DM halo. That is to say, whether we can predict +the fiducial classification (Sec. 2.2.1) based on the dynamical state +indicators, when confronting our method with haloes from a differ- +ent simulation (corresponding to different resolution, gravity solver, +etc.). +We use public simulation data from the suite CAMELS +(Villaescusa-Navarro et al. 2021, 2022), which contains over 4000 +simulations of 25ℎ−1 Mpc cubic, periodic volumes run with dif- +ferent physics, cosmological and astrophysical parameters, and nu- +merical codes. In particular, we have analysed the haloes in the +IllustrisTNG-DM CV-0 simulation, which corresponds to a DM- +only simulation run with Arepo (Springel 2010; Weinberger et al. +2020). Arepo implements a Tree+Particle-Mesh approach (Bagla +2002) for solving the evolution of DM, thus providing a high dy- +namical range even though the number of particles (𝑁part = 2563) +is rather small. The CV-0 simulation corresponds to a background +cosmology with ℎ = 0.6711, Ω𝑚 = 0.3, 𝑛𝑠 = 0.9624, and 𝜎8 = 0.8, +the initial conditions having been set up at 𝑧ini = 127. +We have extracted the halo catalogues and merger trees with +ASOHF (Vallés-Pérez et al. 2022) by following the exact same pro- +cedure described in Sec. 2.2, and determined the dynamical state +indicators (Sec. 2.3). For our analyses, we have considered the 30 +most massive haloes at each time, which corresponds to a similar +mass limit as in the main analysis (cf. Fig. 1). Out of these 30 haloes, +we have dropped the ones that we are not able to trace back in time +for at least one dynamical time (which is most usually none or one +halo, at the considered epochs). +In Fig. 9, we assess the performance of our classification +scheme at three cosmological epochs (𝑧 = 0, 1, and 2, for the pan- +els left to right) by computing, at each time, the fraction of haloes +in each class (totally relaxed, marginally relaxed and unrelaxed) +which have recently suffered mergers or strong accretion (red), or +has undergone a quiet evolution (green). We note that, when evalu- +ating the dynamical state criteria, at any given 𝑧 we only apply the +indicators with weight 𝑤𝑖(𝑧) > 0.05. Otherwise, we consider the +given dynamical state indicator as not meaningful at that particular +epoch. While this particular threshold is arbitrary, it is a sensible +choice and the results do not depend strongly on variations around +this value. According to Fig. 5, this only removes the virial ratio, 𝜂, +at 𝑧 ≳ 1.9. +The totally relaxed subsample is, naturally, the smallest one, +since it is defined rather conservatively as the set of haloes simul- +taneously fulfilling all five relaxation criteria. It typically contains +∼ 10% of the haloes (slightly lower in this case; nevertheless, the +statistics are small). Within this test, all haloes being classified as +totally relaxed have not suffered any major (minor) merger within +one (half) 𝜏dyn or built up more than half of their mass in the last +dynamical time, thus proving to be a selection of the relaxed sample +with high specificity. +The marginally relaxed sample is the most numerous at low +redshifts (𝑧 ≲ 1) and mostly contains objects which have not ex- +perienced any relevant merging or accretion activity, although the +fraction of objects having experienced it increases with increasing +redshift (from ∼ 15% at 𝑧 ≃ 0 to ∼ 40% at 𝑧 ≃ 2). The unrelaxed +subsample, which is especially numerous at high redshift when +merger rates are higher (see, e.g., Wetzel et al. 2009), contains +mostly merging objects, although a small fraction (10 − 25%) of +objects not experiencing mergers or accretion seem to fall into this +category. This may happen because a halo appears to be disturbed, +even when not merging or accreting intensely, due to environmental +effects (e.g., strong tidal field due to the presence of another nearby +massive halo, for instance in a pre-merger state), or even numerical +effects (mainly associated to low resolution). Naturally, it may also +be the case that unrelaxedness after a major merger is last for longer +than 1𝜏dyn(𝑧), since the fiducial classification in Sec. 2.2.1 was only +a rough estimation. +Expanding upon the previous figure, in Fig. 10 we focus on +some quantities tied to the merger and accretion history of the +haloes. In particular, the left panel represents, at three redshifts +(𝑧 = 0, 1, and 2, respectively, from left to right) and for the three +subsamples, the distribution of the time since the last merger (either +major or minor) in units of the dynamical time. Haloes classified as +unrelaxed have usually suffered some merger recently while, on the +other hand, the totally relaxed sample has typically not experienced +any merging activity since several dynamical times ago. A similar +situation is seen for the major mergers (middle panel), although +naturally not all unrelaxed haloes have suffered a major merger +(the unrelaxedness can be due to one or several minor mergers, or +smooth accretion, as well). However, the same trend holds, with +MNRAS 000, 1–15 (2023) + +On the assembly state of DM haloes +11 +0 +1 +2 +3 +4 +5 +Redshift, z +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +Relative weights +Low mass sumbsample +0 +1 +2 +3 +4 +5 +Redshift, z +High mass subsample +r +vr +s200c, 500c +Figure 8. Mass dependence of the redshift evolution of the weights on the dynamical state indicators. Each panel is analogous to Fig. 5 (also using the same +colour coding), showing the results for each of the mass samples defined in Fig. 6: low-mass (left panel) and high-mass (right panel) subsamples. +Totally + relaxed + (N = 2) +Marginally + relaxed + (N = 20) +Unrelaxed + (N = 8) +0% +20% +40% +60% +80% +100% +z = 0 +Totally + relaxed + (N = 2) +Marginally + relaxed + (N = 16) +Unrelaxed + (N = 11) +0% +20% +40% +60% +80% +100% +z = 1 +Totally + relaxed + (N = 1) +Marginally + relaxed + (N = 7) +Unrelaxed + (N = 21) +0% +20% +40% +60% +80% +100% +z = 2 +Figure 9. Classification summary at redshifts 𝑧 = 0 (left panel), 𝑧 = 1 (middle panel), and 𝑧 = 2 (right panel). Within each panel, each of the columns +corresponds to one of the possible classifications (totally relaxed, if 𝑋𝑖 < 𝑋thr +𝑖 +∀𝑖; marginally relaxed, if the previous condition fails but 𝜒 ≥ 1; or unrelaxed +otherwise). Within each bar, the green portion represents the fraction of the haloes which have not suffered any mergers nor strong accretion, while the red +portion corresponds to the fraction of haloes having suffered mergers or strong accretion (i.e., the colour encodes the fiducial classification). Below each +column, 𝑁 indicates the number of objects falling into each category. +0 +1 +2 +Redshift, z +0 +1 +2 +3 +tLM/ +dyn +Totally relaxed +Marginally relaxed +Unrelaxed +0 +1 +2 +Redshift, z +0 +1 +2 +3 +4 +5 +tLMM/ +dyn +0 +1 +2 +Redshift, z +0 +1 +2 +3 +4 +5 +vir +Figure 10. Trends of the evolutionary properties of the haloes according to their dynamical state classification. In each panel, red, orange and green dots +correspond, respectively, to the unrelaxed, marginally relaxed and totally relaxed subsamples. At each redshift (encoded in the horizontal axis) the dot and the +error bars represent, respectively, the mean and the 1𝜎 dispersion of the distribution of the given variable within the subsample. The left panel presents the +time since the last merger (either major or minor) in units of the dynamical time (Δ𝑡LM/𝜏dyn), the central panel corresponds to the time since the last major +merger (Δ𝑡LMM/𝜏dyn), and the right panel shows the accretion rate Γvir. +MNRAS 000, 1–15 (2023) + +12 +Vallés-Pérez et al. +major mergers having occurred a longer time ago as we move from +unrelaxed to marginally relaxed, and to totally relaxed haloes. +Finally, the third panel presents, in a similar way, the distribu- +tion of accretion rates Γvir, which are computed according to the +prescription of Diemer & Kravtsov (2014), +Γvir = Δ log 𝑀vir +Δ log 𝑎 +(25) +with 𝑎 being the scale factor of the Universe, and the increments +computed over a dynamical time following its definition in Eq. 2. +The figure shows, in line with the previous results, an increasing +trend of the accretion rate when moving from the relaxed to the +more disturbed subsamples, within a wide redshift interval. This re- +flects how the dynamical state classification presented in this work, +which only uses information at a fixed time, can offer insight on the +temporal evolution of the systems over the last dynamical time. +3.2.1 +Does a smaller set of indicators provide similar insight? +Lastly, it might be interesting to assess whether a single dynamical +state indicator, or a combination of them, is capable of providing a +similarly accurate classification; i.e., to motivate why it is important +to involve a high number of indicators. This is briefly exemplified in +Table 2, where we show the classification summary for each indica- +tor (or combination). In particular, for each classification class we +give the number of haloes falling into this class (and its percentage +with respect to the total), and the fraction of them which is unre- +laxed according to the fiducial classification ( 𝑓merging). Ideally, this +fraction would be 0 for the totally relaxed class and 1 for the un- +relaxed class. Naturally, when using only one indicator, there is no +marginally relaxed or intermediate category, since the value of the +dynamical state indicator can only be above or below the threshold. +In the case of using two indicators, we have defined the marginally +relaxed sample as the set of haloes fulfilling only one of the two +relaxedness conditions, as it is often done in the literature (e.g. Biffi +et al. 2016; Planelles et al. 2017). +Generally speaking, involving only one dynamical state indi- +cator leads to far poorer results, since the relaxed sample gets often +contaminated (around ∼ 40%) by haloes which have suffered merg- +ers. Likewise, the unrelaxed sample may end up containing a high +fraction of haloes undergoing quiescent evolution for some indica- +tors (e.g., 𝜖; although the particular results have to be considered +carefully due to the reduced statistics). +Interestingly, when using a combination of virial ratio and cen- +tre offset, which is a common option in the literature (e.g., Power +et al. 2012), we are still not able to pick out all merging haloes with +these criteria and even the totally relaxed subsample gets contami- +nated with ∼ 40% of merging haloes. Other common options in the +literature are the combination of mass ratio and centre offset (De +Luca et al. 2021), or centre offset, virial ratio and mass ratio (Cui +et al. 2017; Haggar et al. 2020). We have also tested these com- +binations, taking 𝑓 thr +sub = 0.1 from the aforementioned references, +since we have not involved this indicator in our previous analyses. +In these cases, the results are similar to the Δ𝑟 & 𝜂 combination. +This highlights the necessity of involving and combining as many +indicators of the dynamical state as possible. When using the full +set of indicators derived in this work, the totally relaxed subsample +is rather small, due to its conservative definition. However, even +our marginally relaxed subsample is purer (contains a smaller frac- +tion of merging/accreting haloes) than the totally relaxed sample of +the previous combinations, proving to provide a robust splitting of +haloes according to their dynamical state. +4 +DISCUSSION AND CONCLUSIONS +To fully exploit the capabilities of ongoing surveys (e.g., eROSITA; +Ghirardini et al. 2022), and in the advent of upcoming instruments +over the electromagnetic spectrum (from X-ray, e.g., ATHENA, Nan- +dra et al. 2013; to radio, e.g. SKA, Acosta-Pulido et al. 2015; going +through the optical, e.g. EUCLID, Sartoris et al. 2016; Euclid Col- +laboration et al. 2019), which will provide samples of galaxies and +galaxy clusters unprecedented in size and depth, it remains crucial +to provide reliable indicators of the dynamical state (which in turns +is a fast proxy of the –recent– evolution of the system). As a first +step towards that aim, using 𝑁-Body+hydrodynamics simulations, +in this work we have systematically analysed how to best combine +a series of quantities which can be measured from simulation data +at a given time in order to be able to detect the presence of mergers +and/or ongoing strong accretion. +As a result, we have built an algorithm that combines a series of +different indicators of the dynamical state of a DM halo (namely, its +centre offset Δ𝑟, the virial ratio 𝜂, the mean radial velocity ⟨ � +𝑣𝑟⟩, the +sparsity 𝑠200𝑐,500𝑐, and the ellipticity 𝜀) in order to classify haloes +within three classes. The totally relaxed haloes, comprising the ob- +jects simultaneously fulfilling all relaxedness conditions (which are +redshift-dependent, in general), is a conservatively defined subsam- +ple which, therefore, only contains around ∼ 10% of the haloes at a +given time. Haloes where some relaxedness condition may fail, but +are remarkably relaxed according to the rest of indicators may be +categorised in the marginally relaxed class, using a criterion simi- +lar to Haggar et al. (2020), but allowing different indicators to have +different (redshift-dependent) weights, which are tuned based on +the performance of each indicator on telling relaxed and unrelaxed +haloes apart. Thus, we defined a relaxedness parametre (𝜒), which +tells marginally relaxed (𝜒 ≥ 1) and unrelaxed (𝜒 < 1) apart. The +fits for the redshift dependence of the thresholds and weights are +given in Eqns. 15-24, while equivalent results for massive haloes +are provided in App. A. +Furthermore, we have confronted our classification scheme +against an independent DM-only simulation from the CAMELS suite +(Villaescusa-Navarro et al. 2021, 2022), corresponding to different +input physics, initial conditions and numerical solvers. Using it, we +find that our algorithm performs a clean splitting of relaxed and +unrelaxed haloes across a wide cosmic time interval, and that this +classification improves upon the usage of any single indicator or +some widely used combinations (Δ𝑟 & 𝜂; 𝑓sub & 𝜂; or Δ𝑟, 𝜂 & +𝑓sub). +As a qualitative summary of the main highlights of the classi- +fication scheme, we can mention: +• Placing fix thresholds (which do not evolve with redshift) is +generally undesirable. While some indicators do not show strong +evolution of their optimal thresholds with redshift (e.g., ellipticity, +centre offset), others do (e.g., sparsity, mean radial velocity; all +tending to increase with redshift). This has important consequences, +since it implies that classification schemes for the dynamical state of +haloes that are set at 𝑧 = 0 cannot be directly used at high redshifts. +– At high halo mass (see the precise definition of the high- +mass subsample in Fig. 6), however, the results are slightly +changed: in particular, the redshift dependence of the thresholds +MNRAS 000, 1–15 (2023) + +On the assembly state of DM haloes +13 +Table 2. Classification properties using only one or a combination of dynamical state indicators, exemplified at 𝑧 = 1. Each row corresponds to one dynamical +indicator or combination, and for each classification class we give the number of objects falling into the class (𝑁 , and the percentage with respect to the +total) and the fraction of them which is unrelaxed according to the fiducial classification ( 𝑓merging). When only one indicator is used, there is no intermediate +(marginally relaxed) class. The first block corresponds to the individual indicators involved in this work. The second block contains several combinations +widely used in the literature, with the fitted thresholds in this work (for 𝑓sub, not involved in this work, we use 𝑓 thr +sub = 0.1). The last row corresponds to the +complete method introduced here, using the five indicators (therefore, these results are the same shown in the central panel of Fig. 9). +Totally relaxed +Marginally relaxed +Unrelaxed +Indicator(s) +𝑁 +𝑓merging +𝑁 +𝑓merging +𝑁 +𝑓merging +Δ𝑟 +20 (69%) +0.40 +– +– +9 (31%) +0.78 +𝜂 +25 (86%) +0.48 +– +– +4 (14%) +0.75 +⟨ ˜𝑣𝑟 ⟩ +21 (72%) +0.43 +– +– +8 (28%) +0.75 +𝑠200𝑐,500𝑐 +17 (59%) +0.47 +– +– +12 (41%) +0.58 +𝜖 +10 (34%) +0.50 +– +– +19 (66%) +0.53 +Δ𝑟 & 𝜂 +19 (66%) +0.42 +7 (24%) +0.57 +3 (10%) +1.00 +𝑓sub +23 (79%) +0.48 +– +– +6 (21%) +0.67 +𝜂 & 𝑓sub +18 (62%) +0.44 +7 (24%) +0.43 +4 (14%) +1.00 +Δ𝑟, 𝜂 & 𝑓sub +17 (59%) +0.47 +3 (10%) +0.33 +9 (31%) +0.67 +Full set of indicators +2 (7%) +0.00 +16 (55%) +0.38 +11 (38%) +0.82 +on ⟨ � +𝑣𝑟⟩ and 𝑠200𝑐,500𝑐 is not significant anymore, while the clas- +sification based on the Δ𝑟 benefits from an increasing trend with +increasing redshift. This warns us that the classification cannot +be universal, and that haloes on different mass scales may need +slightly modified criteria. +• At low redshift (𝑧 ≲ 1), even though all indicators offer insight +into the merging state of the halo, it is sparsity and ellipticity of +the DM halo the ones which provide the most valuable information, +well beyond other, more widely used indicators such as centre offset +or virial ratio. Nevertheless, the fact that all relative weights are not +very dissimilar at this epoch (see Fig. 5) means that the classification +scheme can importantly benefit from combining as many indicators +as possible. +– The difference in weights amongst the different observables +(except ⟨ � +𝑣𝑟⟩) is importantly reduced when looking at the high +mass sample (right panel in Fig. 8), reinforcing that, for group- +and cluster-sized haloes at low redshift, it may be important to +combine all indicators suggested in this work. +• At high redshifts (𝑧 ≳ 3), 𝜂 becomes irrelevant for the deter- +mination of the assembly state of the halo, while centre offset and +mean radial velocity become, by far, the dominant indicators. +– Again, the differences are lower for the high-mass subsam- +ple, but the prevalence of Δ𝑟 and ⟨ � +𝑣𝑟⟩ still holds. +In this work, we have focused on the determination of the +assembly state of DM haloes using the full information contained +in a snapshot of a numerical simulation. The motivation for this is +two-fold. On the one hand, it is important to devise efficient methods +to classify large samples of simulated haloes, especially given the +ever-growing trend of simulations, both in size and resolution (see, +e.g., Angulo & Hahn 2022, their table 1), made possible by the +increasing computational power available. On the other hand, it +serves as a first step, which can be further connected to observations +using projected data or, more realistically, mock multiwavelength +observations (e.g., Planelles et al. 2018). +Much of the information comprised in the dynamical state indi- +cators we involve in this work can be lost, or at least hindered, when +moving from the 3-dimensional description to the 2-dimensional +observed data. The first, most natural consequence is the effect of +projection on any geometrical indicator, such as the centre offset, +ellipticity or the mean radial velocity. For the case of centre off- +set and ellipticity, the measured values will only be a lower limit, +with the actual 3-dimensional value depending on the inclination +between the direction of the offset, or the plane containing the major +and minor axis, with the line of sight. +Regarding the mean radial velocity, which is especially impor- +tant for determining the dynamical state at high redshift, besides +the difficulty induced by projection (only velocities along the line +of sight, and distances on the plane of the sky, can be measured), +future kinetic Sunyaev-Zel’dovich (kSZ) observations could be able +to provide some constraints on proper velocities of the intra-cluster +medium (ICM; see, for instance, the estimates of Baldi et al. 2018 +about the kSZ effect due to the coherent rotation of the ICM), even +for high-redshift objects since the SZ effect is essentially distance- +independent (e.g., Voit 2005). Even though the dynamics of the +ICM, especially in the inner regions of haloes, may differ signifi- +cantly from those of the DM halo, probing the velocity field of the +diffuse gas in haloes could supply useful insight onto the dynamical +state of haloes at high redshift. +Lastly, sparsity may be a suitable option for observations, given +its good performance shown across the whole redshift span consid- +ered here (especially, for high-mass haloes). However, care must +be taken when using this quantity: here, we have defined sparsity +from the DM masses obtained from the full, 3-dimensional infor- +mation. However, in observations, masses can be obtained from +several methods (e.g., hydrostatic, lensing, caustic masses), and bi- +ases amongst them are non-negligible (see, for instance, Lovisari +et al. 2020). Moreover, mass biases tend to correlate with the merg- +ing state (Bennett & Sijacki 2022; cf. Gianfagna et al. 2022) and, +while the quotient of two masses at different apertures derived from +the same method may cancel out part of these biases, the fact that +the bias itself depends on the aperture and the large object-to-object +scatter still make the interpretation non-trivial and deserve further +attention themselves. +This work provides a motivated definition of a scheme for +classifying DM haloes according to their dynamical status, based on +simple properties which can be readily extracted from the outputs +of typical halo finders. Future work will need to deal with the +connection of these dynamical and morphological properties of the +DM halo with the baryonic component, as well as the application +to observations, in order to being able to extract the largest possible +amount of information about the assembly state of haloes from +future observational campaigns. +MNRAS 000, 1–15 (2023) + +14 +Vallés-Pérez et al. +ACKNOWLEDGEMENTS +We gratefully thank the anonymous referee for their valuable feed- +back, which has helped us to improve the quality of this manuscript. +This work has been supported by the Agencia Estatal de Inves- +tigación Española (AEI; grant PID2019-107427GB-C33), by the +Ministerio de Ciencia e Innovación (MICIN) en el marco del Plan +de Recuperación, Transformación y Resiliencia del Gobierno de +España through the project ASFAE/2022/001 and by the General- +itat Valenciana (grant PROMETEO/2019/071). DV acknowledges +support from Universitat de València through an Atracció de Talent +fellowship, and gratefully thanks the hospitality of the Dipartimento +di Fisica e Astronomia of the Università di Bologna, where part of +this work was done during a research stay funded by Universitat +de València. We also thank F. Vazza and A. Ragagnin for fruitful +scientific conversations. Simulations have been carried out with the +supercomputer Lluís Vives at the Servei d’Informàtica of the Uni- +versitat de València. This research has made use of the following +open-source packages: NumPy (Harris et al. 2020), SciPy (Virtanen +et al. 2020), matplotlib (Hunter 2007), statsmodels (Seabold +& Perktold 2010), scikit-learn (Pedregosa et al. 2011), and +Colossus (Diemer 2018). +DATA AVAILABILITY +The data underlying this article will be shared upon reasonable +request to the corresponding author. +REFERENCES +Acosta-Pulido J. 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Thus, the sample corresponds to massive groups +and clusters at 𝑧 ∼ 0; to objects above 1013𝑀⊙ at 𝑧 ∼ 2; and to a +mass limit of ∼ 3 × 1012𝑀⊙ at 𝑧 ∼ 5, which might most often be +the progenitors of the massive haloes we find at 𝑧 ∼ 0. +Within this sample, the evolution with redshifts of the thresh- +olds on the dynamical state indicators, shown in Fig. 7, can be given +by the following polynomial fits: +Δthr +𝑟 (𝑧) +��massive = 0.0863(39) + 0.0066(23)𝑧 +(A2) +𝜂thr(𝑧) +��massive = 1.3371(88) + 0.151(14)𝑧 − 0.0139(37)𝑧2 +(A3) +⟨ � +𝑣𝑟⟩thr +DM(𝑧) +��massive = 0.0842(32) +(A4) +𝑠thr +200𝑐,500𝑐(𝑧) +��massive = 1.495(10) +(A5) +𝜀thr(𝑧) +��massive = 0.2710(33) +(A6) +The weights on these indicators, as they appear on the relaxed- +ness parameter (Eq. 14), are fitted by: +𝑤[Δ𝑟](𝑧) +��massive ∝ 0.218(16) − 0.134(26)𝑧 + 0.0356(69)𝑧2 (A7) +𝑤[𝜂](𝑧) +��massive ∝ 0.250(11) − 0.0603(66)𝑧 +(A8) +𝑤[⟨ � +𝑣𝑟⟩DM](𝑧) +��massive ∝ 0.092(17) + 0.109(27)𝑧 − 0.0141(71)𝑧2 +(A9) +𝑤[𝑠200𝑐,500𝑐](𝑧) +��massive ∝ 0.2251(87) +(A10) +𝑤[𝜀](𝑧) +��massive ∝ 0.2537(86) +(A11) +This paper has been typeset from a TEX/LATEX file prepared by the author. +MNRAS 000, 1–15 (2023) + diff --git a/StE0T4oBgHgl3EQfUwAJ/content/tmp_files/load_file.txt b/StE0T4oBgHgl3EQfUwAJ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9e462c5e80254c588f050d56f835cee51f712bbf --- /dev/null +++ b/StE0T4oBgHgl3EQfUwAJ/content/tmp_files/load_file.txt @@ -0,0 +1,1414 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf,len=1413 +page_content='MNRAS 000, 1–15 (2023) Preprint 9 January 2023 Compiled using MNRAS LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='0 On the choice of the most suitable indicator for the assembly state of dark matter haloes through cosmic time David Vallés-Pérez,1★ Susana Planelles,1,2 Óscar Monllor-Berbegal,1 Vicent Quilis1,2 1Departament d’Astronomia i Astrofísica, Universitat de València, E-46100 Burjassot (València), Spain 2Observatori Astronòmic, Universitat de València, E-46980 Paterna (València), Spain Accepted XXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Received YYY;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' in original form ZZZ ABSTRACT The dynamical state and morphological features of galaxies and galaxy clusters, and their high-redshift precursors, are tightly connected with their assembly history, encoding crucial information about the formation and evolution of such cosmic structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' As a first step towards finding an optimal indicator of the assembly state of observed structures, we use a cosmological simulation of a moderate volume to critically examine the best definition of an indicator that is able to discriminate dark matter haloes undergoing mergers and/or strong accretion from haloes experimenting a relaxed evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Using a combination of centre offset, virial ratio, mean radial velocity, sparsity and ellipticity of the dark matter halo, we study how the thresholds on these parameters, as well as their relative weights, should evolve with redshift to provide the best classification possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' This allows us to split a sample of haloes in a totally relaxed, a marginally relaxed and an unrelaxed subsamples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' The resulting classification strongly correlates with the merging activity obtained from the analysis of complete merger trees extracted from whole simulation data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' The results on how the different indicators depend on redshift and halo mass, and their optimal combination to better match the true assembly history of haloes, could constitute relevant hints to find a suitable set of indicators applicable to observational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Key words: large-scale structure of Universe – dark matter – galaxies: clusters: general — methods: numerical 1 INTRODUCTION Deeply interwoven through a complex network of filaments and sheets, dark matter (DM) haloes are bound, diffuse structures which result from the gravitational collapse of the primordial density fluc- tuations and a hierarchical merging history (Zel’dovich 1970;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Press & Schechter 1974;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Gott & Rees 1975).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' DM haloes constitute the fundamental building blocks of the large-scale structure (LSS) of the Universe, and host their baryonic counterparts that we observe over the electromagnetic spectrum (see, for instance, Planelles et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2015, for a review).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' At the galactic scale, the current theories of galaxy formation typically assume DM haloes to be virialised (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=', White & Rees 1978), although this does not necessarily hold for each galactic DM halo;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' while, at larger masses (at the galaxy cluster scale), most DM haloes are still expected to be in the process of viri- alisation, since they are the latest objects to have assembled (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=', Kravtsov & Borgani 2012, for a review on galaxy cluster formation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' However, the dynamical state of individual haloes is tightly connected to their assembly history and, in particular, to the pres- ence of mergers and the accretion rates in the last one or few dynam- ical times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' A merger or a period of intense accretion usually triggers ★ E-mail: david.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='valles-perez@uv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='es many morphological and dynamical disturbances in the halo (as- phericity, higher velocity dispersions, abundance of substructures, changes to the internal structure, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' ), which gradually fade away once the assembly episode is over (see, for example, Poole et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2006 for a thorough analysis of the disturbances and the subsequent relaxation after a merger event at cluster scales).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Since dynamically relaxed and disturbed structures often present fundamentally different properties, a characterisation of the dynamical state of the sample of cosmic structures is often a neces- sary procedure in many analyses of very different natures, such as in studies about the geometry of the cosmic web (Gouin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2021), statistical properties of the population of galaxy clusters (scaling relations, mass functions, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=', Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Seppi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2021), hydrostatic mass bias (Nelson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Biffi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Angelinelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2020), turbulence (Vazza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Valdarnini 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Vallés-Pérez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2021a,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Simonte et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2022), or galactic environments (Kuchner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2022), just to mention a few.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Even though we usually define haloes using the virial radius prescription of Eke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' (1996) and Bryan & Norman (1998), based on the spherical collapse model, this does not imply that, in general, three-dimensional haloes (non necessarily spherical, in a full-cosmological, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=', not isolated environment) defined this way are necessarily in virial equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' While in simulations one can © 2023 The Authors arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='02253v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='CO] 5 Jan 2023 2 Vallés-Pérez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' access the whole temporal evolution of the objects, and thus recover the assembly history of the halo under study in order to assess the dynamical state, this is not possible in observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Thus, for the sake of a more direct comparison with observational works, simple schemes for characterising the dynamical state using halo properties at a given time are usually involved in many analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' For the time being, most works have relied on placing a thresh- old on some halo property expected to correlate with the dynamical state, in order to split the relaxed and unrelaxed subsamples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Per- haps, the most direct of such indicators is the virial ratio, usually defined as 𝜂 ≡ 2𝑇/|𝑊|, where 𝑇 is the intrinsic kinetic energy of the halo and 𝑊 is its gravitational potential energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 𝜂 would be expected to be 1 for an isolated system in a steady state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' However, different works have found different thresholds to best suite their particular classification (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=', Shaw et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Neto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Knebe & Power 2008, see also the discussion in Cui et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Similarly, there is debate about the necessity of including a surface tension term to account for the fact that haloes are not isolated (Poole et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Shaw et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Knebe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Another frequently used indicator, both in simulations and observations, is the centre offset, which quantifies the departure from smoothness and spherical symmetry of the matter distribution, and serves as an indicator of substructure (Crone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' In practice, however, there are many possibilities regarding the choice of centres (see Cui et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2016) and how to set the thresholds (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' D’Onghia & Navarro 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Macciò et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Additionally, in observations the centre offset may depend crucially on the orientation, posing an additional challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Last, other authors have used the fraction of mass in sub- structures as a measure of the dynamical unrelaxedness of a DM halo (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=', Neto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' other recently suggested approaches, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Kimmig et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' While the election of this magnitude is well-motivated, the mass contained in substructures in simulated haloes depends critically on numerical resolution and the precise definition of the substructure extent (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='g, the discussion in Vallés-Pérez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2022), making this criterion less comparable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Since it is difficult that a single property can reflect the complex picture of the dynamical state of a halo, many recent studies have used combinations of these indicators, either by considering as relaxed the haloes which simultaneously fulfil several relaxation criteria (Neto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Biffi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2016), or by defining some combined indicator (Haggar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2021a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' De Luca et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Finally, other metrics of the dynamical state are based on the X-ray morphology, such as the centroid shift 𝜔 (Mohr et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 1993), or the power ratio, 𝑃3/𝑃0 (Buote & Tsai 1995, see also the review of Rasia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2013 on X-ray morphological estimators for galaxy clusters);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' or more sophisticated ones such as those involving Fourier analyses of the fluctuations in mass and X-ray maps (Cerini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2022), or the expansion of the Compton 𝑦-maps in Zernike polynomials (Capalbo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' However, in most of the previous works, the parameters being used and, especially, the thresholds imposed on them have been tuned in a somewhat empirical way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' This has lead to variations in the criteria from work to work, even though the underlying idea is kept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Furthermore, a possible redshift evolution of these thresholds or of their very relevance has been devoted marginal attention, either because the studies were focused on a particular cosmic epoch or because it had been implicitly assumed that these criteria should not evolve with redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' In this work, we intend to critically examine a set of possi- ble indicators of the assembly state, all of which can be obtained from the complete three-dimensional information in simulations, and develop a criterion which accommodates redshift-dependent thresholds and the possibility that different indicators have more or less relevance at different cosmic epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' We note the reader that, while in the following we may refer to the dynamical state of haloes, our main focus is oriented towards the dynamical disturbances asso- ciated to the assembly history of haloes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=', the presence of merger events or episodes of strong accretion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' rather than a more general sense of dynamical unrelaxedness which could include, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=', the presence of substructures even when they are not associated to a merger episode, since they have an impact on properties such as the hydrostatic equilibrium).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' The rest of the manuscript is organised as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2, we introduce our simulation, halo sample and the methodology that we employ for setting the thresholds and relative weights of the different dynamical state indicators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Our resulting criterion is presented in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 3, including the analysis of the mass dependence of our results and a validation of our method with a different sim- ulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Finally, we discuss the applicability of our results in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Appendix A contains the fitting formulae for the thresholds and weights applicable for massive haloes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2 METHODS The results reported in this paper have been extracted from the analysis of a ΛCDM cosmological simulation tracking the coupled evolution of baryons and DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' We describe the relevant details of the simulation in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='1, then cover the halo catalogues and merger tree elaboration in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='2, and discuss how do we compute the dynamical state indicators in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Finally, we introduce our classification strategy in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='1 The simulation The haloes we analyse in this paper are extracted from a numerical simulation run with MASCLET (Quilis 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Quilis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2020), a (magneto-)hydrodynamics and 𝑁-Body code primarily designed for cosmological applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' For evolving the DM component, which is the primary focus of this work, MASCLET implements a multilevel Particle-Mesh (PM) scheme (Hockney & Eastwood 1988), which takes advantage of the adaptive-mesh refinement (AMR) strategy (Berger & Colella 1989) to gain spatial, temporal and force resolu- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' We have simulated a periodic, cubic (𝐿 = 100 ℎ−1 Mpc) do- main, under the assumption of a flat, ΛCDM cosmology specified by the matter density parameter Ω𝑚 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='31 (ΩΛ = 1 − Ω𝑚), baryon density parameter Ω𝑏 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='048, and Hubble parameter ℎ ≡ 𝐻0/(100 km s−1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='678.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' The initial conditions stem from a realisation of the primordial gaussian random field assuming a spectral index 𝑛𝑠 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='96 and an amplitude yielding 𝜎8 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='82, and are set up at redshift 𝑧ini = 100 using a CDM transfer function (Eisenstein & Hu 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' The values selected for the cosmological parameters are consistent with the latest results reported by Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' A first simulation is run at low resolution, using a fix grid of 𝑁3𝑥 = 2563 cells and the same number of equal-mass particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' This is used to identify the Lagrangian regions in the initial conditions which will evolve into dense structures by 𝑧 = 0, and mapping them with enhanced numerical resolution already at 𝑧ini.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' We use this approach to establish three nested levels of initial conditions, resulting in a best mass resolution of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='48 × 107 𝑀⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Using these high-resolution initial conditions, the simulation MNRAS 000, 1–15 (2023) On the assembly state of DM haloes 3 is evolved again using AMR based on gas/DM overdensities, con- verging flows, and Jeans length criteria, achieving a peak resolution of Δ𝑥8 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='3 kpc at the maximum (ℓ = 𝑛ℓ ≡ 8) level of refinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' While the baryonic component is not the primary focus of this work, the simulation includes gas cooling, but no other baryonic effect or feedback mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='2 Halo catalogue and merging history For each snapshot of the simulation, we have identified the DM haloes using the public halo finder ASOHF (Planelles & Quilis 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Knebe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Vallés-Pérez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2022)1, which is based on the spherical-overdensity definition and uses the virial radius (ac- cording to the prescription of Bryan & Norman 1998) to delimit the extent of the haloes that are not substructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' After determining the halo catalogues, these are linked in be- tween snapshots using the merger tree code presented by Vallés- Pérez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' (2022, their section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='2), which identifies all the haloes at a given code output which have contributed to an object in a fol- lowing one, allowing to skip an arbitrary number of snapshots, if necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Using it, we determine the main evolutionary line of each halo, as well as the presence and characterisation of mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Following Planelles & Quilis (2009), Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' (2019), and Vallés-Pérez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' (2020), we have classified each merger event in the sample as either a major merger (if the mass ratio, 𝑀min/𝑀max, between the two haloes involved exceeds 1/3), or a minor merger (1/3 > 𝑀min/𝑀max ≥ 1/10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Mergers below a mass ratio of 1/10 are disregarded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' The merger time is determined as the moment in which the centre of the infalling (the least massive) halo crosses the virial boundary of the host (the most massive) halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='1 Fiducial classification: assembly history of the haloes In order to determine the optimal thresholds on the dynamical state indicators (see below, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='3 and therein), we compare with a reference, or fiducial, classification of the dynamical state based on the full assembly history of haloes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=', the presence of past or ongoing mergers, as well as the accretion rates).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' As a tentative classification of the unrelaxedness induced by a merger event, we will assume that a typical halo remains in a disturbed state for one dynamical time after a major merger, or half a dynamical time after a minor merger, with the dynamical time 𝜏dyn being defined as 𝜏dyn(𝑧) ≡ 1 √︁ 𝐺𝜌 = 1 √︁ 𝐺𝜌𝐵(𝑧)Δvir(𝑧) , (1) with 𝐺 the gravitational constant, 𝜌 the density of the halo, 𝜌𝐵(𝑧) the background matter density and Δvir(𝑧) the virial overdensity (Bryan & Norman 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' While the choice of the timespan is a crude approximation, it responds to the fact that many works have shown that the disturbance triggered by a minor merger is, in general terms, smaller than the effect of a major merger, both for the dark and for the baryonic components (Planelles & Quilis 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Vallés-Pérez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2021b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' In practical terms, since 𝜏dyn(𝑧) varies strongly with redshift and reaches considerable fractions of the age of the Universe, especially at low redshift, we choose to 1 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='com/dvallesp/ASOHF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 0 1 2 3 4 5 Redshift, z 1012 1013 1014 1015 Mass, Mvir (M ) Median Mean (16% 84%) CI (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='5% 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='5%) CI Min & max Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Evolution of the distribution of halo masses in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' The solid line indicates the median mass of the sample, with the dark and light shaded regions enclosing the 16% − 84% (dark blue) and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='5% − 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='5% (light blue) confidence intervals (CIs) around it, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' The dashed line corresponds to the mean mass, while the dotted lines correspond to the maximum and minimum masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' define the number of dynamical times between two moments, 𝑡1 and 𝑡2, as in Jiang & van den Bosch (2016) and Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' (2020): 𝑁𝜏 (𝑡1, 𝑡2) = ∫ 𝑡2 𝑡1 d𝑡 𝜏dyn(𝑧) (2) Additionally, it might be the case that a halo is accreting strongly, but without undergoing any significant merger (either phys- ically or due to the finite resolution of a simulation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Thus, we also consider as unrelaxed, for the purpose of the fiducial classification, any halo which has assembled more than 50% of their mass in the last dynamical time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' For the analyses within this work, all the 28 snapshots of the simulation since redshift 𝑧 = 5 are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' We select the 1000 most massive haloes at each epoch, and discard all those which cannot be reliably traced back in time for at least one 𝜏dyn(𝑧).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' We show, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 1, the redshift evolution of the median mass in the sample (solid line), together with shaded regions enclosing the confidence intervals corresponding to the 16% − 84% (dark blue) and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='5% − 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='5% (light blue) percentiles of the distribution of masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' The dotted lines mark the maximum mass (upper line) and the minimum mass, or mass limit (lower line) in the sample at each time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Thus, the mass limit in our sample evolves from ∼ 1012𝑀⊙ at 𝑧 = 5 to ∼ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='5 × 1012𝑀⊙ at 𝑧 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' The wide redshift interval considered in this study includes from the cluster-, group- and massive galaxy-sized haloes at 𝑧 ≃ 0, to the DM counterpart of galaxies and the progenitors of low-redshift clusters at the high- redshift end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' The results of the fiducial classification are summarised in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2, where we show the number of haloes which are finally considered at each snapshot (blue line, referring to the axis on the left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Only at high redshift (𝑧 ≳ 3), a large fraction (10% to 25%) of the pre- liminary haloes get discarded because they cannot be tracked back in time for at least one dynamical time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' The fraction of unrelaxed MNRAS 000, 1–15 (2023) 4 Vallés-Pérez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 0 1 2 3 4 5 Redshift, z 0 200 400 600 800 1000 Number of haloes 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='0 Fraction of haloes Unrelaxed Merging Accreting Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Fiducial classification of dynamical states of the halo sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' The blue line represents the number of haloes selected per snapshot, according to the left axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' The dashed/dotted lines, according to the right axis, describe the evolution with redshift of the fraction of unrelaxed haloes (green), which can have been labelled as such due to mergers (purple) or strong accretion/mass growth (orange).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' haloes according to the fiducial classification (green, dashed line;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' referring to the axis on the right) varies from ∼ 80% to ∼ 30% through the considered redshift interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Purple and orange, dotted lines show the number of haloes, as a fraction of the total, which are unrelaxed due to either the condition on recent mergers or the con- dition on the accretion rate, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Most of the low-redshift haloes which are labelled unrelaxed are undergoing mergers, while at high redshift the cause for unrelaxedness is more usually a high level of smooth accretion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' This may be due to several reasons, amongst which we can mention the higher density in the vincinity of haloes at high redshift, or resolution limitations of the simulation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=', at high redshift, a halo may be accreting small, underresolved structures, which are therefore not accounted as mergers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='3 Indicators for the assembly state Many possible proxies for the dynamical and assembly state of a DM halo, or their corresponding baryonic structure (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=', a galaxy or a galaxy cluster) have been proposed in the literature (see, for instance, Cole & Lacey 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Crone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Shaw et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Haggar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2021a, just to cite a few).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' While simulations allow to access the complete three-dimensional picture, the lack of the whole information in observations (due to, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=', projection or the inability to observe the dark component, or even the plasma out to large radii) requires that, generally, different quantities are used for assessing the dynamical state in simulations and in observations (Rasia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' De Luca et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Yuan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' While comparison with observations is crucial and will be dealt with in future work, here we shall focus on the dynamical state indicators extracted from the complete, three-dimensional data in simulations as a first step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Unless otherwise specified, all quantities below are referred to the virial volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Centre offset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' The centre offset is usually defined as the distance between two different choices of centre, in units of some aperture radius (typically, the virial radius of the halo, 𝑅vir).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Many exam- ples for the choices of centre pair exist in the literature, such as centre of mass (CM) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' density peak (Baldi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2017) or CM vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' potential minimum (Biffi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2016), extracted from the three- dimensional description in simulations, or the morphological offset of the BCG location vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' X-ray surface brightness peak (Rossetti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2016), amongst many others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' We address the interested reader to Cui et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' (2016), who compare many different choices of observable for defining the centre of galaxy clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' In this work, we have tested the three possible combinations between the minimum of gravitational potential (defined as the location of the most-bound DM particle, as obtained by ASOHF and described in detail in Vallés-Pérez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2022), the DM density peak, and the DM centre-of-mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' We find that the most robust results are obtained for the Peak-CM pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Therefore, we defined the centre offset parametre as: Δ𝑟 = ��𝒓peak,DM − 𝒓CM,DM �� 𝑅vir (3) Virial ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' For a gravitational system in steady state, the virial theorem predicts 2𝑇 + 𝑊 − 𝐸𝑠 = 0, where 𝑇 is the kinetic energy, 𝑊 is the gravitational binding energy, and 𝐸𝑠 is the surface energy term (Chandrasekhar 1961).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Neglecting the surface term, the virial ratio is usually defined as 𝜂 ≡ 2𝑇 |𝑊| , (4) and it is expected that 𝜂 → 1 for isolated systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' However, haloes are not generally isolated systems, and therefore there is not a good a priori reason to drop the surface term in the virial theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Thus, many works define the virial ratio as 𝜂′ = (2𝑇 − 𝐸𝑠)/|𝑊| (Shaw et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2006), while others claim that the surface term overcorrects the virial ratio (Power et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' As the latter, we find that correcting the virial ratio by the surface term wipes out the correlation with merging activity, and thus we shall use the definition in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 4 in the remainder of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Mean radial velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' In a relaxed object, we do not expect impor- tant changes in the radial structure, while an unrelaxed system will experience significant disturbances as it settles down to equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' This motivates the consideration of the mean radial velocity of DM particles, ⟨𝑣𝑟⟩DM = � 𝑖 𝑚𝑖𝑣𝑟,𝑖 � 𝑖 𝑚𝑖 , (5) being 𝑚𝑖 the mass of the 𝑖-th DM particle, and 𝑣𝑟,𝑖 its radial velocity relative to the halo reference frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' In practical terms, we scale this quantity by the circular velocity at the virial radius, 𝑉circ,vir ≡ √︁ 𝐺𝑀vir/𝑅vir, and define the corresponding normalised indicator as ⟨ � 𝑣𝑟⟩DM ≡ |⟨𝑣𝑟⟩DM| 𝑉circ,vir .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' (6) Sparsity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Systems which have experienced recent significant merg- ers tend to display shallower central density profiles due to the MNRAS 000, 1–15 (2023) On the assembly state of DM haloes 5 disturbance caused by the infalling halo, and thus are less concen- trated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Many works (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=', Neto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2020) have pointed out the relation between the time spanned since the last major merger and halo concentration, 𝑐vir = 𝑅vir/𝑅s, being 𝑅s the scale radius of the Navarro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' (1997) profile (or the radius where the logarithmic slope of the DM density profile equals −2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' More recently, sparsity has been suggested as a non-parametric alternative to concentration, which reduces the scatter with halo mass (Balmès et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Corasaniti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2018), and has also been found to correlate with the timing since the last relevant merger (Richardson & Corasaniti 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' While sparsity is generally defined as the quotient between the masses at different spherical overdensi- ties, we find that the one maximising the correlation with merging activity is 𝑠200𝑐,500𝑐 ≡ 𝑀200𝑐 𝑀500𝑐 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' (7) Ellipticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' DM haloes are generally triaxial (Frenk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 1988;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Knebe & Wießner 2006), with significant scatter in halo shape at a given mass and redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Many recent studies have pointed out at the correlation between triaxiality and/or ellipticity of the halo shape and the formation history of a halo, with relaxed haloes tending to be rounder (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Lau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' We define the overall shape of the DM halo by finding the eigenvalues of the shape tensor, defined as 𝑆𝛼𝛽 = ∑︁ 𝑖 𝑚𝑖 𝑟𝑖,𝛼𝑟𝑖,𝛽 𝑟2 𝑖 , (8) which are proportional to the semiaxes squared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' The positions, 𝒓𝑖, are relative to the cluster centre (defined as the location of the density peak), and we choose to normalise them to be unit length to prevent the shape to be dominated by the particles in the outskirts of the halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Note this corresponds to the E2 method introduced by Zemp et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' If 𝑎, 𝑏 and 𝑐 are the semiaxes sorted in non-increasing order, we define the ellipticity of the halo, 𝜖, as: 𝜖 = 1 − 𝑐 𝑎 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' (9) Other indicators not considered in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Amongst the most widely used proxies for the dynamical state of DM haloes in the literature, we have not included the fraction of substructures, 𝑓sub, in this study (neither defined as the mass in substructures as a fraction of the host mass, nor as the ratio between the mass of the heaviest substructure and the host mass, as in Cialone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' While 𝑓sub should naturally correlate with the assembly state (especially, with the merging state), its interpretation is very subtle due to several factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' First, there is not a unique way to define the extent of a subhalo, and differences amongst halo finders have a dramatic impact on the recovered masses of substructures (see Vallés-Pérez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2022, their figures 5 and 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' In second place, the amount of substructure produced in simulations depends strongly, not only on resolution, but also on the numerical scheme employed to solve gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' This introduces strong mass biases (while the most massive haloes in our simulation may host well-resolved substructure, haloes with less than a few ten thousands particles are likely to be substructure-deficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' These obscure dependencies with mass, resolution and numerical scheme limit our ability to consistently incorporate this indicator in our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Simulations with enhanced resolution, capable of fully resolving rich substructure in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Summary of the redshift binning considered for the subsequent analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Each bin contains the haloes extracted from the 𝑁snaps available with 𝑧 ∈ [𝑧min, 𝑧max].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' The mean redshift of the 𝑁haloes haloes in the bin is ¯𝑧, with a fraction 𝑓unrelaxed of them being unrelaxed (either merging or experiencing intense accretion) according to the fiducial classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Note we report ¯𝑧, instead of the median, because 𝑧 is not continuously distributed (at each redshift bin, there are only 𝑁snaps different values of 𝑧).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 𝑧min 𝑧max 𝑁snaps ¯𝑧 𝑁haloes 𝑓unrelaxed 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='2 4 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='562 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='0 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='536 2742 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='600 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='0 2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='350 1791 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='657 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='0 2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='443 1564 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='751 our wide range of masses could be able to overcome this limitation of our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Regarding the indicators describing the shape of the mass dis- tribution, while 𝜖 alone does not fully characterise the shape of an ellipsoid, we have not considered any additional parameter, such as triaxiality𝑇 ≡ 𝑎2−𝑏2 𝑎2−𝑐2 (Franx et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' While ellipticity measures directly the deviation from sphericity, which is expected during as- sembly episodes, the same is not true for triaxiality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' As a matter of fact, triaxiality is undefined for spherical objects, and we do not find a clear reason to have a preference towards either prolate- ness/oblateness during mergers or strong accretion periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='4 Classification strategy 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='1 Redshift binning A total of 28 snapshots of the simulation, since 𝑧 = 5, are saved and used in this analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' To augment the statistics, we have grouped the snapshots in several redshift bins, which are described in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='2 Optimising the thresholds In a first step, we place a threshold, 𝑋thr 𝑖 , for each of the dynamical state indicators, 𝑋𝑖, described in the previous section (𝑖 = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' , 5, for the five dynamical state indicators).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' This is performed indepen- dently at each redshift bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' To do so, we vary 𝑋thr 𝑖 from the minimum to the maximum value of 𝑋𝑖 through the sample, and identify how well does 𝑋thr 𝑖 separate the relaxed and the unrelaxed samples of the fiducial classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' For each value of 𝑋thr 𝑖 , we compute two complementary met- 2 Not all bins contain the same number of snapshots (or haloes): higher redshift bins comprise less snapshots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' While this may increase the scatter in our results at high redshift, grouping more snapshots together at high redshift would increase the systematic uncertainty due to stacking objects of more different epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' MNRAS 000, 1–15 (2023) 6 Vallés-Pérez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' rics of the goodness of the classification3, namely the efficiency in discriminating the unrelaxed haloes, 𝜖unrelaxed(𝑋thr 𝑖 ) = # of unrelaxed haloes properly identified # of unrelaxed haloes (fiducial) (10) and the efficiency in discriminating the relaxed haloes, 𝜖relaxed(𝑋thr 𝑖 ) = # of relaxed haloes properly identified # of relaxed haloes (fiducial) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' (11) Out of all the possible values of 𝑋thr 𝑖 , we choose the one which maximises the product of both metrics, that is to say: ˆ𝑋thr 𝑖 = argmax𝑋thr 𝑖 � 𝜖unrelaxed(𝑋thr 𝑖 ) · 𝜖relaxed(𝑋thr 𝑖 ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' (12) Since 𝜖unrelaxed (𝜖relaxed) can be thought, in a frequentist ap- proach, as the probability of correctly identifying an unrelaxed (re- laxed) halo as such, our choice of threshold in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 12 corresponds to picking the one which enhances the likelihood of correctly clas- sifying both an unrelaxed and a relaxed halo, and thus serves as a compromise between too generous and too stringent thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='3 Totally relaxed, marginally relaxed and disturbed haloes Once the final (redshift-dependent) thresholds, � 𝑋thr 𝑖 (𝑧) �5 𝑖=1, are established, any halo will be regarded as totally relaxed if 𝑋𝑖 < 𝑋thr 𝑖 (𝑧) ∀𝑖 = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' , 5, (13) that is, if it has a low value of all the dynamical state indicators (low centre offset, mean radial velocity and ellipticity, virial ratio and sparsity close to unity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' This allows a very conservative definition of the most relaxed haloes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' However, it may be the case that a halo has a high value of one of the parametres, but is relaxed according to the rest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' This might be the case for a variety of reasons, ranging from physical (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=', a halo with high ellipticity due to a strong tidal field generated by the surrounding large-scale structure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2016) to numerical (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=', underresolved haloes with higher sparsities, misidentification of the centre, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Thus, we deal with all haloes not falling into the totally relaxed category by defining a combined relaxedness indicator, in the manner of Haggar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' (2020) (see also Kuchner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2021a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Gouin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2022) but adding weights which account for the fact that some dynamical state indicators can be more insightful than others at any given particular epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 𝜒 = ������ 𝑤1 � Δ𝑟 Δthr 𝑟 �2 + 𝑤2 � 𝜂 − 1 𝜂thr − 1 �2 + 𝑤3 � ⟨ � 𝑣𝑟⟩DM ⟨ � 𝑣𝑟⟩thr DM �2 + 𝑤4 � 𝑠200𝑐,500𝑐 − 1 𝑠thr 200𝑐,500𝑐 − 1 �2 + 𝑤5 � 𝜖 𝜖thr �2������ −1/2 (14) The weights, {𝑤𝑖}5 𝑖=1, are normalised so that �5 𝑖=1 𝑤𝑖 = 1, and 3 Note that the metrics introduced in Eqns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 10 and 11 also correspond, respectively, to the True Positive Rate (TPR) or sensitivity, and the True Negative Rate (TNR) or specificity in the usual jargon of binary classifica- tions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=', Fawcett 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' However, we choose this notation here for better readability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' are fixed at each redshift bin to be proportional to the performance of their corresponding indicator in splitting the merging and non- merging subsamples of the fiducial classifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' In particular, we set 𝑤𝑖 ∝ 𝜖relaxed𝜖unrelaxed − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='25 (the absolute values being set by the closure relation �5 𝑖=1 𝑤𝑖 = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' If, at a given redshift bin, 𝜖relaxed𝜖unrelaxed ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='25, we consider that the particular indicator is not meaningful and its weight is set to 𝑤𝑖 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' A particular halo which does not belong to the totally relaxed category will be classified as marginally relaxed if 𝜒 ≥ 1, and dis- turbed whenever 𝜒 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Additionally, this classification scheme can naturally handle missing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' For instance, if 𝑠200𝑐,500𝑐 is missing (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=', due to a low resolution not enabling to resolve 𝑅500𝑐), one can simply evaluate 𝜒 neglecting the sparsity term (and multiplying 𝜒 by a factor √1 − 𝑤4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' or, alternatively, renormalising the weights after setting 𝑤4 = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='4 Redshift evolution of the thresholds and weights With the procedure outlined in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='2 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='3, we obtain a threshold and a weight for each dynamical state indicator at each of the redshift bins specified in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' In order to obtain a continu- ous trend for each of these parameters, we fit them to polynomial functions of arbitrary degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' First, we estimate the uncertainties in the thresholds (𝑋thr 𝑖 ) and weights (𝑤𝑖) by computing the standard deviation of the distribution of these parametres obtained in 1000 bootstrap resamplings (Efron 1979).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Then, we fit the redshift evolution of the given parameter to polynomial functions of increasing degree, until the 𝑝-value of the highest degree coefficient falls above 𝑝 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='046 (low significance), or the reduced chi-squared falls below 1 (indicating possible overfit- ting of the model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Fits are performed using least squares weighted to the inverse of the variance of each data point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 3 RESULTS Following the procedure described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='2 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='4 over the whole sample, we have found the optimal thresholds for the dynam- ical state indicators, and fitted them to the best possible polynomial models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' The results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 3, from top to bottom, for the centre offset, virial ratio, mean radial velocity, sparsity and elliptic- ity thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Most of the thresholds on the assembly state indicators present a clear redshift evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' At earlier times, the thresholds on the dynamical state indicators tend to take higher values, reflecting the fact that haloes at earlier times were more irregular or exhibited more disturbed features, even when not having experienced any relevant merging activity or growth during the last dynamical time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' The evolution of the thresholds ranges from very mild or al- most nonexistent (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=', Δthr 𝑟 , 𝜀thr) to noticeable (and definitely worth taking into account;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=', 𝜂thr, 𝑠thr 200𝑐,500𝑐, ⟨ � 𝑣𝑟⟩thr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' This unequivo- cally evidences that fixed, set thresholds on certain parameters may not be able to correctly discriminate relaxed from merging haloes through the whole evolutionary history of the objects, especially when delving into the realm of high-redshift haloes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' The thresholds can be fitted by the following equations (solid lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 3, whose uncertainties are represented by the shaded regions), valid for 0 ≤ 𝑧 ≤ 5, where the figures in parentheses cor- respond to the uncertainty in the two last digits of each coefficient: Δthr 𝑟 (𝑧) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='0849(13) (15) MNRAS 000, 1–15 (2023) On the assembly state of DM haloes 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='080 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='085 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='090 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='095 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='105 thr r (z) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='7 thr(z) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='12 vr thr(z) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='55 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='60 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='65 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='70 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='80 sthr 200c, 500c(z) 0 1 2 3 4 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='28 thr(z) 0 1 2 3 4 5 Redshift, z Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Redshift evolution of the thresholds on the dynamical state indi- cators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' From top to bottom, the panels refer to the centre offset (Δthr 𝑟 ), virial ratio (𝜂thr), mean radial velocity (⟨� 𝑣𝑟 ⟩thr DM), sparsity (𝑠thr 200𝑐,500𝑐), and ellip- ticity (𝜀thr) thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Dots correspond to the optimal threshold obtained within the redshift bin, with the error bars obtained by means of bootstrap resampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Solid lines correspond to the best polynomial fits, with their (16-84)% confidence interval as the shaded region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 𝜂thr(𝑧) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='3383(56) + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='197(11)𝑧 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='0276(32)𝑧2 (16) ⟨ � 𝑣𝑟⟩thr DM(𝑧) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='0718(22) + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='0056(14)𝑧 (17) 𝑠thr 200𝑐,500𝑐(𝑧) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='491(16)+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='064(37)𝑧−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='031(22)𝑧2+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='0060(35)𝑧3 (18) 𝜀thr(𝑧) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='2696(27) (19) Based on the performance of each assembly state indicator in matching the fiducial classification, we fix the weights of each indi- cator in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 14 as described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' The results are summarised in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 4, which is analogous to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 3 but this time showing the weights instead of the thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Note that, if all indicators were equivalently important, 𝑤𝑖 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='2∀𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Thus, 𝑤𝑖 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='2 (𝑤𝑖 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='2) implies above-average (below-average) performance for the given dynamical state indicator at the given epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Interestingly, the importance of each indicator in determining the dynamical state of DM haloes varies strongly with redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' For example, one of the most widely used indicators, the centre offset Δ𝑟, is exceedingly effective in discriminating the disturbed haloes at high redshift, but its effectiveness declines steeply with decreasing redshift and has slightly below-average performance at 𝑧 ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' As an example of the opposite trend, the virial ratio, 𝜂, appears to be irrelevant at high redshift (𝑧 ≳ 2), and is only useful at low redshifts (≲ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' This dissimilar behaviour between centre offset and virial ratio is also reported by the analyses at high redshift of Davis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' As a purely dynamical parameter, the mean radial velocity ⟨ � 𝑣𝑟⟩ is especially relevant at high redshift, likely due to the fact that smooth (nearly radial) accretion could be more important at these stages given the relatively higher density in the surroundings of the halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Sparsity, as well as ellipticity, are especially correlated with the fiducial dynamical state classification at more recent redshifts, although they cannot generally be neglected at any epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' As a matter of fact, at low redshift, 𝜀 is the most relevant indicator of the dynamical state of haloes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' With the same procedure as above, we have fitted the weights to polynomial functions capturing their evolution (solid lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 4, whose uncertainties are represented by the shaded regions), valid for 0 ≤ 𝑧 ≤ 5: 𝑤[Δ𝑟](𝑧) ∝ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='1679(70) + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='0423(50)𝑧 (20) 𝑤[𝜂](𝑧) ∝ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='1965(78) − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='1037(60)𝑧 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='0134(11)𝑧2 (21) 𝑤[⟨ � 𝑣𝑟⟩DM](𝑧) ∝ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='1370(70) + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='0364(48)𝑧 (22) 𝑤[𝑠200𝑐,500𝑐](𝑧) ∝ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='2327(97) +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='051(14)𝑧 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='0153(38)𝑧2 (23) 𝑤[𝜀](𝑧) ∝ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='2603(75) − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='0181(51)𝑧 (24) We note that, while at any epoch the data points fulfilled MNRAS 000, 1–15 (2023) 8 Vallés-Pérez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='40 w[ r](z) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='20 w[ ](z) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='35 w[ vr ](z) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='30 w[s200c, 500c](z) 0 1 2 3 4 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='30 w[ ](z) 0 1 2 3 4 5 Redshift, z Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Redshift evolution of the weights on the dynamical state indicators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' From top to bottom, the panels refer to the centre offset (𝑤 [Δ𝑟 ]), virial ratio (𝑤 [𝜂]), mean radial velocity (𝑤 [⟨ ˜𝑣𝑟 ⟩]), sparsity (𝑤 [𝑠200𝑐,500𝑐 ]), and ellipticity (𝑤 [𝜀]) weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Dots correspond to the weights obtained within the redshift bin, with the error bars obtained by means of bootstrap resampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Solid lines correspond to the best polynomial fits, with their (16-84)% confidence interval as the shaded region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 0 1 2 3 4 5 Redshift, z 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='40 Relative weights r vr s200c, 500c Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Overall fitted redshift evolution of the weights of the dynamical state indicators, using the complete sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' �5 𝑖=1 𝑤𝑖 = 1, this is not necessarily true for the fitting polynomials evaluated at any arbitrary redshift (although it holds to a few per- cents).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Thus, they must be normalised by their sum before plugging them into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' For better comparison of the relative importance of each of the indicators, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 5 presents the fits of the weights on each dynamical state indicator, as a function of decreasing redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' The behaviour can be roughly summarised as: At high redshift (𝑧 ∈ [2, 5]), the centre offset provides the most insightful information about the recent assembly activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' This can be primarily complemented by the mean radial velocity (especially at 𝑧 ≳ 3), or sparsity and ellipticity (at 𝑧 ≲ 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' The virial ratio does not seem to provide any insight on the dynamical state at high redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' At intermediate redshifts, (𝑧 ∈ [1, 2]), sparsity, ellipticity and centre offset provide similarly useful information about the dynam- ical state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' The relevance of the virial ratio is still limited at this epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' At low redshifts (𝑧 ≲ 1), the ellipticity of the DM halo corre- lates exceptionally well with the dynamical state, as well as sparsity does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' While no dynamical state indicator is negligible at this stage, centre offset and virial ratio also present reasonable performances, while mean radial velocity is the least useful indicator at this time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='1 Dependence on halo mass The previous analyses have considered all haloes on an equal footing, despite their broad distribution in masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Invoking self- similarity (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=', Navarro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2010 for a thorough analysis on the level of self-similarity of haloes), it may be argued that the same thresholds and weights could be used for all halo masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' How- ever, the fact that many halo properties (related to the dynamical state indicators in our work) scale with mass demands to explicitly check how our results depend on the mass scale of the haloes being considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' We have split the complete sample, introduced in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='3, in two subsamples, namely a low-mass and a high-mass subsample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' The high-mass subsample contains, at each redshift, the 20% most massive haloes in the complete sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' This is chosen so as to MNRAS 000, 1–15 (2023) On the assembly state of DM haloes 9 0 1 2 3 4 5 Redshift, z 1012 1013 1014 1015 Mvir (M ) Low mass subsample High mass subsample Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Definition of the mass subsamples in terms of mass, as a function of redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' The gray (salmon) shaded regions correspond to the low-mass (high-mass) subsamples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' The dotted lines mark the mass limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' contain, at redshift 0, all the haloes associated to massive groups and clusters (𝑀DM > 3 × 1013𝑀⊙).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 6 shows the evolution of the mass limits on each subsample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Note that, therefore, our mass groups do not correspond to fixed-mass ranges, but rather to two sub-populations with a redshift-dependent mass threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' While it would definetely be interesting to explore the dependence of our thresholds and weights with actual mass ranges, our limited statistics prevent us from this goal and we may defer this for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Repeating the analyses above separately for each mass sub- sample, we can infer the mass dependence of the thresholds on the dynamical state indicators, and that of their corresponding weights in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' The redshift evolution of the thresholds for each mass subsample is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 7, which is analogous to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 3 but displaying only the fits for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' The same colour coding as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 6 is used here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' For some indicators, such as ellipticity, there is no hint for any significant trend of the evolution of the threshold with mass, at least within the statistical uncertainties given by our sample size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' That is to say, at least within the mass range considered in this work (roughly, [1012 − 1015] 𝑀⊙), the relaxation criteria based on this indicator can be used regardless of the scale of the objects (as customarily done with many indicators, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Power et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' However, the rest of indicators of the dynamical state do present significant dependence on halo mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' In particular, the threshold on Δ𝑟 remains constant with redshift for the low-mass subsample, while it increases linearly with increasing redshift for group and cluster- sized DM haloes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' This would suggest that imposing a constant threshold on the centre offset may be too conservative and could, for massive haloes at high redshift, artificially increase in excess the number of disturbed haloes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Regarding virial ratio, there is a minor trend with mass at intermediate and low redshifts (𝑧 ≲ 3), with higher-mass haloes preferring a slightly more stringent threshold to separate dynami- cally relaxed and unrelaxed haloes, but this difference corresponds to a small variation on the value of the parameter (Δ𝜂thr ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='05).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' The most striking difference appears at high redshift, but is not rel- evant since we have found that 𝜂 itself is not meaningful at high redshift (see the second panel in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 4, as well as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 8 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='13 thr r (z) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='8 thr(z) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='10 vr thr(z) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='55 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='60 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='65 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='70 sthr 200c, 500c(z) 0 1 2 3 4 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='2625 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='2650 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='2675 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='2700 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='2725 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='2750 thr(z) 0 1 2 3 4 5 Redshift, z Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Mass dependence of the redshift evolution of the thresholds on the dynamical state indicators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' The figure is analogous to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 3, with each line corresponding to the fit performed with a mass subsample (the same colour coding as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 6 is used: gray [salmon] lines correspond to the low-mass [high-mass] subsamples), and the shaded regions enclosing 1𝜎 confidence intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' MNRAS 000, 1–15 (2023) 10 Vallés-Pérez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' The mass dependence of ⟨ � 𝑣𝑟⟩ is moderate, with massive haloes preferring a constant threshold around ⟨ � 𝑣𝑟⟩ ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='085 and low-mass systems displaying a decreasing trend with decreasing redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Last, the two mass subsamples present a different behaviours with respect to the threshold on halo sparsity, 𝑠thr 200𝑐,500𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' In this case, lower mass haloes require consistently larger thresholds on sparsity to discriminate relaxed and merging objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' While smaller haloes tend to be more concentrated (see, for instance, Dutton & Macciò 2014), the mass-dependence of sparsity is much more contained (Corasaniti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Corasaniti & Rasera 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' It might be the case that, both for physical (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=', stronger influence of the environ- ment) and numerical (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=', less mass resolution elements leading to more unresolved central regions) reasons, low-mass haloes present a broader distribution in sparsities (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=', figure 10 in Balmès et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2014) and thus a larger sparsity threshold is required at the low mass end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Finally, we show in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 8 the evolution of the weights on each of the dynamical state indicators (as they appear in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 14), for each of the mass subsamples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Besides the general trends already analysed for the whole sample when Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 4 was presented, several differences emerge between the high-mass and the low-mass subsamples, espe- cially at intermediate and high redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' At low redshift, however, the weights are essentially compatible amongst the two subsam- ples, with only a small hint of virial ratio and centre offset being –comparatively– more effective in higher-mass haloes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' At intermediate redshifts, 𝑧 ∼ 2-3, high-mass haloes find el- lipticity and mean radial velocity to be better indicators of their dynamical state, and centre offset and sparsity comparatively worse ones, when confronted to the low-mass sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' At high redshifts, 𝑧 ∼ 5, the performance of sparsity gets more penalised for lower- mass haloes and, in these objects, centre offset can be relatively more important than in higher-mass haloes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' This further highlights that, besides not being able to put fix criteria (in the sense of them not evolving with redshift) for assessing the dynamical state of dark matter haloes, they have to be carefully chosen depending on the scale of the object being studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' We provide fits for the thresholds and weights for the high-mass (group and cluster-sized) subsample in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='2 Classification assessment The aim of this section is to validate to which extent the dynam- ical state classification introduced in this work, which only uses information at a given timestep, is capable of predicting the merg- ing state of the DM halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' That is to say, whether we can predict the fiducial classification (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='1) based on the dynamical state indicators, when confronting our method with haloes from a differ- ent simulation (corresponding to different resolution, gravity solver, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' We use public simulation data from the suite CAMELS (Villaescusa-Navarro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2021, 2022), which contains over 4000 simulations of 25ℎ−1 Mpc cubic, periodic volumes run with dif- ferent physics, cosmological and astrophysical parameters, and nu- merical codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' In particular, we have analysed the haloes in the IllustrisTNG-DM CV-0 simulation, which corresponds to a DM- only simulation run with Arepo (Springel 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Weinberger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Arepo implements a Tree+Particle-Mesh approach (Bagla 2002) for solving the evolution of DM, thus providing a high dy- namical range even though the number of particles (𝑁part = 2563) is rather small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' The CV-0 simulation corresponds to a background cosmology with ℎ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='6711, Ω𝑚 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='3, 𝑛𝑠 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='9624, and 𝜎8 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='8, the initial conditions having been set up at 𝑧ini = 127.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' We have extracted the halo catalogues and merger trees with ASOHF (Vallés-Pérez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2022) by following the exact same pro- cedure described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='2, and determined the dynamical state indicators (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' For our analyses, we have considered the 30 most massive haloes at each time, which corresponds to a similar mass limit as in the main analysis (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Out of these 30 haloes, we have dropped the ones that we are not able to trace back in time for at least one dynamical time (which is most usually none or one halo, at the considered epochs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 9, we assess the performance of our classification scheme at three cosmological epochs (𝑧 = 0, 1, and 2, for the pan- els left to right) by computing, at each time, the fraction of haloes in each class (totally relaxed, marginally relaxed and unrelaxed) which have recently suffered mergers or strong accretion (red), or has undergone a quiet evolution (green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' We note that, when evalu- ating the dynamical state criteria, at any given 𝑧 we only apply the indicators with weight 𝑤𝑖(𝑧) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Otherwise, we consider the given dynamical state indicator as not meaningful at that particular epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' While this particular threshold is arbitrary, it is a sensible choice and the results do not depend strongly on variations around this value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' According to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 5, this only removes the virial ratio, 𝜂, at 𝑧 ≳ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' The totally relaxed subsample is, naturally, the smallest one, since it is defined rather conservatively as the set of haloes simul- taneously fulfilling all five relaxation criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' It typically contains ∼ 10% of the haloes (slightly lower in this case;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' nevertheless, the statistics are small).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Within this test, all haloes being classified as totally relaxed have not suffered any major (minor) merger within one (half) 𝜏dyn or built up more than half of their mass in the last dynamical time, thus proving to be a selection of the relaxed sample with high specificity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' The marginally relaxed sample is the most numerous at low redshifts (𝑧 ≲ 1) and mostly contains objects which have not ex- perienced any relevant merging or accretion activity, although the fraction of objects having experienced it increases with increasing redshift (from ∼ 15% at 𝑧 ≃ 0 to ∼ 40% at 𝑧 ≃ 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' The unrelaxed subsample, which is especially numerous at high redshift when merger rates are higher (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=', Wetzel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2009), contains mostly merging objects, although a small fraction (10 − 25%) of objects not experiencing mergers or accretion seem to fall into this category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' This may happen because a halo appears to be disturbed, even when not merging or accreting intensely, due to environmental effects (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=', strong tidal field due to the presence of another nearby massive halo, for instance in a pre-merger state), or even numerical effects (mainly associated to low resolution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Naturally, it may also be the case that unrelaxedness after a major merger is last for longer than 1𝜏dyn(𝑧), since the fiducial classification in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='1 was only a rough estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Expanding upon the previous figure, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 10 we focus on some quantities tied to the merger and accretion history of the haloes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' In particular, the left panel represents, at three redshifts (𝑧 = 0, 1, and 2, respectively, from left to right) and for the three subsamples, the distribution of the time since the last merger (either major or minor) in units of the dynamical time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Haloes classified as unrelaxed have usually suffered some merger recently while, on the other hand, the totally relaxed sample has typically not experienced any merging activity since several dynamical times ago.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' A similar situation is seen for the major mergers (middle panel), although naturally not all unrelaxed haloes have suffered a major merger (the unrelaxedness can be due to one or several minor mergers, or smooth accretion, as well).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' However, the same trend holds, with MNRAS 000, 1–15 (2023) On the assembly state of DM haloes 11 0 1 2 3 4 5 Redshift, z 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='5 Relative weights Low mass sumbsample 0 1 2 3 4 5 Redshift, z High mass subsample r vr s200c, 500c Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Mass dependence of the redshift evolution of the weights on the dynamical state indicators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Each panel is analogous to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 5 (also using the same colour coding), showing the results for each of the mass samples defined in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 6: low-mass (left panel) and high-mass (right panel) subsamples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Totally relaxed (N = 2) Marginally relaxed (N = 20) Unrelaxed (N = 8) 0% 20% 40% 60% 80% 100% z = 0 Totally relaxed (N = 2) Marginally relaxed (N = 16) Unrelaxed (N = 11) 0% 20% 40% 60% 80% 100% z = 1 Totally relaxed (N = 1) Marginally relaxed (N = 7) Unrelaxed (N = 21) 0% 20% 40% 60% 80% 100% z = 2 Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Classification summary at redshifts 𝑧 = 0 (left panel), 𝑧 = 1 (middle panel), and 𝑧 = 2 (right panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Within each panel, each of the columns corresponds to one of the possible classifications (totally relaxed, if 𝑋𝑖 < 𝑋thr 𝑖 ∀𝑖;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' marginally relaxed, if the previous condition fails but 𝜒 ≥ 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' or unrelaxed otherwise).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Within each bar, the green portion represents the fraction of the haloes which have not suffered any mergers nor strong accretion, while the red portion corresponds to the fraction of haloes having suffered mergers or strong accretion (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=', the colour encodes the fiducial classification).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Below each column, 𝑁 indicates the number of objects falling into each category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 0 1 2 Redshift, z 0 1 2 3 tLM/ dyn Totally relaxed Marginally relaxed Unrelaxed 0 1 2 Redshift, z 0 1 2 3 4 5 tLMM/ dyn 0 1 2 Redshift, z 0 1 2 3 4 5 vir Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Trends of the evolutionary properties of the haloes according to their dynamical state classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' In each panel, red, orange and green dots correspond, respectively, to the unrelaxed, marginally relaxed and totally relaxed subsamples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' At each redshift (encoded in the horizontal axis) the dot and the error bars represent, respectively, the mean and the 1𝜎 dispersion of the distribution of the given variable within the subsample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' The left panel presents the time since the last merger (either major or minor) in units of the dynamical time (Δ𝑡LM/𝜏dyn), the central panel corresponds to the time since the last major merger (Δ𝑡LMM/𝜏dyn), and the right panel shows the accretion rate Γvir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' MNRAS 000, 1–15 (2023) 12 Vallés-Pérez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' major mergers having occurred a longer time ago as we move from unrelaxed to marginally relaxed, and to totally relaxed haloes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Finally, the third panel presents, in a similar way, the distribu- tion of accretion rates Γvir, which are computed according to the prescription of Diemer & Kravtsov (2014), Γvir = Δ log 𝑀vir Δ log 𝑎 (25) with 𝑎 being the scale factor of the Universe, and the increments computed over a dynamical time following its definition in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' The figure shows, in line with the previous results, an increasing trend of the accretion rate when moving from the relaxed to the more disturbed subsamples, within a wide redshift interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' This re- flects how the dynamical state classification presented in this work, which only uses information at a fixed time, can offer insight on the temporal evolution of the systems over the last dynamical time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='1 Does a smaller set of indicators provide similar insight?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Lastly, it might be interesting to assess whether a single dynamical state indicator, or a combination of them, is capable of providing a similarly accurate classification;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=', to motivate why it is important to involve a high number of indicators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' This is briefly exemplified in Table 2, where we show the classification summary for each indica- tor (or combination).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' In particular, for each classification class we give the number of haloes falling into this class (and its percentage with respect to the total), and the fraction of them which is unre- laxed according to the fiducial classification ( 𝑓merging).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Ideally, this fraction would be 0 for the totally relaxed class and 1 for the un- relaxed class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Naturally, when using only one indicator, there is no marginally relaxed or intermediate category, since the value of the dynamical state indicator can only be above or below the threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' In the case of using two indicators, we have defined the marginally relaxed sample as the set of haloes fulfilling only one of the two relaxedness conditions, as it is often done in the literature (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Biffi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Planelles et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Generally speaking, involving only one dynamical state indi- cator leads to far poorer results, since the relaxed sample gets often contaminated (around ∼ 40%) by haloes which have suffered merg- ers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Likewise, the unrelaxed sample may end up containing a high fraction of haloes undergoing quiescent evolution for some indica- tors (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=', 𝜖;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' although the particular results have to be considered carefully due to the reduced statistics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Interestingly, when using a combination of virial ratio and cen- tre offset, which is a common option in the literature (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=', Power et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2012), we are still not able to pick out all merging haloes with these criteria and even the totally relaxed subsample gets contami- nated with ∼ 40% of merging haloes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Other common options in the literature are the combination of mass ratio and centre offset (De Luca et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2021), or centre offset, virial ratio and mass ratio (Cui et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Haggar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' We have also tested these com- binations, taking 𝑓 thr sub = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='1 from the aforementioned references, since we have not involved this indicator in our previous analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' In these cases, the results are similar to the Δ𝑟 & 𝜂 combination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' This highlights the necessity of involving and combining as many indicators of the dynamical state as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' When using the full set of indicators derived in this work, the totally relaxed subsample is rather small, due to its conservative definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' However, even our marginally relaxed subsample is purer (contains a smaller frac- tion of merging/accreting haloes) than the totally relaxed sample of the previous combinations, proving to provide a robust splitting of haloes according to their dynamical state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 4 DISCUSSION AND CONCLUSIONS To fully exploit the capabilities of ongoing surveys (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=', eROSITA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Ghirardini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2022), and in the advent of upcoming instruments over the electromagnetic spectrum (from X-ray, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=', ATHENA, Nan- dra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' to radio, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' SKA, Acosta-Pulido et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' going through the optical, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' EUCLID, Sartoris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Euclid Col- laboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2019), which will provide samples of galaxies and galaxy clusters unprecedented in size and depth, it remains crucial to provide reliable indicators of the dynamical state (which in turns is a fast proxy of the –recent– evolution of the system).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' As a first step towards that aim, using 𝑁-Body+hydrodynamics simulations, in this work we have systematically analysed how to best combine a series of quantities which can be measured from simulation data at a given time in order to be able to detect the presence of mergers and/or ongoing strong accretion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' As a result, we have built an algorithm that combines a series of different indicators of the dynamical state of a DM halo (namely, its centre offset Δ𝑟, the virial ratio 𝜂, the mean radial velocity ⟨ � 𝑣𝑟⟩, the sparsity 𝑠200𝑐,500𝑐, and the ellipticity 𝜀) in order to classify haloes within three classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' The totally relaxed haloes, comprising the ob- jects simultaneously fulfilling all relaxedness conditions (which are redshift-dependent, in general), is a conservatively defined subsam- ple which, therefore, only contains around ∼ 10% of the haloes at a given time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Haloes where some relaxedness condition may fail, but are remarkably relaxed according to the rest of indicators may be categorised in the marginally relaxed class, using a criterion simi- lar to Haggar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' (2020), but allowing different indicators to have different (redshift-dependent) weights, which are tuned based on the performance of each indicator on telling relaxed and unrelaxed haloes apart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Thus, we defined a relaxedness parametre (𝜒), which tells marginally relaxed (𝜒 ≥ 1) and unrelaxed (𝜒 < 1) apart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' The fits for the redshift dependence of the thresholds and weights are given in Eqns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 15-24, while equivalent results for massive haloes are provided in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Furthermore, we have confronted our classification scheme against an independent DM-only simulation from the CAMELS suite (Villaescusa-Navarro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2021, 2022), corresponding to different input physics, initial conditions and numerical solvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Using it, we find that our algorithm performs a clean splitting of relaxed and unrelaxed haloes across a wide cosmic time interval, and that this classification improves upon the usage of any single indicator or some widely used combinations (Δ𝑟 & 𝜂;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 𝑓sub & 𝜂;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' or Δ𝑟, 𝜂 & 𝑓sub).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' As a qualitative summary of the main highlights of the classi- fication scheme, we can mention: Placing fix thresholds (which do not evolve with redshift) is generally undesirable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' While some indicators do not show strong evolution of their optimal thresholds with redshift (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=', ellipticity, centre offset), others do (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=', sparsity, mean radial velocity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' all tending to increase with redshift).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' This has important consequences, since it implies that classification schemes for the dynamical state of haloes that are set at 𝑧 = 0 cannot be directly used at high redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' – At high halo mass (see the precise definition of the high- mass subsample in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 6), however, the results are slightly changed: in particular, the redshift dependence of the thresholds MNRAS 000, 1–15 (2023) On the assembly state of DM haloes 13 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Classification properties using only one or a combination of dynamical state indicators, exemplified at 𝑧 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Each row corresponds to one dynamical indicator or combination, and for each classification class we give the number of objects falling into the class (𝑁 , and the percentage with respect to the total) and the fraction of them which is unrelaxed according to the fiducial classification ( 𝑓merging).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' When only one indicator is used, there is no intermediate (marginally relaxed) class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' The first block corresponds to the individual indicators involved in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' The second block contains several combinations widely used in the literature, with the fitted thresholds in this work (for 𝑓sub, not involved in this work, we use 𝑓 thr sub = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' The last row corresponds to the complete method introduced here, using the five indicators (therefore, these results are the same shown in the central panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Totally relaxed Marginally relaxed Unrelaxed Indicator(s) 𝑁 𝑓merging 𝑁 𝑓merging 𝑁 𝑓merging Δ𝑟 20 (69%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='40 – – 9 (31%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='78 𝜂 25 (86%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='48 – – 4 (14%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='75 ⟨ ˜𝑣𝑟 ⟩ 21 (72%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='43 – – 8 (28%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='75 𝑠200𝑐,500𝑐 17 (59%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='47 – – 12 (41%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='58 𝜖 10 (34%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='50 – – 19 (66%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='53 Δ𝑟 & 𝜂 19 (66%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='42 7 (24%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='57 3 (10%) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='00 𝑓sub 23 (79%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='48 – – 6 (21%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='67 𝜂 & 𝑓sub 18 (62%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='44 7 (24%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='43 4 (14%) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='00 Δ𝑟, 𝜂 & 𝑓sub 17 (59%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='47 3 (10%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='33 9 (31%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='67 Full set of indicators 2 (7%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='00 16 (55%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='38 11 (38%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='82 on ⟨ � 𝑣𝑟⟩ and 𝑠200𝑐,500𝑐 is not significant anymore, while the clas- sification based on the Δ𝑟 benefits from an increasing trend with increasing redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' This warns us that the classification cannot be universal, and that haloes on different mass scales may need slightly modified criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' At low redshift (𝑧 ≲ 1), even though all indicators offer insight into the merging state of the halo, it is sparsity and ellipticity of the DM halo the ones which provide the most valuable information, well beyond other, more widely used indicators such as centre offset or virial ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Nevertheless, the fact that all relative weights are not very dissimilar at this epoch (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 5) means that the classification scheme can importantly benefit from combining as many indicators as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' – The difference in weights amongst the different observables (except ⟨ � 𝑣𝑟⟩) is importantly reduced when looking at the high mass sample (right panel in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 8), reinforcing that, for group- and cluster-sized haloes at low redshift, it may be important to combine all indicators suggested in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' At high redshifts (𝑧 ≳ 3), 𝜂 becomes irrelevant for the deter- mination of the assembly state of the halo, while centre offset and mean radial velocity become, by far, the dominant indicators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' – Again, the differences are lower for the high-mass subsam- ple, but the prevalence of Δ𝑟 and ⟨ � 𝑣𝑟⟩ still holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' In this work, we have focused on the determination of the assembly state of DM haloes using the full information contained in a snapshot of a numerical simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' The motivation for this is two-fold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' On the one hand, it is important to devise efficient methods to classify large samples of simulated haloes, especially given the ever-growing trend of simulations, both in size and resolution (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=', Angulo & Hahn 2022, their table 1), made possible by the increasing computational power available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' On the other hand, it serves as a first step, which can be further connected to observations using projected data or, more realistically, mock multiwavelength observations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=', Planelles et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Much of the information comprised in the dynamical state indi- cators we involve in this work can be lost, or at least hindered, when moving from the 3-dimensional description to the 2-dimensional observed data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' The first, most natural consequence is the effect of projection on any geometrical indicator, such as the centre offset, ellipticity or the mean radial velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' For the case of centre off- set and ellipticity, the measured values will only be a lower limit, with the actual 3-dimensional value depending on the inclination between the direction of the offset, or the plane containing the major and minor axis, with the line of sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Regarding the mean radial velocity, which is especially impor- tant for determining the dynamical state at high redshift, besides the difficulty induced by projection (only velocities along the line of sight, and distances on the plane of the sky, can be measured), future kinetic Sunyaev-Zel’dovich (kSZ) observations could be able to provide some constraints on proper velocities of the intra-cluster medium (ICM;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' see, for instance, the estimates of Baldi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2018 about the kSZ effect due to the coherent rotation of the ICM), even for high-redshift objects since the SZ effect is essentially distance- independent (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=', Voit 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Even though the dynamics of the ICM, especially in the inner regions of haloes, may differ signifi- cantly from those of the DM halo, probing the velocity field of the diffuse gas in haloes could supply useful insight onto the dynamical state of haloes at high redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Lastly, sparsity may be a suitable option for observations, given its good performance shown across the whole redshift span consid- ered here (especially, for high-mass haloes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' However, care must be taken when using this quantity: here, we have defined sparsity from the DM masses obtained from the full, 3-dimensional infor- mation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' However, in observations, masses can be obtained from several methods (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=', hydrostatic, lensing, caustic masses), and bi- ases amongst them are non-negligible (see, for instance, Lovisari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Moreover, mass biases tend to correlate with the merg- ing state (Bennett & Sijacki 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Gianfagna et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2022) and, while the quotient of two masses at different apertures derived from the same method may cancel out part of these biases, the fact that the bias itself depends on the aperture and the large object-to-object scatter still make the interpretation non-trivial and deserve further attention themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' This work provides a motivated definition of a scheme for classifying DM haloes according to their dynamical status, based on simple properties which can be readily extracted from the outputs of typical halo finders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Future work will need to deal with the connection of these dynamical and morphological properties of the DM halo with the baryonic component, as well as the application to observations, in order to being able to extract the largest possible amount of information about the assembly state of haloes from future observational campaigns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' MNRAS 000, 1–15 (2023) 14 Vallés-Pérez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' ACKNOWLEDGEMENTS We gratefully thank the anonymous referee for their valuable feed- back, which has helped us to improve the quality of this manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' This work has been supported by the Agencia Estatal de Inves- tigación Española (AEI;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' grant PID2019-107427GB-C33), by the Ministerio de Ciencia e Innovación (MICIN) en el marco del Plan de Recuperación, Transformación y Resiliencia del Gobierno de España through the project ASFAE/2022/001 and by the General- itat Valenciana (grant PROMETEO/2019/071).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' DV acknowledges support from Universitat de València through an Atracció de Talent fellowship, and gratefully thanks the hospitality of the Dipartimento di Fisica e Astronomia of the Università di Bologna, where part of this work was done during a research stay funded by Universitat de València.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' We also thank F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Vazza and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Ragagnin for fruitful scientific conversations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Simulations have been carried out with the supercomputer Lluís Vives at the Servei d’Informàtica of the Uni- versitat de València.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' This research has made use of the following open-source packages: NumPy (Harris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2020), SciPy (Virtanen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2020), matplotlib (Hunter 2007), statsmodels (Seabold & Perktold 2010), scikit-learn (Pedregosa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 2011), and Colossus (Diemer 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' DATA AVAILABILITY The data underlying this article will be shared upon reasonable request to the corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' REFERENCES Acosta-Pulido J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' A.' metadata={'source': 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+page_content=' 6, corresponds to a redshift-dependent mass limit, which can be parametrised by log10 𝑀lim(𝑧) 𝑀⊙ = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='49 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='21𝑧 (A1) within ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='05 dex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Thus, the sample corresponds to massive groups and clusters at 𝑧 ∼ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' to objects above 1013𝑀⊙ at 𝑧 ∼ 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' and to a mass limit of ∼ 3 × 1012𝑀⊙ at 𝑧 ∼ 5, which might most often be the progenitors of the massive haloes we find at 𝑧 ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' Within this sample, the evolution with redshifts of the thresh- olds on the dynamical state indicators, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content=' 7, can be given by the following polynomial fits: Δthr 𝑟 (𝑧) ��massive = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='0863(39) + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQfUwAJ/content/2301.02253v1.pdf'} +page_content='0066(23)𝑧 (A2) 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Coronary Angiograms + +Chen Zhao1, Zhihui Xu2, Jingfeng Jiang3, Michele Esposito4, Drew Pienta5, Guang-Uei Hung6, Weihua +Zhou1,7* + +1. Department of Applied Computing, Michigan Technological University, Houghton, MI, USA +2. Department of Cardiology, The First Affiliated Hospital of Nanjing Medical University, Nanjing, China +3. Department of Biomedical Engineering, Michigan Technological University, Houghton, MI, USA +4. Department of Cardiology, Medical University of South Carolina, Charleston, SC, USA +5. Mechanical Engineering-Engineering Mechanics, Michigan Technological University, Houghton, MI, +USA +6. Department of Nuclear Medicine, Chang Bing Show Chwan Memorial Hospital, Changhua, Taiwan +7. Center for Biocomputing and Digital Health, Institute of Computing and Cyber-systems, and Health +Research Institute, Michigan Technological University, Houghton, MI, USA + +Corresponding author: +Weihua Zhou, PhD, Tel: +1 906-487-2666 +E-mail address: whzhou@mtu.edu +Mailing address: 1400 Townsend Dr, Houghton, MI 49931 + + + + +Abstract +Semantic labeling of coronary arterial segments in invasive coronary angiography (ICA) is important for +automated assessment and report generation of coronary artery stenosis in the computer-aided diagnosis of +coronary artery disease (CAD). However, separating and identifying individual coronary arterial segments +is challenging because morphological similarities of different branches on the coronary arterial tree and +human-to-human variabilities exist. Inspired by the training procedure of interventional cardiologists for +interpreting the structure of coronary arteries, we propose an association graph-based graph matching +network (AGMN) for coronary arterial semantic labeling. We first extract the vascular tree from invasive +coronary angiography (ICA) and convert it into multiple individual graphs. Then, an association graph is +constructed from two individual graphs where each vertex represents the relationship between two arterial +segments. Thus, we convert the arterial segment labeling task into a vertex classification task; ultimately, +the semantic artery labeling becomes equivalent to identifying the artery-to-artery correspondence on +graphs. More specifically, using the association graph, the AGMN extracts the vertex features by the +embedding module, aggregates the features from adjacent vertices and edges by graph convolution network, +and decodes the features to generate the semantic mappings between arteries. By learning the mapping of +arterial branches between two individual graphs, the unlabeled arterial segments are classified by the +labeled segments to achieve semantic labeling. A dataset containing 263 ICAs was employed to train and +validate the proposed model, and a five-fold cross-validation scheme was performed. Our AGMN model +achieved an average accuracy of 0.8264, an average precision of 0.8276, an average recall of 0.8264, and +an average F1-score of 0.8262, which significantly outperformed existing coronary artery semantic labeling +methods. In conclusion, we have developed and validated a new algorithm with high accuracy, +interpretability, and robustness for coronary artery semantic labeling on ICAs. + +Keywords: coronary artery disease, invasive coronary angiography, coronary arterial anatomy, +semantic labeling, graph matching network + + + + +1. Introduction +Coronary artery disease (CAD), caused by narrowing or blockages of the coronary arteries, is the most +common cardiovascular disease in the United States [1,2]. The narrowing is due to the buildup of fatty +plaque along the artery walls, composed of cholesterol, lipids, and fibrous tissue [3]. If one or more of these +arteries become severely obstructed, thereby reducing downstream blood flow, this may have the +deleterious consequence of resulting in myocardial ischemia or infarction [4]. +Invasive coronary angiography (ICA) remains the gold standard for diagnosing CAD [5]. ICA involves the +injection of contrast media into the epicardial arteries with the acquisition of continuous fluoroscopy +angiograms. For clinical decision-making, a trained cardiologist diagnoses CAD by subjectively assessing +the percent stenosis, or narrowing, of diseased arterial segments compared to normal arterial segments. +Automatic labeling of anatomical branches provides critical information for the diagnosis, report generation, +and region of interest visualization [6]. Meanwhile, measurement of coronary artery stenosis influences +patient management while improving diagnostic efficiency and confidence [7]. In clinical practice, +cardiologists and radiologists report the pathological findings for each arterial segment according to the +American Heart Association guidelines [8]. +The coronary vascular tree is complex and contains two major systems: the left coronary artery (LCA) and +the right coronary artery (RCA) systems. The LCA is more clinically relevant, given it provides most of +the blood supply to the left ventricle and is therefore associated with higher in-hospital mortality among +patients undergoing percutaneous coronary intervention [9]. The LCA system contains the left main artery +(LMA), left descending artery (LAD), and left circumflex artery (LCX), as well as multiple diagonal +arteries (D) and obtuse marginal (OM) branches [10]. The LMA arises from the aorta above the left cusp +of the aortic valve and perfuses the left ventricle's anterior, septal, and lateral walls. As shown in Figure +1(a), the LMA bifurcates into two main branches: the left anterior descending artery (LAD), which courses +between the left and right ventricles towards the apex along the anterior interventricular sulcus, and the left +circumflex artery (LCX), which runs laterally along the atrioventricular groove. The D and OM branches +originate from the LAD and LCX, respectively. It is worth noting that the main branches of the coronary +arterial tree are the LMA, LAD, and LCX, while the D and OM arteries are the side branches. Compared +to brain vessels and airways, semantic labeling of coronary arteries is more challenging due to variations in +length, size, and position [10,11]. + +Figure 1. (a) Visualization of coronary arterial tree in LCA system along with left ventricle and (b) +challenging examples for coronary artery semantic labeling. The yellow rectangles indicate the overlay of +arteries in 2D. For these three examples in (b), the LAD branches from the first two ICAs showed similarity + +LCX +LMA +OM +ICAI +ICA2 +ICA3 +LAD +LAD +LAD +D +Leftventricle +D +(a)VisualizationofLCAsystem +(b) Challenging examples for semantic labeling of the coronary arteriesin morphological and pixel-wise features with the D branch of the third ICA, demonstrating some +challenges accociated with coronary semantic segmentation. +Pixel-intensity-based models have difficulties distinguishing each arterial segment and generating semantic +segmentation because of the morphological similarity among different branches in the coronary vascular +tree and the overlap of the arteries in 2D (projection) ICA, as shown in Figure 1 (b). Also, the coronary +arteries not only span over a long distance but also show similar semantic features with each other [12], +making it challenging to associate them with the exact branches. False identifications of arterial segments +not only impair the understanding of the structure of the arterial tree but also causes inaccurate assessments +of vascular stenosis in the clinical workflow [13]. Existing methods only relying on position and imaging +features may produce unsatisfactory results when processing complicated coronary vasculature [14]. The +topology is a crucial factor in arterial identification, inspiring us to convert arteries and their connectivity +into graphs and perform coronary artery semantic labeling using graphs. +In this paper, we propose a novel algorithm to perform coronary artery semantic labeling using ICAs. The +key innovation of our coronary artery semantic labeling method is the use of a graph-matching network +(GMN) to build the semantic correspondence between arterial segments from different ICAs. Formally, +coronary artery semantic labeling is equivalent to finding the corresponding one-to-one mapping for arterial +segments from two different arterial graphs generated from the coronary vascular trees. This study proposes +an association-graph-based graph matching network (AGMN) to learn the similarities between arterial +segments from two (arterial) individual graphs. Consequently, the labeled segments classify the unlabeled +arterial segments to achieve semantic labeling by learning the matching of arterial branches between two +individual graphs. +The workflow of the proposed graph-matching approach for coronary artery semantic labeling is shown in +Figure 2. We first performed the coronary artery tree extraction as preprocessing the original ICA image +(see Figure 2 (a)). The entire vascular tree is extracted using FP-U-Net++, a coronary artery binary +segmentation model from a prior publication [15]. Then each centerline is extracted, and key points within +the centerline are detected. Each set of centerlines, including its affiliated key points, is further simplified +and denoised by applying a set of designated rules. Collectively, the vascular tree can be converted into an +individual graph. +For each edge and node in the individual graph, we extract the pixel-wise, positional, and topological +features for feature representation. An association graph is constructed from two individual graphs to +convert the arterial segment labeling task into a vertex classification task where each vertex represents the +relationship between two arterial segments, as shown in Figure 2 (b). The AGMN is then proposed to +perform the vertex classification as well as graph matching and coronary artery semantic labeling. +Extensive experiments, including the comparative experiments established by deep-learning and traditional +machine-learning methods for coronary artery semantic labeling, confirmed that the proposed AGMN +quantitatively and qualitatively outperformed other state-of-the art methods in accuracy and robustness. +The contributions and innovations of this work are as follows: +1) We designe five specific rules and a pipeline to preprocess the vascular tree and generate the +individual graphs according to ICAs. +2) We convert the pixel-to-pixel semantic segmentation task into a graph-matching task for coronary +artery semantic labeling. Our designed association graph-based graph-matching neural network +significantly outperforms the existing coronary arterial semantic labeling methods. +3) We exploit the robustness of the proposed method for coronary artery semantic labeling using the +corrupted datasets. + + + +Figure 2. Workflow of the proposed coronary artery preprocessing and Association-graph-based Graph +Matching Network (AGMN) for coronary artery semantic labeling. (a) Coronary arterial tree extraction and +key point detection. (b) Graph generation and graph matching for coronary artery semantic labeling. Each +ICA forms an individual graph consisting of nodal features extracted from each arterial segment. An +association graph is built by considering the connectivity of those two individual graphs. The AGMN +classifies vertices in the association graph into positive (in green) and negative vertices (in yellow). The +positive vertex represents the corresponding nodes in the individual graphs that are matched and have +identical semantic labels. Note that a node (i.e., an arterial segment) in the individual graph is referred to +as a node, while a node in the association graph is referred to as a vertex. + +2. Related Work +Coronary artery semantic segmentation. Coronary semantic segmentation is a challenging problem +because coronary arteries show similar features in pixel-wise appearance and morphology. According to +the problem definition, existing methods in the literature can be divided into two categories: 1) pixel-to- +pixel-based image semantic segmentation approaches and 2) multi-class classification-based arterial +segment labeling approaches. +The pixel-to-pixel-based image semantic segmentation model aims to assign a label to each pixel, a.k.a. +pixel-level classification [16]. Xian et al. proposed an end-to-end attention residual U-Net for major artery +segmentation on ICAs [17], where those major arteries include LAD, LXA, and RCA. Their model achieved +an average F1-score > 0.8 among 89% of selected ICAs. Silva et al. designed an EfficientU-Net++ and +achieved a Dice score of 0.8904 for major artery segmentation [18]. Zhang et al. proposed a progressive +perception learning framework containing the context, interference, and boundary perception modules for +major coronary artery segmentation. Although Zhang et al. achieved an average Dice score of greater than +0.95 for all types of major arteries [12], their model omitted the side branches, such as D and OM branches, +limiting its clinical use. + +(a) Coronaryarterial tree extraction +Centerline : +I Key Point : +FP-U-Net++ +Extraction +I Detection : +ICA +VascularTree +VascularCenterline +Key Points +Graph matching result +Arterial +node +segment +Visualize +feature +matching +extraction +result +GA +ICAI +Individualgraph +ICAI +ICA +generation +AGMNfor +Arteria +vertex +vertex +segment +classification +feature +ICA +node +extraction +Classification +Results +SemanticLabelins +.(b)GraphgenerationandgraphmatchingforcoronaryarterysemanticlabelingThe multi-class classification-based arterial labeling model is to assign a label to each arterial segment. +Compared to the end-to-end pixel-level classification, this approach focuses on segment-level classification. +Our previous work [14] is a machine-learning-based coronary artery semantic labeling method on ICAs, +which extracted the hand-crafted features for each arterial segment and employed the support vector +machine (SVM) for segment classification. Without using deep learning, the performance of feature +embedding was limited and thus induced a low segment classification performance. Cao et al. employed +prior knowledge and designed an iterative algorithm to label coronary arteries from main branches to side +branches on 3D Cardiac CT angiograms (CCTA) [19]. Wu et al. proposed a bi-directional tree-based long +short-term memory (LSTM) network for arterial segment semantic labeling on CCTAs [6]. The arterial +spatial locations and directions were used as the features for arterial segment classification. Yang et al. +converted the coronary artery semantic labeling task into a graph edge classification task and designed a +partial-residual graph convolutional network (GCN) for artery classification on CCTAs [20]. However, the +three methods mentioned above were developed for CCTA images instead of using ICA images. In short, +the arterial segment classification method using ICA images is underdeveloped. +Graph-matching using deep learning. Graph-matching aiming to establish a meaningful node and edge +correspondence between two graphs is often formulated as a graph edit distance problem [21], a maximum +common subgraph problem [22], or a quadratic assignment problem [23]. However, all these solutions are +NP-hard. Instead of solving the NP-hard problem, a graph-matching neural network is proposed to find the +node correspondence between graphs. GCN has been proposed to process the non-Euclidean data to +aggregate features from the adjacent nodes and edges [24]. Nowak et al. presented a GCN-based method +for solving the quadratic assignment problem by learning the pre-defined affinity matrix [25]. Wang et al. +proposed a cross-graph affinity learning approach for permutation learning to solve the graph-matching +problem [26]. Specifically, in Wang et al.'s approach, the graph-matching problem was relaxed to the linear +assignment problem, and a Sinkhorn classifier was adopted as the combinatorial solver. It is worth noting +that all three approaches above used an exact matching criterion, requiring the number of nodes between +graphs to be identical. However, the anatomical structures between coronary vascular trees from different +subjects vary; thus, an inexact graph-matching algorithm is required for coronary artery semantic labeling +[27]. +3. Methodology +The approach presented in this study focuses on segment-level classification for coronary artery semantic +labeling. Using AGMN, the problem of the semantic labeling task is converted into an equivalent problem +of finding one-to-one or one-to-zero mapping for arterial segments from two different vascular trees. Figure +1 illustrates the overall workflow, followed by details in subsequent sections below. +3.1. Coronary arterial tree extraction and individual graph generation +Coronary arterial contours are extracted using our previously developed Feature Pyramid U-Net++ (FP-U- +Net++) [15]. Then, each centerline is extracted to reduce redundant foreground pixels in a binary image +while preserving the connectivity and topology of the vascular tree. As a result, the vascular tree's centerline +and the arterial segments' diameters are calculated, as shown in Figure 2 (a). The workflow of individual +graph generation is shown in Figure 3. + + + +LCX +LCX +OM +LAD +LMA +LMA +LAD +18 +D +eFigure 3. Workflow of individual graph generation and vascular centerline post-processing: (a) original +ICA image; (b) centerline extraction and key point detection. The bifurcation points with degree > 3 are +marked in red, and the endpoints with degree = 1 are marked in green; (c) vascular structure after merging +the splitting points in the yellow region in (b) and deleting the cycle in the blue region in (b). In this sub- +figure, the bifurcation points with degree > 3 are marked in red, the endpoints with degree = 1 are marked +in green, and the degree two points generated by merging the splitting points are marked in blue; (d) +vascular structure after merging the degree two points in (c); (e) generated individual graph; (f) the final +individual graph by switching nodes and edges of the individual graph in (e). Since the graph matching is +to find the node (arterial segment) correspondence rather than the edge correspondence, we switch node +and edge in the individual graph in (e). In (c), (d), and (f), the classes of the arterial segments are labeled; +in (c), (d), and (e), the node indices are annotated. + +To convert the vascular tree into an arterial graph, we define two types of points: bifurcation points and +endpoints. A bifurcation point is an intersection point that bifurcates the arterial segments into sub-segments. +An endpoint represents the end of an arterial segment. This conversion process iterates through all points +in a centerline until all bifurcations and endpoints are extracted. As shown in Figure 3 (b), each centerline +contains multiple arterial segments and multiple bifurcations and endpoints. To find the links between one +bifurcation point and one endpoint, or links between two bifurcation points, each bifurcation and endpoint +is removed from the vascular centerline. Consequently, each vascular centerline is separated into several +arterial segments. +We design several rules and an algorithm to eliminate errors and build the individual graph. +i) Delete the capillary segments. If the maximum diameter associated with a point in the centerline is +smaller than the threshold 𝑇𝑑, then it has no significance for clinical analysis and is removed [15]. In +addition, any short arterial segments with less than 𝑇𝑐 pixels in the centerline are also removed. +ii) Merge splitting points. Typical errors in an automatically generated arterial graph are induced by splitting +points [28]. If two bifurcation points are located closely, then the splitting points affect graph topology +creation, as shown in the yellow circle in Figure 3 (b). A threshold was set to remove two splitting points +if the Euclidian distance between two bifurcation points is smaller than the threshold 𝑇𝑠𝑝. As a result, those +two points and related edges are merged. +iii) Delete cycles. To build an undirected acyclic graph for graph convolution, cycles in the graph must be +deleted. If a cycle exists, the arterial segment with a smaller diameter is removed, as shown in Figure 3 (b). +iv) Merge degree two points. After removing the capillary arterial segments and the splitting points, the +degrees of the bifurcation points are reduced to two. Then those bifurcation points are merged into the +connected two arterial segments, as shown in Figure 3 (c) and (d). +v) Switch nodes and edges. The bifurcation points and endpoints are nodes in a graph, and the edges are the +link between nodes. Naturally, each node in the individual graph represents a bifurcation point or an +endpoint, and each edge represents an arterial segment. However, our graph-matching network aims at +building node correspondence rather than edge correspondence. Semantic labeling requires the network to +build relationships between arterial segments rather than the detected key points. Thus, we switch nodes +and edges in the individual graph so that each node represents an arterial segment, and each edge represents +a bifurcation node (with degree ≥ 3). Consequently, a graph is generated for each vascular tree, denoted as +𝐺 = (𝑉, 𝐸) where 𝑉 is the node set, and 𝐸 is the edge set. The generated graph is shown in Figure 3 (f). +The adjacent matrix of the graph is generated by the connectivity of the vascular tree. +Our algorithm for arterial tree modification and individual graph generation is shown in Algorithm 1. + +Algorithm 1. Individual graph generation from a vascular tree. +3.2. Graph matching neural network for arterial segment labeling +Graph matching aims to find the node correspondence between two individual graphs 𝐺1 = (𝑉1, 𝐸1) and +𝐺2 = (𝑉2, 𝐸2), where |𝑉1| = 𝑛1, |𝑉2| = 𝑛2, |𝐸1| = 𝑛𝑒1 and |𝐸2| = 𝑛𝑒2. Without loss of generality, we +assume 𝑛1 ≤ 𝑛2 in this study. The two-graph matching problem can be written as a quadratic assignment +programming (QAP) [29,30], defined as +𝐽(𝑋) = 𝑣𝑒𝑐(𝑀)𝑇𝐾𝑣𝑒𝑐(𝑀), 𝑠. 𝑡. 𝑀 ∈ {0,1}𝑛1×𝑛2 +(1) +where 𝑀 ∈ ℝ𝑛1×𝑛2 is the permutation matrix encoding node-to-node correspondence and 𝐾 ∈ ℝ𝑛1𝑛2×𝑛1𝑛2 +is the affinity matrix of the association graph 𝐺𝐴 = (𝑉𝐴, 𝐸𝐴) generated by the connectivity of the graphs +𝐺1 and 𝐺2 , where |𝑉𝐴| = 𝑛1 × 𝑛2 and |𝐸𝐴| = 2 × 𝑛𝑒1 × 𝑛𝑒2 . 𝑣𝑒𝑐 indicates the vectorization. To +eliminate ambiguity, the node (an arterial segment) in the individual graph is defined as node, while the +node in the association graph is defined as vertex. +The vertices of the association graph 𝑉𝐴 = 𝑉1 × 𝑉2 encode the node-to-node correspondence, denoted as +𝑉𝑖,𝑎 +𝐴 = (𝑉𝑖 +1,𝑉𝑎2). The diagonal elements in 𝐾 represent the node-to-node matching level. For vertex +generation, each candidate's correspondence (𝑉𝑖 +1,𝑉𝑎2) ∈ 𝑉1 × 𝑉2 will be considered a vertex 𝑉𝑖𝑎 +𝐴. The edge +in the association graph is denoted as 𝐸𝑖𝑎,𝑗𝑏 +𝐴 +, where 𝑖, 𝑗 ∈ {1, … , 𝑛1} and 𝑎, 𝑏 ∈ {1, … , 𝑛2}. For edge +generation, we build an edge between a pair of vertices 𝑉𝑖𝑎 +𝐴 and 𝑉𝑗𝑏 +𝐴 if and only if there are two edges in +their own graphs that (𝑉𝑖 +1,𝑉𝑗 +1) ∈ 𝐸1 and (𝑉𝑎2, 𝑉𝑏 +2) ∈ 𝐸2. Thus, the graph-matching problem is converted to +classify vertices in the association graph 𝐺𝐴 into either positive vertices or negative vertices. For example, +if the node 𝑉𝑖 +1 ∈ 𝐺1 and 𝑉𝑎2 ∈ 𝐺2 have the same semantic labels, then the vertex 𝑉𝑖𝑎 +𝐴 ∈ 𝐺𝐴 is a positive +vertex, and vice versa. The overall association graph-based graph-matching network contains the following +five modules, as shown in Figure 4. +Input: +• +𝑋: input ICA image. +• +𝑇𝑑: threshold for removing an arterial segment if its diameter is smaller than 𝑇𝑑. +• +𝑇𝑐: threshold for removing an arterial segment if its centerline length is shorter than 𝑇𝑐. +• +𝑇𝑠𝑝: distance threshold for removing splitting points. +Output: 𝐺 = (𝑉, 𝐸): Generated individual graph. +1. Extract a vascular tree by FP-U-Net++ using 𝑋 and generate the vascular centerline; +2. Find bifurcation points and end points to separate the vascular centerline to arterial segments; +3. Calculate diameters and remove capillary segments by 𝑇𝑑; +4. Merge splitting points by 𝑇𝑠𝑝; +5. Delete cycles if two arterial segments stem from the same point and end at the same point; +6. Merge degree two points if the degrees of splitting points are reduced into two; +7. Generate the individual graph and switch nodes and edges to create the final individual graph 𝐺. + + +Figure 4. A flowchart showing the architecture of the association graph-based graph matching network: (a) +individual graph generation and node feature extraction; (b) association graph generation, ground truth +generation, and feature concatenation for vertices in the association graph; (c) feature representation +learning, including feature embedding, graph convolution network and feature decoding; (d) vertex +classification by major probability voting; (e) loss calculation and model training. +i) Extracting features for nodes in individual graphs. Using the processing algorithm, the individual graph +is generated. Each node in an individual graph is an arterial segment. We extract the pixel-derived features, +position-based features, and topological features for each arterial segment; all features are denoted as the +features for node in an individual graph, as 𝑥𝑖 +𝑔, s.t. 𝑔 ∈ {1,2} and 𝑖 ∈ {1, … , 𝑛𝑔}. For pixel-derived features, +we extract the radiomics features as in our previous work [14] and [31]. For position-based features and +topological features, we designed 22 hand-crafted features, as shown in Table 1. + + + +(a) + (b) +(c) +(d) +- +Feature +Vertex +Arterial +GAii +embedding +classification +segment +yia +feature +OM2 +extraction +Graph +convolution +ICA +Association +Nmp +Individual graph +graph +x +a +generation +Feature +G +GA: +decoding +Classification +Results +Arterial +segment +(e) +feature +ICA +Loss +extraction +Ground Truth +Y +Y +Table 1. Hand-crafted features for each arterial segment +Type +Index +Feature description +Pixel +feature +1 +Number of pixels in the artery segment +2 +Length of centerline +3-6 +Standard deviation, mean, the minimum and maximum radius of the artery segment +7-24 +First-order statistical radiomics features describing the distribution of pixel intensities +within the arterial segments +25-48 +Gray-level co-occurrence matrix (GLCM) features describing the second-order joint +probability function of an arterial segment +49-62 +Gray level dependence matrix (GLDM) features, counting the number of connected pixels +within distance 𝛿 that are dependent on the center pixel +63-78 +A gray level run length matrix (GLRLM) feature quantifies gray level runs, represented +by the length in the number of pixels that have the same gray level value +79-94 +Gray level size zone (GLSZM) features, counting the gray level zones in the arterial +segment +95-99 +Neighboring gray-tone difference matrix (NGTDM) features, counting the difference +between the gray value of a pixel and the average of its neighbors +Position +feature +100-103 +Weighted and absolute centers of the segment positions related to the center of the vascular +tree +104-111 +Weighted and absolute positions of the two key points related to the vascular tree center +112-119 +Weighted and absolute positions of the two key points related to the artery segment center +Topology +feature +120-121 +Degree of the two key points +ii) Extracting features for vertices in the association graph. The vertex features are generated by +concatenating node features from individual graphs 𝐺1 and 𝐺2, as shown in Eq. 2. +𝑥𝑖𝑎 = [𝑥𝑖 +1, 𝑥𝑎2], s.t. 𝑖 ∈ {1, ⋯ , 𝑛1}, 𝑎 ∈ {1, ⋯ , 𝑛2} +(2) +where [⋅] is the concatenation operator. For the edges in association graphs, the features are generated by +concatenating features of the edges in the individual graphs, as shown in Eq. 3. +𝑒(𝑖𝑎,𝑗𝑏) = [𝑒𝑖𝑗 +1 , 𝑒𝑎𝑏 +2 ], s.t. 𝑖, 𝑗 ∈ {1, ⋯ , 𝑛1}, 𝑎, 𝑏 ∈ {1, ⋯ , 𝑛2} +(3) +where 𝑒𝑖𝑗 +𝑔 = [𝑥𝑖 +𝑔, 𝑥𝑗 +𝑔], 𝑔 ∈ {1,2} and 𝑖, 𝑗 ∈ {1, … , 𝑛𝑔} represents features of edge in the individual graph, +constituted by the concatenation of the features of two connected nodes 𝑉𝑖 +𝑔 and 𝑉𝑗 +𝑔. Then, 𝑒(𝑖𝑎,𝑗𝑏) indicates +the features of edges in association graphs. +iii) Feature representation learning. We first develop a feature embedding module to embed the node and +edge features in the association graph into latent representations by multi-layer perceptions (MLPs). In this +study, the feature embedding is performed separately on vertices and edges, denoted as 𝑓𝑒𝑚𝑏 +𝑣 + and 𝑓𝑒𝑚𝑏 +𝑒 +. +Formally, the feature embedding module is defined in Eq. 4. +𝑥𝑖𝑎 +𝐴 =𝑓𝑒𝑚𝑏 +𝑣 +(𝑥𝑖𝑎) +𝑒(𝑖𝑎,𝑗𝑏) +𝐴 += 𝑓𝑒𝑚𝑏 +𝑒 +(𝑒(𝑖𝑎,𝑗𝑏)) +𝐺𝑒𝑚𝑏 +𝐴 += [𝑓𝑒𝑚𝑏 +𝑣 +(𝑥𝑖𝑎 +𝐴 ), 𝑓𝑒𝑚𝑏 +𝑒 +(𝑒(𝑖𝑎,𝑗𝑏) +𝐴 +)], s.t. 𝑖, 𝑗 ∈ {1, ⋯ , 𝑛1}, 𝑎, 𝑏 ∈ {1, ⋯ , 𝑛2} +(4) +After performing feature embedding, a GCN is employed to aggregate features from the adjacent vertices +for message passing [32,33]. We adopt the message-passing neural network to aggregate features from the +adjacent vertices [34], which contains a message-passing phase for feature aggregation and a readout phase +for feature update. In detail, the edge convolution layer first aggregates features from the two connected +vertices, and then updates its features iteratively, as shown in Eq. 5. +𝑒(𝑖𝑎,𝑗𝑏) +𝑡+1 += 𝜙𝑒([𝑒(𝑖𝑎,𝑗𝑏) +𝑡 +, 𝑥𝑖𝑎 +𝑡 , 𝑥𝑗𝑏 +𝑡 ]), 𝑠. 𝑡. 𝑖, 𝑗 ∈ {1, ⋯ , 𝑛1}, 𝑎, 𝑏 ∈ {1, ⋯ , 𝑛2}, and 𝑡 ∈ [1, ⋯ , 𝑁𝑚𝑝] +(5) + +where 𝜙𝑒 is the edge convolution layer implemented by MLP and 𝑒(𝑖𝑎,𝑗𝑏) +𝑡+1 + becomes the updated edge +features. 𝑡 indicates the index of the message passing, and 𝑁𝑚𝑝 is the total number of message-passing +steps. If 𝑡 = 1, then 𝑒(𝑖𝑎,𝑗𝑏) +𝑡 += 𝑒(𝑖𝑎,𝑗𝑏) +𝐴 + and 𝑥𝑖𝑎 +𝑡 = 𝑥𝑖𝑎 +𝐴 as defined in Eq. 4. +For each vertex, a vertex convolution layer is employed to aggregate features from the adjacence edges. +The vertex convolution layer first aggregates features from the adjacent edges in the association graph and +then updates its features iteratively, as shown in Eq. 6. +𝑥𝑖𝑎 +𝑡+1 = 𝜙𝑣 ([ ∑ 𝑒(𝑖𝑎,𝑗𝑏) +𝑡+1 +𝑗𝑏∈𝐸𝑖𝑎 +, 𝑥𝑖𝑎 +𝑡 ]) , 𝑠. 𝑡. 𝑖, 𝑗 ∈ {1, ⋯ , 𝑛1}, 𝑎, 𝑏 ∈ {1, ⋯ , 𝑛2} and 𝑡 ∈ [1, ⋯ , 𝑁𝑚𝑝] +(6) +where 𝐸𝑖𝑎 is a set containing the connected edges of vertex 𝑥𝑖𝑎 +𝐴 , and ∑ +⋅ +𝑗𝑏∈𝐸𝑖𝑎 indicates the element-wise +summation of the features from the adjacent edges. 𝜙𝑣 is the vertex convolution layer implemented by +MLPs. The per-vertex and per-edge features are computed independently, and the weights of vertex +convolution layer and edge convolution layer are shared to calculate per-vertex and per-edge affinity. +According to Eqs. 5 and 6, the updated vertex and edge features are denoted as 𝑥𝑖𝑎 +𝑁𝑚𝑝 and 𝑒(𝑖𝑎,𝑗𝑏) +𝑁𝑚𝑝 . +Unlike nature images, the visibility and anatomy between ICA images are different. Thus, the number of +nodes in two individual graphs may be different. Since we assume 𝑛1 ≤ 𝑛2, we manually select 𝐺1 and 𝐺2 +so that the number of nodes in 𝐺1 is smaller than that in 𝐺2. The selected 𝐺1 and 𝐺2 are used as a pair for +training the AGMN. Because the per-edge embedding, per-vertex embedding, and graph convolution layers +are reused across all edges and vertices, the designed AGMN automatically supports a form of +combinatorial optimization for graphs with a varying number of nodes, which is suitable and feasible for +coronary artery graph matching. Thus, the proposed AGMN is independent of the input individual graphs, +allowing it to perform inexact graph matching rather than the exact mapping problem for individual graphs +with the same number of nodes [35]. +After iteratively updating the edge features and vertex features, an MLP decoder module is employed to +convert the learned feature representation to vertex classification probability, denoted as 𝜙𝑑. Formally, the +output of the AGMN is shown in Eq. 7. +𝑦̂𝑖𝑎 = 𝜙𝑑 (𝑥𝑖𝑎 +𝑁𝑚𝑝), 𝑠. 𝑡. 𝑖 ∈ {1, ⋯ , 𝑛1},𝑎 ∈ {1, ⋯ , 𝑛2} +(7) +where 𝑦̂𝑖𝑎 indicates the probability of vertex 𝑉𝑖𝑎 +𝐴 belonging to a positive vertex. +iv) Vertex classification. Graph matching is equivalent to vertex classification, so a vertex classifier is +adopted to predict the matching results. According to the decoder, the matching probability between 𝑥𝑖 +1 and +𝑥𝑎2 is calculated by the decoder 𝜙𝑑 as 𝑦̂𝑖𝑎. Since each arterial branch is only matched with one arterial +branch, a major probability voting strategy is employed to generate the final prediction. Formally, the vertex +classification result is defined as: +𝑦̂𝑖𝑎 = { +1, 𝑖𝑓 argmax +𝑘∈{1,…,𝑛2} +𝑦̂𝑖𝑘 = 𝑎 +0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 + +(8) +In other words, the vertex 𝑦̂𝑖𝑎 in the vertex set {𝑥𝑖1 +𝐴 , 𝑥𝑖2 +𝐴 , ⋯ , 𝑥𝑖𝑛2 +𝐴 } is selected as the positive vertex if 𝑦̂𝑖𝑎 has +the highest probability among other vertices {𝑥𝑖1 +𝐴 , ⋯ , 𝑥𝑖,(𝑎−1) +𝐴 +, 𝑥𝑖,(𝑎+1) +𝐴 +⋯ , 𝑥𝑖𝑛2 +𝐴 }. +v) Loss function. The node-to-node correspondence between the vertex classification results and the ground +truth is used to guide the model training. The ground truth and classification results are denoted as two +permutation matrix 𝑀 ∈ ℝ𝑛1×𝑛2 and 𝑀̂ ∈ ℝ𝑛1×𝑛2, where the element in i-th row and a-th column indicates +the relationship between node 𝑉𝑖 +1 ∈ 𝐺1 and node 𝑉𝑎2 ∈ 𝐺2. The permutation loss [26] computed by the + +cross entropy between the predicted vertex class and the ground truth is used as an objective function, as +shown in Eq. 9. +𝐿𝑝𝑒𝑟𝑚 = 𝑐𝑟𝑜𝑠𝑠_𝑒𝑛𝑡𝑟𝑜𝑝𝑦(𝑀, 𝑀̂) = − (∑ ∑((1 − 𝑦̂𝑖𝑎)log(1 − 𝑦𝑖𝑎) + 𝑦̂𝑖𝑎 log 𝑦𝑖𝑎) +𝑛2 +𝑎=1 +𝑛1 +𝑖=1 +) +(9) +where 𝑦𝑖𝑎 is the ground truth of the vertex 𝑥𝑖𝑎 +𝐴 . If 𝑦𝑖𝑎 = 1, then the arterial segment 𝑉𝑖 +1 in 𝐺1 and the +arterial segment 𝑉𝑎2 in 𝐺2 have the same semantic labels. +3.3. Training and testing +Before introducing the training and testing strategies, we first demonstrate the method to generate the +labeled dataset. We manually annotated the ICA images with semantic labels for each arterial segment. +Then, the semantic label was assigned to each node (i.e., each arterial segment) in the individual graph. +During the creation of the database, the node correspondences between arterial segments are automatically +identified based on the semantic labels, i.e. 𝑦𝑖𝑎 = 1 if two arterial segments have the same types. However, +the main branches, such as LCX and LAD, are separated into several small branches during the individual +graph generation. Then, the arterial branches with the same semantic labels may have more than one node +in the individual graph. For example, in Figure 3 (d), the LAD branch has two segments due to the +bifurcation points for side branch D1. These segments are matched with the LAD segments from another +individual graph. If we build the ground truth of the association graph without re-naming semantic labels, +a complete bipartite graph or biclique will be built; in the bipartite graph or biclique, every node of the first +set is connected to every node of the second set [36]. In this example, the first LAD branch will have two +matched nodes, and two vertices connecting the first LAD branch are positive vertices in the association +graph. If a well-trained AGMN is obtained, the major probability voting strategy will fail to generate the +final decision because these two vertices' probabilities are equal to 1. +Our designed AGMN requires the individual graph and the matching relationship to satisfy the constraint +of one-to-one or one-to-zero mapping. One arterial segment or one node 𝑉𝑖 +1 in the individual graph 𝐺1 +should only have one or zero matched node in the graph 𝐺2. In this study, we annotate the arteries into +several sub-classes. For example, the LAD branch in Figure 3 (d) is separated into two segments named +LAD1 and LAD2. The LMA is upstream of the blood flow, and we follow this flow to assign the indices +of the main and side branches sequentially. Therefore, the LCX and LAD segments are separated into +several sub-segments. Each node of the first graph is only connected to one node of the second graph, and +the major probability voting strategy is then usable for this task. +To train the model, at each training step, two individual graphs from the same view were randomly selected +from the training set 𝐷𝑡𝑟. Only left coronary arteries (LCAs) were used for model training and validation +in this study. In addition, only ICAs from two regular views, left anterior oblique (LAO) and right anterior +oblique (RAO), were enrolled. Because of the anatomical difference between ICAs from different views, +the two selected individual graphs were from the ICAs with the same view. Since we assumed the number +of nodes 𝑛1 ≤ 𝑛2 in this study, we had to switch 𝐺1 and 𝐺2 according to the number of nodes during the +training. A batch of graph pairs was randomly generated for each training iteration to accelerate the model +training [37] and prevent the weights from trapping into the local minimum [38]. The association graph +was built according to the two selected graphs' semantic labels. The difference between the AGMN +prediction and the ground truth was used to calculate the loss and train the model, as defined in Eq. 9. +For the model testing, each individual graph in the test set 𝐷𝑡𝑒 is used to perform graph matching with a set +of graphs. The set is denoted as the template set, 𝐷𝑡𝑝. Cardiologists learn to read and understand the ICA +in clinical practice by comparing it with the representative ICAs. When making decisions, a set of +representative ICAs can be used as templates for reference. Our designed testing strategy imitates this +procedure. Each individual graph from the test set is paired with the individual graph from one +representative subject in the template set for graph matching. In the template set, each arterial segment is + +labeled for reference. Using the well-trained AGMN, the mapping relationship between unlabeled arteries +in the test subject and the labeled arteries from the template subject are obtained. The vertex classification +result for the test subject among the subjects in the template set is voted based on maximum voting. For +example, if LMA in the test subject is matched with LMA branches from five subjects in the template set +and is matched with LAD branches from two subjects in the template set, then this arterial segment in the +test subject is labeled as the LMA branch. +The designed training and testing strategies of our AGMN is shown in Algorithm 2. +Algorithm 2. Training and testing strategies of the proposed GMN for coronary arterial semantic labeling +3.4. Performance evaluation +The semantic labeling problem is converted into a multi-class classification problem among arterial +segments. As a classification problem, the weighted accuracy (ACC), weighted precision (PRE), weighted +recall (REC), and weighted F1-score (F1) are used to evaluate the model performance. We separate the +LAD and LCX branches into sub-segments during the model training. However, in the evaluation process, +we group the sub-segments into their original classes. The weighted ACC, SP, SN, and F1 definitions are +shown in Eqs. 10 to 13. +𝐴𝐶𝐶 = 1 +𝑛 ∑ +𝑇𝑃𝑐 + 𝑇𝑁𝑐 +𝑇𝑃𝑐 + 𝑇𝑁𝑐 + 𝐹𝑁𝑐 + 𝐹𝑃𝑐 +× 𝑛𝐶 +𝐶 +𝑐=1 + +(10) +𝑃𝑅𝐸 = 1 +𝑛 ∑ +𝑇𝑃𝑐 +𝑇𝑃𝑐 + 𝐹𝑃𝑐 +× 𝑛𝐶 +𝐶 +𝑐=1 + +(11) +Input: +• +𝐷𝑡𝑟: training set, contains 𝑁𝑡𝑟 labeled individual graphs. +• +𝐷𝑡𝑒: test set, contains 𝑁𝑡𝑒 unlabeled individual graphs. +• +𝐷𝑡𝑝: template set, contains 𝑁𝑡𝑝 labeled individual graphs. +• +𝑁: number of training steps. +• +𝑁𝑚𝑝: number of the message passing times. +Output: 𝑁𝑡𝑝 labeled individual graphs in 𝐷𝑡𝑒. +Training: +For 𝑖𝑡𝑒𝑟 = 1 ⋯ 𝑁 do +1. Random select two individual graphs 𝐺1 and 𝐺2 from 𝐷𝑡𝑟 from the same view; +2. Extract features for each arterial segment in 𝐺1 and 𝐺2; +3. Build association graph 𝐺𝐴 and extract features using Eqs. 2 and 3; +4. Update vertex and edge features of 𝐺𝐴 using Eqs. 4 to 6 for 𝑁𝑚𝑝 iterations by GCN; +5. Decode features and calculate the vertex class using Eqs. 7 and 8; +6. Calculate the loss function 𝐿𝑝𝑒𝑟𝑚 defined in Eq. 9 and optimize the AGMN. +Testing: +For each individual graph 𝐺𝑖 +𝑡𝑒 (𝑖 ∈ [1, ⋯ , 𝑁𝑡𝑒]) in 𝐷𝑡𝑒: +1. Extract features for each arterial segment in 𝐺𝑖 +𝑡𝑒; +For each individual graph 𝐺𝑗 +𝑡𝑝 (𝑗 ∈ [1, ⋯ , 𝑁𝑡𝑝]) in 𝐷𝑡𝑝: +2. Extract features for each arterial segment in 𝐺𝑗 +𝑡𝑝; +3. Build the association graph 𝐺𝐴 using 𝐺𝑖 +𝑡𝑒 and 𝐺𝑗 +𝑡𝑝; +4. Update vertex and edge features of 𝐺𝐴 using Eqs. 4 to 6 for 𝑁𝑚𝑝 iterations; +5. Decode features and calculate the vertex class using Eqs. 7 and 8; +6. Assign labels for nodes in 𝐺𝑖 +𝑡𝑒 according to major voting among 𝐺𝑗 +𝑡𝑝, 𝑗 ∈ [1, ⋯ , 𝑁𝑡𝑝]. + + +𝑅𝐸𝐶 = 1 +𝑛 ∑ +𝑇𝑁𝑐 +𝑇𝑁𝑐 + 𝐹𝑁𝑐 +× 𝑛𝐶 +𝐶 +𝑐=1 + +(12) +𝐹1 = 1 +𝑛 ∑ +𝑇𝑃𝑐 +𝑇𝑃𝑐 + 1 +2 (𝐹𝑃𝑐 + 𝐹𝑁𝑐) +𝐶 +𝑐=1 + × 𝑛𝑐 +(13) + +where 𝑇𝑃𝑐, 𝑇𝑁𝑐, 𝐹𝑃𝑐 and 𝐹𝑁𝑐 represent the true positive arterial segment, true negative arterial segment, +false positive arterial segment, and false negative arterial segment, respectively. 𝑐 refers to the class index +of arterial segments, and 𝐶 is the total number of classes. 𝑛𝐶 is the number of arterial segments in class 𝑐 +and 𝑛 is the total number of arterial segments. +4. Experiments and results +In this section, we conduct experiments to demonstrate the effectiveness of the proposed AGMN for +coronary artery semantic labeling. The enrolled subjects, experimental settings, and results will be +presented. +4.1. Dataset and enrolled subjects +In this study, we manually annotated 204 and 59 ICAs from site 1 [15] at The First Affiliated Hospital of +Nanjing Medical University and site 2 at Chang Bing Show Chwan Memorial Hospital, respectively. In +total, this retrospective study enrolled 263 ICA images. For site 1, subjects who received ICA from February +26, 2019, to July 18, 2019, were enrolled. The ICAs were performed using an interventional angiography +system (AXIOM-Artis, Siemens, Munich) and were acquired at 15 frames/sec. The image size of ICA +videos ranged from 512×512 to 864×864, and the pixel spacing ranged from 0.2 mm to 0.39 mm. For site +2, the ICAs were performed using an interventional angiography system (AlluraClarity, Philips Healthcare, +Eindhoven, Netherlands) and were acquired at 3.75, 7.5, 15, and 30 frames/sec. The image size of ICA +videos was 1024×1024, and the pixel spacing was 0.184 mm. Table 1 shows the number of images in each +ICA view used in this study. +Table 1. Views and corresponding image numbers in this paper. LCA, left coronary artery. LAO, left +anterior oblique; RAO, right anterior oblique. +Site +LAO +RAO +Total +Site 1 +55 +149 +204 +Site 2 +23 +36 +59 +For each patient, a frame that was used for anatomical structure analysis in clinical practice was selected +from the view video for semantic labeling. In this study, we only focus on semantic labeling for the main +branches of LMA, LAD, and LCX, and the side branches of D and OM. +4.2. Implementation details +We implemented our designed AGMN using TensorFlow and GraphNets [33] on an NVIDIA RTX 3090 +GPU card. The thresholds used in Algorithm 1 were set as 𝑇𝑑 = 1.8 mm, 𝑇𝑐 = 15 pixels, and 𝑇𝑠𝑝 = 8 +pixels. For the 263 ICA images, we selected first 𝑁𝑡𝑝 images as the labeled individual graphs for the +template set, while the rest ICAs were used for a five-fold cross-validation with the stratified sampling +according to the view angles of ICAs. Consequently, for each experiment, the training set contained 𝑁𝑡𝑟 = +(263 − 𝑁𝑡𝑝) × 0.8 samples, and the test set contained 𝑁𝑡𝑒 = (263 − 𝑁𝑡𝑝) × 0.2 samples. All images were +resized to 512 × 512 before extracting the features. Each model was fine-tuned for 100,000 training steps +using a batch size of 32. The Adam optimizer [39], with an initial learning rate of 0.0001, was employed as +the optimizer. We used the exponential decay strategy in the training phase to adjust the learning rate, and +we set the decay rate as 0.98 for each 2000 training steps. Each training step took 0.202 seconds, and the + +total training time one-fold was 5.6 hours. The hyperparameters were tuned on the test set during the cross- +validation for each hyperparameter setting. The grid search settings are shown in Table 2. +Table 2. Hyperparameter settings in the grid search +Hyperparameter +Search space +Description +Number of hidden units in +MLP +[16, 32, 64] +The MLP includes the feature embedding +module, GCN, and feature decoder module. For +each experiment, these two hyperparameters are +identically set to all MLP layers. +Number of MLP layers +[2, 3, 4] +Number of the message passing +steps (𝑁𝑚𝑝) +[2, 3, 4] +The number of the message passing steps +indicates the update iterations of the GCN +module in GMN. +Number of samples in the +template set (𝑁𝑡𝑝) +[27, 40, 52, 79] +10%, 15%, 20% and 30% of the ICA images +were selected using the stratified sampling +according to the view angles as the template set. + +More specifically, we first fixed the hyperparameters of 𝑁𝑡𝑝 as 40 (15% of the dataset) and 𝑁𝑚𝑝 as 3 before +we tuned the MLP layer size hyperparameters and the number of MLP layers. Then, we fixed the number +of hidden units in MLP, and the number of MLP layers, then tuned the hyperparameters of 𝑁𝑡𝑝 and 𝑁𝑚𝑝. +We selected the best parameter for each hyperparameter according to the highest average accuracy among +the five-fold evaluation. +4.3. Experimental results of AGMN +The best performance was achieved under the settings that the number of hidden units was 64 with 4 MLP +layers. The AGMN was trained using 40 samples (15%) as the template set. And the number of message- +passing steps was set as 4. Under this setting, the results for each type of arterial segment are listed in Table +3. +Table 3. A summary of the best performance achieved by our proposed AGMN for coronary artery +semantic labeling. LMA, left main artery; LAD, left descending artery; LCX, left circumflex artery; D, +diagonal artery; OM obtuse margin. +Artery type +ACC +PRE +REC +F1 +LMA +0.9956±0.0089 +0.9911±0.0109 +0.9956±0.0089 +0.9933±0.0089 +LCX +0.8432±0.0306 +0.8476±0.0481 +0.8432±0.0306 +0.8452±0.0386 +LAD +0.8046±0.0452 +0.8256±0.0307 +0.8046±0.0452 +0.8143±0.0310 +D +0.7956±0.0412 +0.7536±0.0493 +0.7956±0.0412 +0.7736±0.0424 +OM +0.7565±0.0825 +0.7613±0.0319 +0.7565±0.0825 +0.7569±0.0508 +All +0.8264±0.0302 +0.8276±0.0298 +0.8264±0.0302 +0.8262±0.0301 +Three examples are visualized for the graph-matching results using our proposed AGMN in Figure 5. + + +Figure 5. Graph matching results for three examples. The left ICAs are from the testing set, and the right +ICAs are from the template set. The green line indicates a correct match, and the red line represents a wrong +match. The corresponding arterial semantic labels are annotated. + +AD +01 +OM +LCXZ +OM2 +LCX3 +AD +OM1 +CX2 +0M2 +CX2 +QM +LCX3 +D1 +LCX2 +OM2 +CX34.4. Comparison with other coronary artery semantic labeling methods +We compared our proposed AGMN approach to four other coronary artery semantic labeling approaches. +Those benchmark approaches include: +• +Machine learning-based method [14]. Our previous work employed machine learning based +methods to perform coronary artery semantic labeling using ICAs. Each artery is extracted using +our FP-U-Net++ model and converted to vascular centerlines using step 1 to 3 in Algorithm 1. We +adopted the same pipeline, extracted the same features, and employed support vector machine with +radial basis function kernel as the classifier to perform arterial segment classification. +• +Bi-directional tree long short-term memory (BiTreeLSTM) [6]. Wu et al. developed a tree- +structured LSTM neural network for coronary artery labeling using CCTA. Arterial spatial and +directions in 3D polar coordination were used as the arterial segment features. The graph-structured +coronary artery was converted to a tree-structured arterial tree, and a bi-directional tree LSTM was +adopted for segment classification. The bi-directional sequence input stems from LMA to side +branches and from side branches to LMA. Since our ICA images are in 2D, we only extracted +arterial spatial and directions in 2D polar coordination as features. We adopted the same +architecture used in their paper for arterial segment classification. +• +Up-to-down (UTD) and down-to-up (DTU) nets [6]. These two baseline models were implemented +as the ablation study for BiTreeLSTM. Both UTD and DTU nets were implemented by a single +tree LSTM. However, the UTD net used the arterial segments from the root of the coronary vascular +tree, LMA, to any side branches for training, while the DTU net used the segments from the side +branches to LMA for training. +• +Conditional partial-residual GCN (CPR-GCN) [20]. Yang et al. developed a GCN neural network +embedded with a partial residual network for coronary artery semantic labeling using CCTA. The +positional features contained the arterial segment direction and position, and the imaging features +were extracted by convolution neural networks (CNN) and LSTM. All features were concatenated +and used for training. The output of the GCN is the segment classification result. We replaced the +3D position features with 2D features and replaced the 3D CNNs with 2D CNNs as the baseline. +The performance comparisons between the proposed AGMN and baseline models are illustrated in Table +4. We adopted five-fold cross-validation for each model and employed stratified sampling to split the +samples into training and testing datasets. For baseline models, including UTD, DTU, BiTreeLSTM, and +CPR-GCN, we also performed the grid search, and the models with the best performance were used for +comparison. + + + +Table 4. Comparisons between baseline methods and our proposed AGMN for coronary artery semantic +labeling using our ICA dataset. The means and standard deviations of the accuracy, precision, recall, and +F1-scores among the five folds are presented. The bold texts indicate they achieved the best performance +in their corresponding evaluation metrics. +Method +Metric +LMA +LAD +LCX +D +OM +All +Machine +learning [14] +ACC +0.9925±0.0151 +0.6331±0.0540 +0.6388±0.0390 +0.6147±0.0410 +0.5907±0.0419 +0.6651±0.0080 +PRE +0.9778±0.0071 +0.6586±0.0174 +0.6375±0.0378 +0.5554±0.0101 +0.6278±0.0261 +0.6679±0.0081 +REC +0.9925±0.0151 +0.6331±0.0540 +0.6388±0.0390 +0.6147±0.0410 +0.5907±0.0419 +0.6651±0.0080 +F1 +0.9850±0.0076 +0.6437±0.0213 +0.6360±0.0087 +0.5832±0.0234 +0.6071±0.0183 +0.6646±0.0077 +UTD net [6] +ACC +1.0000±0.0000 +0.8828±0.0136 +0.9245±0.0235 +0.0000±0.0000 +0.0000±0.0000 +0.6182±0.0094 +PRE +1.0000±0.0000 +0.7927±0.0167 +0.4629±0.0106 +0.0000±0.0000 +0.0000±0.0000 +0.4562±0.0069 +REC +1.0000±0.0000 +0.8828±0.0136 +0.9245±0.0235 +0.0000±0.0000 +0.0000±0.0000 +0.6182±0.0094 +F1 +1.0000±0.0000 +0.8353±0.0142 +0.6169±0.0143 +0.0000±0.0000 +0.0000±0.0000 +0.5135±0.0075 +DTU net [6] +ACC +0.0000±0.0000 +1.0000±0.0000 +0.7359±0.0194 +0.0000±0.0000 +0.3575±0.2923 +0.5450±0.0516 +PRE +0.0000±0.0000 +0.4284±0.0044 +0.6795±0.1010 +0.0000±0.0000 +0.5924±0.4837 +0.4264±0.1212 +REC +0.0000±0.0000 +1.0000±0.0000 +0.7359±0.0194 +0.0000±0.0000 +0.3575±0.2923 +0.5450±0.0516 +F1 +0.0000±0.0000 +0.5998±0.0043 +0.7018±0.0532 +0.0000±0.0000 +0.4458±0.3643 +0.4491±0.0838 +BiTreeLSTM +[6] +ACC +1.0000±0.0000 +0.8845±0.0150 +0.9871±0.0120 +0.0000±0.0000 +0.5981±0.0165 +0.7492±0.0085 +PRE +1.0000±0.0000 +0.8562±0.0190 +0.5853±0.0099 +0.0000±0.0000 +0.9808±0.0122 +0.6927±0.0074 +REC +1.0000±0.0000 +0.8845±0.0150 +0.9871±0.0120 +0.0000±0.0000 +0.5981±0.0165 +0.7492±0.0085 +F1 +1.0000±0.0000 +0.8699±0.0101 +0.7348±0.0093 +0.0000±0.0000 +0.7429±0.0141 +0.6967±0.0085 +CPR-GCN +[20] +ACC +0.5361±0.2996 +0.5319±0.1239 +0.5072±0.1447 +0.0624±0.0953 +0.5341±0.3045 +0.4581±0.0536 +PRE +0.6208±0.3240 +0.5675±0.0540 +0.3964±0.0570 +0.2802±0.3727 +0.3821±0.0139 +0.4463±0.1075 +REC +0.5361±0.2996 +0.5319±0.1239 +0.5072±0.1447 +0.0624±0.0953 +0.5341±0.3045 +0.4581±0.0536 +F1 +0.5698±0.3026 +0.5455±0.0899 +0.4353±0.0632 +0.0742±0.0957 +0.3924±0.1660 +0.4192±0.0661 +Our AGMN +ACC +0.9956±0.0089 +0.8432±0.0306 +0.8046±0.0452 +0.7956±0.0412 +0.7565±0.0825 +0.8264±0.0302 +PRE +0.9911±0.0109 +0.8476±0.0481 +0.8256±0.0307 +0.7536±0.0493 +0.7613±0.0319 +0.8276±0.0298 +REC +0.9956±0.0089 +0.8432±0.0306 +0.8046±0.0452 +0.7956±0.0412 +0.7565±0.0825 +0.8264±0.0302 +F1 +0.9933±0.0089 +0.8452±0.0386 +0.8143±0.0310 +0.7736±0.0424 +0.7569±0.0508 +0.8262±0.0301 +According to Table 4, our AGMN achieved the highest average accuracy of 0.8264, average precision of +0.8276, average recall of 0.8264, and average F1-score of 0.8262 among all types of coronary arteries, +which outperformed other baseline models significantly. +All models achieved high performance on main branches prediction, except for DTU net. The UTD and +BiTreeLSTM performed the perfect classification on LMA; the accuracy was 100%. Because LMA was +required to be manually assigned to build the tree-structured arteries, it did not reflect its actual performance. +The input sequence of UTD and BiTreeLSTM stems from LMA to the side branches. For the LAD branch, +the BiTreeLSTM achieved a higher precision and F1-score than the proposed AGMN due to the correct +classification of LMA. Since LAD is connected to LMA directly, so the classification task is relaxed. For +the LCX branch, the BiTreeLSTM achieved a higher recall than the proposed AGMN because of the prior +knowledge of the LMA branch. +Our AGMN outperformed other baselines with a large margin for the side branches. By comparing the +similarities of side branches between different individual graphs, the AGMN can differentiate D branches +and OM branches. In addition, the D branches are connected with LAD, and OM branches are connected +with LCX; the performance of side branch classification is guaranteed since the AGMN has achieved a +high classification performance of LAD and LCX branches. +4.5. Feature importance +In this study, we designed 121 hand-craft features, including pixel features, positional features, and +topological features. A leave-one-out technique [40] is adopted to identify the feature significance. A +feature is significant if the performance of semantic labeling decreases significantly when this feature is +replaced by zero. By ranking the accuracy drops, the importance of the feature is obtained. We set the same +hyperparameters demonstrated in section 4.3. We compared the averaged the accuracy changes among five- +folds between using the original dataset and the corrupted datasets, as shown in Figure 6. + + +Figure 6. Feature importance ranking for classifying coronary arterial segments. Feature significance was +determined by the accuracy drops between using raw features and zero-filled features. The vertical axis +indicates the feature names, while the horizontal axis indicates the drops in accuracy. +The top 15 features with the highest accuracy changes are shown in Figure 6. Among these 15 features, 2 +are topological features, 2 are pixel-wise features and the other 11 are positional features. For the +topological features, p1 degree and p2 degree indicate the degrees of the two endpoints of an artery segment. +If we set these two features as zeros, then the accuracy of coronary artery semantic labeling dropped about +30%, indicating the convincible importance of these two features. For the pixel features, +original_glcm_ldmn and original_glcm_lmc2 are the features calculated by the Gray Level Co-occurrence +Matrix, which describes the second-order joint probability function of the region of the arterial segments. +The remaining 11 features belong to hand-craft positional features, representing the weighted or absolute +coordinates of the center pixels within the arterial segments. Due to the anatomical structure of the coronary +artery, the position of the artery is important for classification. +4.6. Data attack +The proposed AGMN was trained and evaluated based only on the 'ideal' individual graphs. However, even +though our previous coronary artery binary segmentation model [12] has achieved the Dice similarity +coefficient of 0.8899, we cannot guarantee that it would generate satisfactory arterial contours for all ICAs. +To test the robustness of the designed model, we created the corrupted datasets by randomly removing parts +of arterial segments from the ICAs in the test set. The removed arterial segment must contain one endpoint +to generate a connected graph. Otherwise, if the arterial segment contains 2 bifurcation points and is +removed, the individual arterial graph would be split into two individual graphs as well as the vascular tree +would be separated. In this situation, human intervention is required. +Using the corrupted dataset, we compared the AGMN with the graph- or tree-based baseline models, +including BiTreeLSTM and CPR-GCN. We compared the performance drops using the corrupted dataset +by randomly removing 5%, 7.5%, 10%, 12.5%, 15%, 17.5%, and 20% arterial segments. The ACC, PREC, +REC, and F1 and their changes using different corrupted datasets are shown in Figure 7. + +p2_degree +pl_degree +weighted_y_center +y_center +Feature Name +x_center +weighted_x_center +pl_x_center +p2_abs_y_center +p2_abs_y_weighted_center +pl_x_weighted center +original_glcm_Idmn +p2_x_center +original_glcm_Imc2 +pl_abs_y_center +pl_abs_x_weighted_center +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +△ +Figure 7. The achieved ACC, PREC, REC, and F1 of the proposed AGMN, CPR-GCN, and BiTreeLSTM +using different corrupted datasets. The horizontal axis indicates the probability of deleting an artery segment +randomly. + +The results demonstrated that our AGMN was robust since the accuracy was above 0.79, even using the +corrupted datasets with 20% of the arterial segments removed. However, the accuracy changes of AGMN +were more significant than those of BiTreeLSTM. According to Table 4, we observed that the BiTreeLSTM +achieved a low performance on side branch classification. When generating the corrupted datasets, parts of +the side branches were removed, which didn't affect the overall performance significantly. We didn't +remove the LMA branches when testing the BiTreeLSTM using the corrupted datasets. The LMA is the + +Model Names +AGMN +CPR-GCN +BiTreeLSTM +0.9 +0.9 +0.8 +0.8 +0.7 +0.7 +0.6 +R +0.6 +A +0.5 +0.5 +0.4 +0.4 +0.3 +0.3 +.0% +od +o0 +o +oo +.0% +od +0 +1: +20. +No +0.9 +0.9 +0.8 +0.8 +0.7 +0.7 +c +E +0.6 +F 0.6 +R +0.5 +0.5 +0.4 +0.4 +0.3 +0.3 +oo +.0% +5% +.0% +.0% +0 +.5% +.0% +.0% +5. +17.5% +20. +NO +20root of the arterial tree. If we removed the LMA branch, the arterial tree would be separated into two sub- +trees. The tree-based model, BiTreeLSTM, failed to perform artery semantic labeling when using the data +without LMA, which is the root of the vascular tree. In addition, our proposed AGMN achieved the highest +ACCs, PRECs, RECs, and F1s using different corrupted datasets than other baseline models, indicating its +robustness and powerful performance. +5. Limitations and future work +The limitation of the proposed AGMN is that during the prediction, the graph matching procedures are +required to be performed between the test subject and every subject in the template set. If we employ fewer +subjects in the template set, the prediction time would be reduced. In the future, graph clustering will be +used to select the most representative subjects in each cluster and then construct the template set to +accelerate the prediction. +We use hand-crafted features as the pixel-level features to reduce the training time and model complexity. +However, the feature representation capability is limited compared to CNN-extracted features. In the future, +a light-weight deep learning-based method is recommended to automatically extract the pixel features for +each segment rather than the hand-crafted radiomics features. +6. Conclusion +In this paper, we developed and validated a new algorithm for coronary artery semantic labeling on ICAs +with high accuracy, interpretability, and robustness. A new workflow for the individual graph generation +and the association graph-based graph matching network was proposed. The association graph-based +approach, per-node, and per-edge feature representation learning network performed inexact graph +matching. The experimental results showed that our AGMN achieved the best performance and +significantly outperformed existing approaches. By analyzing the feature importance, the interpretability of +AGMN is guaranteed. Our AGMN still performed highly in the data attack experiments, even using +corrupted datasets. + +Credit authorship contribution statement +Chen Zhao: Conceptualization, methodology, coding, manuscript writing. +Zhihui Xu: Data management and clinical validation. +Jingfeng Jiang: Methodology and manuscript writing. +Michele Esposito: Clinical validation and manuscript writing. +Drew Pienta: Methodology and manuscript writing. +Guang-Uei Hung: Data management and manuscript writing. +Weihua Zhou: Supervision, project administration, funding acquisition, manuscript writing, review. + +Declaration of Competing Interest +The authors declare no conflicts of interest. + +Acknowledgment +This research was supported by a new faculty startup grant from Michigan Technological University +Institute of Computing and Cybersystems (PI: Weihua Zhou). 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MOGONET integrates multi-omics +data using graph convolutional networks allowing patient classification and biomarker identification. +Nat Commun. 2021 December;12(1):3445. + + diff --git a/UdE3T4oBgHgl3EQf0Atf/content/tmp_files/load_file.txt b/UdE3T4oBgHgl3EQf0Atf/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7a97af0ae8120ac8740ac758f3c0f82aba6d7006 --- /dev/null +++ b/UdE3T4oBgHgl3EQf0Atf/content/tmp_files/load_file.txt @@ -0,0 +1,1235 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf,len=1234 +page_content='AGMN: Association Graph-based Graph Matching Network for Coronary Artery Semantic Labeling on Invasive Coronary Angiograms Chen Zhao1, Zhihui Xu2, Jingfeng Jiang3, Michele Esposito4, Drew Pienta5, Guang-Uei Hung6, Weihua Zhou1,7* 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Department of Applied Computing, Michigan Technological University, Houghton, MI, USA 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Department of Cardiology, The First Affiliated Hospital of Nanjing Medical University, Nanjing, China 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Department of Biomedical Engineering, Michigan Technological University, Houghton, MI, USA 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Department of Cardiology, Medical University of South Carolina, Charleston, SC, USA 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Mechanical Engineering-Engineering Mechanics, Michigan Technological University, Houghton, MI, USA 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Department of Nuclear Medicine, Chang Bing Show Chwan Memorial Hospital, Changhua, Taiwan 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Center for Biocomputing and Digital Health, Institute of Computing and Cyber-systems, and Health Research Institute, Michigan Technological University, Houghton, MI, USA Corresponding author: Weihua Zhou, PhD, Tel: +1 906-487-2666 E-mail address: whzhou@mtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content='edu Mailing address: 1400 Townsend Dr, Houghton, MI 49931 Abstract Semantic labeling of coronary arterial segments in invasive coronary angiography (ICA) is important for automated assessment and report generation of coronary artery stenosis in the computer-aided diagnosis of coronary artery disease (CAD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' However, separating and identifying individual coronary arterial segments is challenging because morphological similarities of different branches on the coronary arterial tree and human-to-human variabilities exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Inspired by the training procedure of interventional cardiologists for interpreting the structure of coronary arteries, we propose an association graph-based graph matching network (AGMN) for coronary arterial semantic labeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' We first extract the vascular tree from invasive coronary angiography (ICA) and convert it into multiple individual graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Then, an association graph is constructed from two individual graphs where each vertex represents the relationship between two arterial segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Thus, we convert the arterial segment labeling task into a vertex classification task;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' ultimately, the semantic artery labeling becomes equivalent to identifying the artery-to-artery correspondence on graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' More specifically, using the association graph, the AGMN extracts the vertex features by the embedding module, aggregates the features from adjacent vertices and edges by graph convolution network, and decodes the features to generate the semantic mappings between arteries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' By learning the mapping of arterial branches between two individual graphs, the unlabeled arterial segments are classified by the labeled segments to achieve semantic labeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' A dataset containing 263 ICAs was employed to train and validate the proposed model, and a five-fold cross-validation scheme was performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Our AGMN model achieved an average accuracy of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content='8264, an average precision of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content='8276, an average recall of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content='8264, and an average F1-score of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content='8262, which significantly outperformed existing coronary artery semantic labeling methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' In conclusion, we have developed and validated a new algorithm with high accuracy, interpretability, and robustness for coronary artery semantic labeling on ICAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Keywords: coronary artery disease, invasive coronary angiography, coronary arterial anatomy, semantic labeling, graph matching network 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Introduction Coronary artery disease (CAD), caused by narrowing or blockages of the coronary arteries, is the most common cardiovascular disease in the United States [1,2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' The narrowing is due to the buildup of fatty plaque along the artery walls, composed of cholesterol, lipids, and fibrous tissue [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' If one or more of these arteries become severely obstructed, thereby reducing downstream blood flow, this may have the deleterious consequence of resulting in myocardial ischemia or infarction [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Invasive coronary angiography (ICA) remains the gold standard for diagnosing CAD [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' ICA involves the injection of contrast media into the epicardial arteries with the acquisition of continuous fluoroscopy angiograms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' For clinical decision-making, a trained cardiologist diagnoses CAD by subjectively assessing the percent stenosis, or narrowing, of diseased arterial segments compared to normal arterial segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Automatic labeling of anatomical branches provides critical information for the diagnosis, report generation, and region of interest visualization [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Meanwhile, measurement of coronary artery stenosis influences patient management while improving diagnostic efficiency and confidence [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' In clinical practice, cardiologists and radiologists report the pathological findings for each arterial segment according to the American Heart Association guidelines [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' The coronary vascular tree is complex and contains two major systems: the left coronary artery (LCA) and the right coronary artery (RCA) systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' The LCA is more clinically relevant, given it provides most of the blood supply to the left ventricle and is therefore associated with higher in-hospital mortality among patients undergoing percutaneous coronary intervention [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' The LCA system contains the left main artery (LMA), left descending artery (LAD), and left circumflex artery (LCX), as well as multiple diagonal arteries (D) and obtuse marginal (OM) branches [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=" The LMA arises from the aorta above the left cusp of the aortic valve and perfuses the left ventricle's anterior, septal, and lateral walls." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' As shown in Figure 1(a), the LMA bifurcates into two main branches: the left anterior descending artery (LAD), which courses between the left and right ventricles towards the apex along the anterior interventricular sulcus, and the left circumflex artery (LCX), which runs laterally along the atrioventricular groove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' The D and OM branches originate from the LAD and LCX, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' It is worth noting that the main branches of the coronary arterial tree are the LMA, LAD, and LCX, while the D and OM arteries are the side branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Compared to brain vessels and airways, semantic labeling of coronary arteries is more challenging due to variations in length, size, and position [10,11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' (a) Visualization of coronary arterial tree in LCA system along with left ventricle and (b) challenging examples for coronary artery semantic labeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' The yellow rectangles indicate the overlay of arteries in 2D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' For these three examples in (b), the LAD branches from the first two ICAs showed similarity LCX LMA OM ICAI ICA2 ICA3 LAD LAD LAD D Leftventricle D (a)VisualizationofLCAsystem (b) Challenging examples for semantic labeling of the coronary arteriesin morphological and pixel-wise features with the D branch of the third ICA, demonstrating some challenges accociated with coronary semantic segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Pixel-intensity-based models have difficulties distinguishing each arterial segment and generating semantic segmentation because of the morphological similarity among different branches in the coronary vascular tree and the overlap of the arteries in 2D (projection) ICA, as shown in Figure 1 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Also, the coronary arteries not only span over a long distance but also show similar semantic features with each other [12], making it challenging to associate them with the exact branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' False identifications of arterial segments not only impair the understanding of the structure of the arterial tree but also causes inaccurate assessments of vascular stenosis in the clinical workflow [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Existing methods only relying on position and imaging features may produce unsatisfactory results when processing complicated coronary vasculature [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' The topology is a crucial factor in arterial identification, inspiring us to convert arteries and their connectivity into graphs and perform coronary artery semantic labeling using graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' In this paper, we propose a novel algorithm to perform coronary artery semantic labeling using ICAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' The key innovation of our coronary artery semantic labeling method is the use of a graph-matching network (GMN) to build the semantic correspondence between arterial segments from different ICAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Formally, coronary artery semantic labeling is equivalent to finding the corresponding one-to-one mapping for arterial segments from two different arterial graphs generated from the coronary vascular trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' This study proposes an association-graph-based graph matching network (AGMN) to learn the similarities between arterial segments from two (arterial) individual graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Consequently, the labeled segments classify the unlabeled arterial segments to achieve semantic labeling by learning the matching of arterial branches between two individual graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' The workflow of the proposed graph-matching approach for coronary artery semantic labeling is shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' We first performed the coronary artery tree extraction as preprocessing the original ICA image (see Figure 2 (a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' The entire vascular tree is extracted using FP-U-Net++, a coronary artery binary segmentation model from a prior publication [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Then each centerline is extracted, and key points within the centerline are detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Each set of centerlines, including its affiliated key points, is further simplified and denoised by applying a set of designated rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Collectively, the vascular tree can be converted into an individual graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' For each edge and node in the individual graph, we extract the pixel-wise, positional, and topological features for feature representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' An association graph is constructed from two individual graphs to convert the arterial segment labeling task into a vertex classification task where each vertex represents the relationship between two arterial segments, as shown in Figure 2 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' The AGMN is then proposed to perform the vertex classification as well as graph matching and coronary artery semantic labeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Extensive experiments, including the comparative experiments established by deep-learning and traditional machine-learning methods for coronary artery semantic labeling, confirmed that the proposed AGMN quantitatively and qualitatively outperformed other state-of-the art methods in accuracy and robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' The contributions and innovations of this work are as follows: 1) We designe five specific rules and a pipeline to preprocess the vascular tree and generate the individual graphs according to ICAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' 2) We convert the pixel-to-pixel semantic segmentation task into a graph-matching task for coronary artery semantic labeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Our designed association graph-based graph-matching neural network significantly outperforms the existing coronary arterial semantic labeling methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' 3) We exploit the robustness of the proposed method for coronary artery semantic labeling using the corrupted datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Workflow of the proposed coronary artery preprocessing and Association-graph-based Graph Matching Network (AGMN) for coronary artery semantic labeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' (a) Coronary arterial tree extraction and key point detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' (b) Graph generation and graph matching for coronary artery semantic labeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Each ICA forms an individual graph consisting of nodal features extracted from each arterial segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' An association graph is built by considering the connectivity of those two individual graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' The AGMN classifies vertices in the association graph into positive (in green) and negative vertices (in yellow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' The positive vertex represents the corresponding nodes in the individual graphs that are matched and have identical semantic labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Note that a node (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=', an arterial segment) in the individual graph is referred to as a node, while a node in the association graph is referred to as a vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Related Work Coronary artery semantic segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Coronary semantic segmentation is a challenging problem because coronary arteries show similar features in pixel-wise appearance and morphology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' According to the problem definition, existing methods in the literature can be divided into two categories: 1) pixel-to- pixel-based image semantic segmentation approaches and 2) multi-class classification-based arterial segment labeling approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' The pixel-to-pixel-based image semantic segmentation model aims to assign a label to each pixel, a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' pixel-level classification [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Xian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' proposed an end-to-end attention residual U-Net for major artery segmentation on ICAs [17], where those major arteries include LAD, LXA, and RCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Their model achieved an average F1-score > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content='8 among 89% of selected ICAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Silva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' designed an EfficientU-Net++ and achieved a Dice score of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content='8904 for major artery segmentation [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' proposed a progressive perception learning framework containing the context, interference, and boundary perception modules for major coronary artery segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Although Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' achieved an average Dice score of greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content='95 for all types of major arteries [12], their model omitted the side branches, such as D and OM branches, limiting its clinical use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' (a) Coronaryarterial tree extraction Centerline : I Key Point : FP-U-Net++ Extraction I Detection : ICA VascularTree VascularCenterline Key Points Graph matching result Arterial node segment Visualize feature matching extraction result GA ICAI Individualgraph ICAI ICA generation AGMNfor Arteria vertex vertex segment classification feature ICA node extraction Classification Results SemanticLabelins .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' (b)GraphgenerationandgraphmatchingforcoronaryarterysemanticlabelingThe multi-class classification-based arterial labeling model is to assign a label to each arterial segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Compared to the end-to-end pixel-level classification, this approach focuses on segment-level classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Our previous work [14] is a machine-learning-based coronary artery semantic labeling method on ICAs, which extracted the hand-crafted features for each arterial segment and employed the support vector machine (SVM) for segment classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Without using deep learning, the performance of feature embedding was limited and thus induced a low segment classification performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' employed prior knowledge and designed an iterative algorithm to label coronary arteries from main branches to side branches on 3D Cardiac CT angiograms (CCTA) [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' proposed a bi-directional tree-based long short-term memory (LSTM) network for arterial segment semantic labeling on CCTAs [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' The arterial spatial locations and directions were used as the features for arterial segment classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' converted the coronary artery semantic labeling task into a graph edge classification task and designed a partial-residual graph convolutional network (GCN) for artery classification on CCTAs [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' However, the three methods mentioned above were developed for CCTA images instead of using ICA images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' In short, the arterial segment classification method using ICA images is underdeveloped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Graph-matching using deep learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Graph-matching aiming to establish a meaningful node and edge correspondence between two graphs is often formulated as a graph edit distance problem [21], a maximum common subgraph problem [22], or a quadratic assignment problem [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' However, all these solutions are NP-hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Instead of solving the NP-hard problem, a graph-matching neural network is proposed to find the node correspondence between graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' GCN has been proposed to process the non-Euclidean data to aggregate features from the adjacent nodes and edges [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Nowak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' presented a GCN-based method for solving the quadratic assignment problem by learning the pre-defined affinity matrix [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' proposed a cross-graph affinity learning approach for permutation learning to solve the graph-matching problem [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Specifically, in Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=" 's approach, the graph-matching problem was relaxed to the linear assignment problem, and a Sinkhorn classifier was adopted as the combinatorial solver." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' It is worth noting that all three approaches above used an exact matching criterion, requiring the number of nodes between graphs to be identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' However, the anatomical structures between coronary vascular trees from different subjects vary;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' thus, an inexact graph-matching algorithm is required for coronary artery semantic labeling [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Methodology The approach presented in this study focuses on segment-level classification for coronary artery semantic labeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Using AGMN, the problem of the semantic labeling task is converted into an equivalent problem of finding one-to-one or one-to-zero mapping for arterial segments from two different vascular trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Figure 1 illustrates the overall workflow, followed by details in subsequent sections below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Coronary arterial tree extraction and individual graph generation Coronary arterial contours are extracted using our previously developed Feature Pyramid U-Net++ (FP-U- Net++) [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Then, each centerline is extracted to reduce redundant foreground pixels in a binary image while preserving the connectivity and topology of the vascular tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=" As a result, the vascular tree's centerline and the arterial segments' diameters are calculated, as shown in Figure 2 (a)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' The workflow of individual graph generation is shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' LCX LCX OM LAD LMA LMA LAD 18 D eFigure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Workflow of individual graph generation and vascular centerline post-processing: (a) original ICA image;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' (b) centerline extraction and key point detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' The bifurcation points with degree > 3 are marked in red, and the endpoints with degree = 1 are marked in green;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' (c) vascular structure after merging the splitting points in the yellow region in (b) and deleting the cycle in the blue region in (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' In this sub- figure, the bifurcation points with degree > 3 are marked in red, the endpoints with degree = 1 are marked in green, and the degree two points generated by merging the splitting points are marked in blue;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' (d) vascular structure after merging the degree two points in (c);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' (e) generated individual graph;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' (f) the final individual graph by switching nodes and edges of the individual graph in (e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Since the graph matching is to find the node (arterial segment) correspondence rather than the edge correspondence, we switch node and edge in the individual graph in (e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' In (c), (d), and (f), the classes of the arterial segments are labeled;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' in (c), (d), and (e), the node indices are annotated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' To convert the vascular tree into an arterial graph, we define two types of points: bifurcation points and endpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' A bifurcation point is an intersection point that bifurcates the arterial segments into sub-segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' An endpoint represents the end of an arterial segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' This conversion process iterates through all points in a centerline until all bifurcations and endpoints are extracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' As shown in Figure 3 (b), each centerline contains multiple arterial segments and multiple bifurcations and endpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' To find the links between one bifurcation point and one endpoint, or links between two bifurcation points, each bifurcation and endpoint is removed from the vascular centerline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Consequently, each vascular centerline is separated into several arterial segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' We design several rules and an algorithm to eliminate errors and build the individual graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' i) Delete the capillary segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' If the maximum diameter associated with a point in the centerline is smaller than the threshold 𝑇𝑑, then it has no significance for clinical analysis and is removed [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' In addition, any short arterial segments with less than 𝑇𝑐 pixels in the centerline are also removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' ii) Merge splitting points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Typical errors in an automatically generated arterial graph are induced by splitting points [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' If two bifurcation points are located closely, then the splitting points affect graph topology creation, as shown in the yellow circle in Figure 3 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' A threshold was set to remove two splitting points if the Euclidian distance between two bifurcation points is smaller than the threshold 𝑇𝑠𝑝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' As a result, those two points and related edges are merged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' iii) Delete cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' To build an undirected acyclic graph for graph convolution, cycles in the graph must be deleted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' If a cycle exists, the arterial segment with a smaller diameter is removed, as shown in Figure 3 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' iv) Merge degree two points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' After removing the capillary arterial segments and the splitting points, the degrees of the bifurcation points are reduced to two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Then those bifurcation points are merged into the connected two arterial segments, as shown in Figure 3 (c) and (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' v) Switch nodes and edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' The bifurcation points and endpoints are nodes in a graph, and the edges are the link between nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Naturally, each node in the individual graph represents a bifurcation point or an endpoint, and each edge represents an arterial segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' However, our graph-matching network aims at building node correspondence rather than edge correspondence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Semantic labeling requires the network to build relationships between arterial segments rather than the detected key points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Thus, we switch nodes and edges in the individual graph so that each node represents an arterial segment, and each edge represents a bifurcation node (with degree ≥ 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Consequently, a graph is generated for each vascular tree, denoted as 𝐺 = (𝑉, 𝐸) where 𝑉 is the node set, and 𝐸 is the edge set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' The generated graph is shown in Figure 3 (f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' The adjacent matrix of the graph is generated by the connectivity of the vascular tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Our algorithm for arterial tree modification and individual graph generation is shown in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Individual graph generation from a vascular tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Graph matching neural network for arterial segment labeling Graph matching aims to find the node correspondence between two individual graphs 𝐺1 = (𝑉1, 𝐸1) and 𝐺2 = (𝑉2, 𝐸2), where |𝑉1| = 𝑛1, |𝑉2| = 𝑛2, |𝐸1| = 𝑛𝑒1 and |𝐸2| = 𝑛𝑒2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Without loss of generality, we assume 𝑛1 ≤ 𝑛2 in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' The two-graph matching problem can be written as a quadratic assignment programming (QAP) [29,30], defined as 𝐽(𝑋) = 𝑣𝑒𝑐(𝑀)𝑇𝐾𝑣𝑒𝑐(𝑀), 𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' 𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' 𝑀 ∈ {0,1}𝑛1×𝑛2 (1) where 𝑀 ∈ ℝ𝑛1×𝑛2 is the permutation matrix encoding node-to-node correspondence and 𝐾 ∈ ℝ𝑛1𝑛2×𝑛1𝑛2 is the affinity matrix of the association graph 𝐺𝐴 = (𝑉𝐴, 𝐸𝐴) generated by the connectivity of the graphs 𝐺1 and 𝐺2 , where |𝑉𝐴| = 𝑛1 × 𝑛2 and |𝐸𝐴| = 2 × 𝑛𝑒1 × 𝑛𝑒2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' 𝑣𝑒𝑐 indicates the vectorization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' To eliminate ambiguity, the node (an arterial segment) in the individual graph is defined as node, while the node in the association graph is defined as vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' The vertices of the association graph 𝑉𝐴 = 𝑉1 × 𝑉2 encode the node-to-node correspondence, denoted as 𝑉𝑖,𝑎 𝐴 = (𝑉𝑖 1,𝑉𝑎2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' The diagonal elements in 𝐾 represent the node-to-node matching level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=" For vertex generation, each candidate's correspondence (𝑉𝑖 1,𝑉𝑎2) ∈ 𝑉1 × 𝑉2 will be considered a vertex 𝑉𝑖𝑎 𝐴." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' The edge in the association graph is denoted as 𝐸𝑖𝑎,𝑗𝑏 𝐴 , where 𝑖, 𝑗 ∈ {1, … , 𝑛1} and 𝑎, 𝑏 ∈ {1, … , 𝑛2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' For edge generation, we build an edge between a pair of vertices 𝑉𝑖𝑎 𝐴 and 𝑉𝑗𝑏 𝐴 if and only if there are two edges in their own graphs that (𝑉𝑖 1,𝑉𝑗 1) ∈ 𝐸1 and (𝑉𝑎2, 𝑉𝑏 2) ∈ 𝐸2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Thus, the graph-matching problem is converted to classify vertices in the association graph 𝐺𝐴 into either positive vertices or negative vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' For example, if the node 𝑉𝑖 1 ∈ 𝐺1 and 𝑉𝑎2 ∈ 𝐺2 have the same semantic labels, then the vertex 𝑉𝑖𝑎 𝐴 ∈ 𝐺𝐴 is a positive vertex, and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' The overall association graph-based graph-matching network contains the following five modules, as shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Input: 𝑋: input ICA image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' 𝑇𝑑: threshold for removing an arterial segment if its diameter is smaller than 𝑇𝑑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' 𝑇𝑐: threshold for removing an arterial segment if its centerline length is shorter than 𝑇𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' 𝑇𝑠𝑝: distance threshold for removing splitting points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Output: 𝐺 = (𝑉, 𝐸): Generated individual graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Extract a vascular tree by FP-U-Net++ using 𝑋 and generate the vascular centerline;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Find bifurcation points and end points to separate the vascular centerline to arterial segments;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Calculate diameters and remove capillary segments by 𝑇𝑑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Merge splitting points by 𝑇𝑠𝑝;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Delete cycles if two arterial segments stem from the same point and end at the same point;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Merge degree two points if the degrees of splitting points are reduced into two;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Generate the individual graph and switch nodes and edges to create the final individual graph 𝐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' A flowchart showing the architecture of the association graph-based graph matching network: (a) individual graph generation and node feature extraction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' (b) association graph generation, ground truth generation, and feature concatenation for vertices in the association graph;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' (c) feature representation learning, including feature embedding, graph convolution network and feature decoding;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' (d) vertex classification by major probability voting;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' (e) loss calculation and model training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' i) Extracting features for nodes in individual graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Using the processing algorithm, the individual graph is generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Each node in an individual graph is an arterial segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' We extract the pixel-derived features, position-based features, and topological features for each arterial segment;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' all features are denoted as the features for node in an individual graph, as 𝑥𝑖 𝑔, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' 𝑔 ∈ {1,2} and 𝑖 ∈ {1, … , 𝑛𝑔}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' For pixel-derived features, we extract the radiomics features as in our previous work [14] and [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' For position-based features and topological features, we designed 22 hand-crafted features, as shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' (a) (b) (c) (d) Feature Vertex Arterial GAii embedding classification segment yia feature OM2 extraction Graph convolution ICA Association Nmp Individual graph graph x a generation Feature G GA: decoding Classification Results Arterial segment (e) feature ICA Loss extraction Ground Truth Y Y Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Hand-crafted features for each arterial segment Type Index Feature description Pixel feature 1 Number of pixels in the artery segment 2 Length of centerline 3-6 Standard deviation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' mean,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' the minimum and maximum radius of the artery segment 7-24 First-order statistical radiomics features describing the distribution of pixel intensities within the arterial segments 25-48 Gray-level co-occurrence matrix (GLCM) features describing the second-order joint probability function of an arterial segment 49-62 Gray level dependence matrix (GLDM) features,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' counting the number of connected pixels within distance 𝛿 that are dependent on the center pixel 63-78 A gray level run length matrix (GLRLM) feature quantifies gray level runs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' represented by the length in the number of pixels that have the same gray level value 79-94 Gray level size zone (GLSZM) features,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' counting the gray level zones in the arterial segment 95-99 Neighboring gray-tone difference matrix (NGTDM) features,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' counting the difference ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content='between the gray value of a pixel and the average of its neighbors ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content='Position ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content='feature ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content='100-103 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content='Weighted and absolute centers of the segment positions related to the center of the vascular ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content='tree ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content='104-111 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content='Weighted and absolute positions of the two key points related to the vascular tree center ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content='112-119 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content='Weighted and absolute positions of the two key points related to the artery segment center ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content='Topology ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content='feature ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content='120-121 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content='Degree of the two key points ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content='ii) Extracting features for vertices in the association graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' The vertex features are generated by concatenating node features from individual graphs 𝐺1 and 𝐺2, as shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' 𝑥𝑖𝑎 = [𝑥𝑖 1, 𝑥𝑎2], s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' 𝑖 ∈ {1, ⋯ , 𝑛1}, 𝑎 ∈ {1, ⋯ , 𝑛2} (2) where [⋅] is the concatenation operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' For the edges in association graphs, the features are generated by concatenating features of the edges in the individual graphs, as shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' 𝑒(𝑖𝑎,𝑗𝑏) = [𝑒𝑖𝑗 1 , 𝑒𝑎𝑏 2 ], s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' 𝑖, 𝑗 ∈ {1, ⋯ , 𝑛1}, 𝑎, 𝑏 ∈ {1, ⋯ , 𝑛2} (3) where 𝑒𝑖𝑗 𝑔 = [𝑥𝑖 𝑔, 𝑥𝑗 𝑔], 𝑔 ∈ {1,2} and 𝑖, 𝑗 ∈ {1, … , 𝑛𝑔} represents features of edge in the individual graph, constituted by the concatenation of the features of two connected nodes 𝑉𝑖 𝑔 and 𝑉𝑗 𝑔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Then, 𝑒(𝑖𝑎,𝑗𝑏) indicates the features of edges in association graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' iii) Feature representation learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' We first develop a feature embedding module to embed the node and edge features in the association graph into latent representations by multi-layer perceptions (MLPs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' In this study, the feature embedding is performed separately on vertices and edges, denoted as 𝑓𝑒𝑚𝑏 𝑣 and 𝑓𝑒𝑚𝑏 𝑒 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Formally, the feature embedding module is defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' 𝑥𝑖𝑎 𝐴 =𝑓𝑒𝑚𝑏 𝑣 (𝑥𝑖𝑎) 𝑒(𝑖𝑎,𝑗𝑏) 𝐴 = 𝑓𝑒𝑚𝑏 𝑒 (𝑒(𝑖𝑎,𝑗𝑏)) 𝐺𝑒𝑚𝑏 𝐴 = [𝑓𝑒𝑚𝑏 𝑣 (𝑥𝑖𝑎 𝐴 ), 𝑓𝑒𝑚𝑏 𝑒 (𝑒(𝑖𝑎,𝑗𝑏) 𝐴 )], s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' 𝑖, 𝑗 ∈ {1, ⋯ , 𝑛1}, 𝑎, 𝑏 ∈ {1, ⋯ , 𝑛2} (4) After performing feature embedding, a GCN is employed to aggregate features from the adjacent vertices for message passing [32,33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' We adopt the message-passing neural network to aggregate features from the adjacent vertices [34], which contains a message-passing phase for feature aggregation and a readout phase for feature update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' In detail, the edge convolution layer first aggregates features from the two connected vertices, and then updates its features iteratively, as shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' 𝑒(𝑖𝑎,𝑗𝑏) 𝑡+1 = 𝜙𝑒([𝑒(𝑖𝑎,𝑗𝑏) 𝑡 , 𝑥𝑖𝑎 𝑡 , 𝑥𝑗𝑏 𝑡 ]), 𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' 𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' 𝑖, 𝑗 ∈ {1, ⋯ , 𝑛1}, 𝑎, 𝑏 ∈ {1, ⋯ , 𝑛2}, and 𝑡 ∈ [1, ⋯ , 𝑁𝑚𝑝] (5) where 𝜙𝑒 is the edge convolution layer implemented by MLP and 𝑒(𝑖𝑎,𝑗𝑏) 𝑡+1 becomes the updated edge features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' 𝑡 indicates the index of the message passing, and 𝑁𝑚𝑝 is the total number of message-passing steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' If 𝑡 = 1, then 𝑒(𝑖𝑎,𝑗𝑏) 𝑡 = 𝑒(𝑖𝑎,𝑗𝑏) 𝐴 and 𝑥𝑖𝑎 𝑡 = 𝑥𝑖𝑎 𝐴 as defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' For each vertex, a vertex convolution layer is employed to aggregate features from the adjacence edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' The vertex convolution layer first aggregates features from the adjacent edges in the association graph and then updates its features iteratively, as shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' 𝑥𝑖𝑎 𝑡+1 = 𝜙𝑣 ([ ∑ 𝑒(𝑖𝑎,𝑗𝑏) 𝑡+1 𝑗𝑏∈𝐸𝑖𝑎 , 𝑥𝑖𝑎 𝑡 ]) , 𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' 𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' 𝑖, 𝑗 ∈ {1, ⋯ , 𝑛1}, 𝑎, 𝑏 ∈ {1, ⋯ , 𝑛2} and 𝑡 ∈ [1, ⋯ , 𝑁𝑚𝑝] (6) where 𝐸𝑖𝑎 is a set containing the connected edges of vertex 𝑥𝑖𝑎 𝐴 , and ∑ ⋅ 𝑗𝑏∈𝐸𝑖𝑎 indicates the element-wise summation of the features from the adjacent edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' 𝜙𝑣 is the vertex convolution layer implemented by MLPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' The per-vertex and per-edge features are computed independently, and the weights of vertex convolution layer and edge convolution layer are shared to calculate per-vertex and per-edge affinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' According to Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' 5 and 6, the updated vertex and edge features are denoted as 𝑥𝑖𝑎 𝑁𝑚𝑝 and 𝑒(𝑖𝑎,𝑗𝑏) 𝑁𝑚𝑝 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Unlike nature images, the visibility and anatomy between ICA images are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Thus, the number of nodes in two individual graphs may be different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Since we assume 𝑛1 ≤ 𝑛2, we manually select 𝐺1 and 𝐺2 so that the number of nodes in 𝐺1 is smaller than that in 𝐺2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' The selected 𝐺1 and 𝐺2 are used as a pair for training the AGMN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Because the per-edge embedding, per-vertex embedding, and graph convolution layers are reused across all edges and vertices, the designed AGMN automatically supports a form of combinatorial optimization for graphs with a varying number of nodes, which is suitable and feasible for coronary artery graph matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Thus, the proposed AGMN is independent of the input individual graphs, allowing it to perform inexact graph matching rather than the exact mapping problem for individual graphs with the same number of nodes [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' After iteratively updating the edge features and vertex features, an MLP decoder module is employed to convert the learned feature representation to vertex classification probability, denoted as 𝜙𝑑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Formally, the output of the AGMN is shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' 𝑦̂𝑖𝑎 = 𝜙𝑑 (𝑥𝑖𝑎 𝑁𝑚𝑝), 𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' 𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' 𝑖 ∈ {1, ⋯ , 𝑛1},𝑎 ∈ {1, ⋯ , 𝑛2} (7) where 𝑦̂𝑖𝑎 indicates the probability of vertex 𝑉𝑖𝑎 𝐴 belonging to a positive vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' iv) Vertex classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Graph matching is equivalent to vertex classification, so a vertex classifier is adopted to predict the matching results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' According to the decoder, the matching probability between 𝑥𝑖 1 and 𝑥𝑎2 is calculated by the decoder 𝜙𝑑 as 𝑦̂𝑖𝑎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Since each arterial branch is only matched with one arterial branch, a major probability voting strategy is employed to generate the final prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Formally, the vertex classification result is defined as: 𝑦̂𝑖𝑎 = { 1, 𝑖𝑓 argmax 𝑘∈{1,…,𝑛2} 𝑦̂𝑖𝑘 = 𝑎 0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 (8) In other words, the vertex 𝑦̂𝑖𝑎 in the vertex set {𝑥𝑖1 𝐴 , 𝑥𝑖2 𝐴 , ⋯ , 𝑥𝑖𝑛2 𝐴 } is selected as the positive vertex if 𝑦̂𝑖𝑎 has the highest probability among other vertices {𝑥𝑖1 𝐴 , ⋯ , 𝑥𝑖,(𝑎−1) 𝐴 , 𝑥𝑖,(𝑎+1) 𝐴 ⋯ , 𝑥𝑖𝑛2 𝐴 }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' v) Loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' The node-to-node correspondence between the vertex classification results and the ground truth is used to guide the model training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' The ground truth and classification results are denoted as two permutation matrix 𝑀 ∈ ℝ𝑛1×𝑛2 and 𝑀̂ ∈ ℝ𝑛1×𝑛2, where the element in i-th row and a-th column indicates the relationship between node 𝑉𝑖 1 ∈ 𝐺1 and node 𝑉𝑎2 ∈ 𝐺2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' The permutation loss [26] computed by the cross entropy between the predicted vertex class and the ground truth is used as an objective function, as shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' 𝐿𝑝𝑒𝑟𝑚 = 𝑐𝑟𝑜𝑠𝑠_𝑒𝑛𝑡𝑟𝑜𝑝𝑦(𝑀, 𝑀̂) = − (∑ ∑((1 − 𝑦̂𝑖𝑎)log(1 − 𝑦𝑖𝑎) + 𝑦̂𝑖𝑎 log 𝑦𝑖𝑎) 𝑛2 𝑎=1 𝑛1 𝑖=1 ) (9) where 𝑦𝑖𝑎 is the ground truth of the vertex 𝑥𝑖𝑎 𝐴 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' If 𝑦𝑖𝑎 = 1, then the arterial segment 𝑉𝑖 1 in 𝐺1 and the arterial segment 𝑉𝑎2 in 𝐺2 have the same semantic labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Training and testing Before introducing the training and testing strategies, we first demonstrate the method to generate the labeled dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' We manually annotated the ICA images with semantic labels for each arterial segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Then, the semantic label was assigned to each node (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=', each arterial segment) in the individual graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' During the creation of the database, the node correspondences between arterial segments are automatically identified based on the semantic labels, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' 𝑦𝑖𝑎 = 1 if two arterial segments have the same types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' However, the main branches, such as LCX and LAD, are separated into several small branches during the individual graph generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Then, the arterial branches with the same semantic labels may have more than one node in the individual graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' For example, in Figure 3 (d), the LAD branch has two segments due to the bifurcation points for side branch D1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' These segments are matched with the LAD segments from another individual graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' If we build the ground truth of the association graph without re-naming semantic labels, a complete bipartite graph or biclique will be built;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' in the bipartite graph or biclique, every node of the first set is connected to every node of the second set [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' In this example, the first LAD branch will have two matched nodes, and two vertices connecting the first LAD branch are positive vertices in the association graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=" If a well-trained AGMN is obtained, the major probability voting strategy will fail to generate the final decision because these two vertices' probabilities are equal to 1." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Our designed AGMN requires the individual graph and the matching relationship to satisfy the constraint of one-to-one or one-to-zero mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' One arterial segment or one node 𝑉𝑖 1 in the individual graph 𝐺1 should only have one or zero matched node in the graph 𝐺2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' In this study, we annotate the arteries into several sub-classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' For example, the LAD branch in Figure 3 (d) is separated into two segments named LAD1 and LAD2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' The LMA is upstream of the blood flow, and we follow this flow to assign the indices of the main and side branches sequentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Therefore, the LCX and LAD segments are separated into several sub-segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Each node of the first graph is only connected to one node of the second graph, and the major probability voting strategy is then usable for this task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' To train the model, at each training step, two individual graphs from the same view were randomly selected from the training set 𝐷𝑡𝑟.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Only left coronary arteries (LCAs) were used for model training and validation in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' In addition, only ICAs from two regular views, left anterior oblique (LAO) and right anterior oblique (RAO), were enrolled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Because of the anatomical difference between ICAs from different views, the two selected individual graphs were from the ICAs with the same view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Since we assumed the number of nodes 𝑛1 ≤ 𝑛2 in this study, we had to switch 𝐺1 and 𝐺2 according to the number of nodes during the training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' A batch of graph pairs was randomly generated for each training iteration to accelerate the model training [37] and prevent the weights from trapping into the local minimum [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=" The association graph was built according to the two selected graphs' semantic labels." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' The difference between the AGMN prediction and the ground truth was used to calculate the loss and train the model, as defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' For the model testing, each individual graph in the test set 𝐷𝑡𝑒 is used to perform graph matching with a set of graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' The set is denoted as the template set, 𝐷𝑡𝑝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Cardiologists learn to read and understand the ICA in clinical practice by comparing it with the representative ICAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' When making decisions, a set of representative ICAs can be used as templates for reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Our designed testing strategy imitates this procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Each individual graph from the test set is paired with the individual graph from one representative subject in the template set for graph matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' In the template set, each arterial segment is labeled for reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Using the well-trained AGMN, the mapping relationship between unlabeled arteries in the test subject and the labeled arteries from the template subject are obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' The vertex classification result for the test subject among the subjects in the template set is voted based on maximum voting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' For example, if LMA in the test subject is matched with LMA branches from five subjects in the template set and is matched with LAD branches from two subjects in the template set, then this arterial segment in the test subject is labeled as the LMA branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' The designed training and testing strategies of our AGMN is shown in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Training and testing strategies of the proposed GMN for coronary arterial semantic labeling 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Performance evaluation The semantic labeling problem is converted into a multi-class classification problem among arterial segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' As a classification problem, the weighted accuracy (ACC), weighted precision (PRE), weighted recall (REC), and weighted F1-score (F1) are used to evaluate the model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' We separate the LAD and LCX branches into sub-segments during the model training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' However, in the evaluation process, we group the sub-segments into their original classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' The weighted ACC, SP, SN, and F1 definitions are shown in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' 10 to 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' 𝐴𝐶𝐶 = 1 𝑛 ∑ 𝑇𝑃𝑐 + 𝑇𝑁𝑐 𝑇𝑃𝑐 + 𝑇𝑁𝑐 + 𝐹𝑁𝑐 + 𝐹𝑃𝑐 × 𝑛𝐶 𝐶 𝑐=1 (10) 𝑃𝑅𝐸 = 1 𝑛 ∑ 𝑇𝑃𝑐 𝑇𝑃𝑐 + 𝐹𝑃𝑐 × 𝑛𝐶 𝐶 𝑐=1 (11) Input: 𝐷𝑡𝑟: training set, contains 𝑁𝑡𝑟 labeled individual graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' 𝐷𝑡𝑒: test set, contains 𝑁𝑡𝑒 unlabeled individual graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' 𝐷𝑡𝑝: template set, contains 𝑁𝑡𝑝 labeled individual graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' 𝑁: number of training steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' 𝑁𝑚𝑝: number of the message passing times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Output: 𝑁𝑡𝑝 labeled individual graphs in 𝐷𝑡𝑒.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Training: For 𝑖𝑡𝑒𝑟 = 1 ⋯ 𝑁 do 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Random select two individual graphs 𝐺1 and 𝐺2 from 𝐷𝑡𝑟 from the same view;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Extract features for each arterial segment in 𝐺1 and 𝐺2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Build association graph 𝐺𝐴 and extract features using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' 2 and 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Update vertex and edge features of 𝐺𝐴 using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' 4 to 6 for 𝑁𝑚𝑝 iterations by GCN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Decode features and calculate the vertex class using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' 7 and 8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Calculate the loss function 𝐿𝑝𝑒𝑟𝑚 defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' 9 and optimize the AGMN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Testing: For each individual graph 𝐺𝑖 𝑡𝑒 (𝑖 ∈ [1, ⋯ , 𝑁𝑡𝑒]) in 𝐷𝑡𝑒: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Extract features for each arterial segment in 𝐺𝑖 𝑡𝑒;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' For each individual graph 𝐺𝑗 𝑡𝑝 (𝑗 ∈ [1, ⋯ , 𝑁𝑡𝑝]) in 𝐷𝑡𝑝: 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Extract features for each arterial segment in 𝐺𝑗 𝑡𝑝;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Build the association graph 𝐺𝐴 using 𝐺𝑖 𝑡𝑒 and 𝐺𝑗 𝑡𝑝;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Update vertex and edge features of 𝐺𝐴 using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' 4 to 6 for 𝑁𝑚𝑝 iterations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Decode features and calculate the vertex class using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' 7 and 8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Assign labels for nodes in 𝐺𝑖 𝑡𝑒 according to major voting among 𝐺𝑗 𝑡𝑝, 𝑗 ∈ [1, ⋯ , 𝑁𝑡𝑝].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' 𝑅𝐸𝐶 = 1 𝑛 ∑ 𝑇𝑁𝑐 𝑇𝑁𝑐 + 𝐹𝑁𝑐 × 𝑛𝐶 𝐶 𝑐=1 (12) 𝐹1 = 1 𝑛 ∑ 𝑇𝑃𝑐 𝑇𝑃𝑐 + 1 2 (𝐹𝑃𝑐 + 𝐹𝑁𝑐) 𝐶 𝑐=1 × 𝑛𝑐 (13) where 𝑇𝑃𝑐, 𝑇𝑁𝑐, 𝐹𝑃𝑐 and 𝐹𝑁𝑐 represent the true positive arterial segment, true negative arterial segment, false positive arterial segment, and false negative arterial segment, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' 𝑐 refers to the class index of arterial segments, and 𝐶 is the total number of classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' 𝑛𝐶 is the number of arterial segments in class 𝑐 and 𝑛 is the total number of arterial segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Experiments and results In this section, we conduct experiments to demonstrate the effectiveness of the proposed AGMN for coronary artery semantic labeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' The enrolled subjects, experimental settings, and results will be presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Dataset and enrolled subjects In this study, we manually annotated 204 and 59 ICAs from site 1 [15] at The First Affiliated Hospital of Nanjing Medical University and site 2 at Chang Bing Show Chwan Memorial Hospital, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' In total, this retrospective study enrolled 263 ICA images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' For site 1, subjects who received ICA from February 26, 2019, to July 18, 2019, were enrolled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' The ICAs were performed using an interventional angiography system (AXIOM-Artis, Siemens, Munich) and were acquired at 15 frames/sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' The image size of ICA videos ranged from 512×512 to 864×864, and the pixel spacing ranged from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content='2 mm to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content='39 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' For site 2, the ICAs were performed using an interventional angiography system (AlluraClarity, Philips Healthcare, Eindhoven, Netherlands) and were acquired at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content='75, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content='5, 15, and 30 frames/sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' The image size of ICA videos was 1024×1024, and the pixel spacing was 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content='184 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Table 1 shows the number of images in each ICA view used in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Views and corresponding image numbers in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' LCA, left coronary artery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' LAO, left anterior oblique;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' RAO, right anterior oblique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Site LAO RAO Total Site 1 55 149 204 Site 2 23 36 59 For each patient, a frame that was used for anatomical structure analysis in clinical practice was selected from the view video for semantic labeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' In this study, we only focus on semantic labeling for the main branches of LMA, LAD, and LCX, and the side branches of D and OM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Implementation details We implemented our designed AGMN using TensorFlow and GraphNets [33] on an NVIDIA RTX 3090 GPU card.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' The thresholds used in Algorithm 1 were set as 𝑇𝑑 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content='8 mm, 𝑇𝑐 = 15 pixels, and 𝑇𝑠𝑝 = 8 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' For the 263 ICA images, we selected first 𝑁𝑡𝑝 images as the labeled individual graphs for the template set, while the rest ICAs were used for a five-fold cross-validation with the stratified sampling according to the view angles of ICAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Consequently, for each experiment, the training set contained 𝑁𝑡𝑟 = (263 − 𝑁𝑡𝑝) × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content='8 samples, and the test set contained 𝑁𝑡𝑒 = (263 − 𝑁𝑡𝑝) × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content='2 samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' All images were resized to 512 × 512 before extracting the features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Each model was fine-tuned for 100,000 training steps using a batch size of 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' The Adam optimizer [39], with an initial learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content='0001, was employed as the optimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' We used the exponential decay strategy in the training phase to adjust the learning rate, and we set the decay rate as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content='98 for each 2000 training steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Each training step took 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content='202 seconds, and the total training time one-fold was 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content='6 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' The hyperparameters were tuned on the test set during the cross- validation for each hyperparameter setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' The grid search settings are shown in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Hyperparameter settings in the grid search Hyperparameter Search space Description Number of hidden units in MLP [16, 32, 64] The MLP includes the feature embedding module, GCN, and feature decoder module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' For each experiment, these two hyperparameters are identically set to all MLP layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Number of MLP layers [2, 3, 4] Number of the message passing steps (𝑁𝑚𝑝) [2, 3, 4] The number of the message passing steps indicates the update iterations of the GCN module in GMN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Number of samples in the template set (𝑁𝑡𝑝) [27, 40, 52, 79] 10%, 15%, 20% and 30% of the ICA images were selected using the stratified sampling according to the view angles as the template set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' More specifically, we first fixed the hyperparameters of 𝑁𝑡𝑝 as 40 (15% of the dataset) and 𝑁𝑚𝑝 as 3 before we tuned the MLP layer size hyperparameters and the number of MLP layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Then, we fixed the number of hidden units in MLP, and the number of MLP layers, then tuned the hyperparameters of 𝑁𝑡𝑝 and 𝑁𝑚𝑝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' We selected the best parameter for each hyperparameter according to the highest average accuracy among the five-fold evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Experimental results of AGMN The best performance was achieved under the settings that the number of hidden units was 64 with 4 MLP layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' The AGMN was trained using 40 samples (15%) as the template set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' And the number of message- passing steps was set as 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Under this setting, the results for each type of arterial segment are listed in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' A summary of the best performance achieved by our proposed AGMN for coronary artery semantic labeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' LMA, left main artery;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' LAD, left descending artery;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' LCX, left circumflex artery;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' D, diagonal artery;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' OM obtuse margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Artery type ACC PRE REC F1 LMA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content='9956±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content='0089 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content='9911±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content='0109 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content='8262±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content='0301 Three examples are visualized for the graph-matching results using our proposed AGMN in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Graph matching results for three examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' The left ICAs are from the testing set, and the right ICAs are from the template set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' The green line indicates a correct match, and the red line represents a wrong match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' The corresponding arterial semantic labels are annotated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' AD 01 OM LCXZ OM2 LCX3 AD OM1 CX2 0M2 CX2 QM LCX3 D1 LCX2 OM2 CX34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Comparison with other coronary artery semantic labeling methods We compared our proposed AGMN approach to four other coronary artery semantic labeling approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Those benchmark approaches include: Machine learning-based method [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Our previous work employed machine learning based methods to perform coronary artery semantic labeling using ICAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Each artery is extracted using our FP-U-Net++ model and converted to vascular centerlines using step 1 to 3 in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' We adopted the same pipeline, extracted the same features, and employed support vector machine with radial basis function kernel as the classifier to perform arterial segment classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Bi-directional tree long short-term memory (BiTreeLSTM) [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' developed a tree- structured LSTM neural network for coronary artery labeling using CCTA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Arterial spatial and directions in 3D polar coordination were used as the arterial segment features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' The graph-structured coronary artery was converted to a tree-structured arterial tree, and a bi-directional tree LSTM was adopted for segment classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' The bi-directional sequence input stems from LMA to side branches and from side branches to LMA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Since our ICA images are in 2D, we only extracted arterial spatial and directions in 2D polar coordination as features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' We adopted the same architecture used in their paper for arterial segment classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Up-to-down (UTD) and down-to-up (DTU) nets [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' These two baseline models were implemented as the ablation study for BiTreeLSTM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Both UTD and DTU nets were implemented by a single tree LSTM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' However, the UTD net used the arterial segments from the root of the coronary vascular tree, LMA, to any side branches for training, while the DTU net used the segments from the side branches to LMA for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Conditional partial-residual GCN (CPR-GCN) [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' developed a GCN neural network embedded with a partial residual network for coronary artery semantic labeling using CCTA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' The positional features contained the arterial segment direction and position, and the imaging features were extracted by convolution neural networks (CNN) and LSTM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' All features were concatenated and used for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' The output of the GCN is the segment classification result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' We replaced the 3D position features with 2D features and replaced the 3D CNNs with 2D CNNs as the baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' The performance comparisons between the proposed AGMN and baseline models are illustrated in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' We adopted five-fold cross-validation for each model and employed stratified sampling to split the samples into training and testing datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' For baseline models, including UTD, DTU, BiTreeLSTM, and CPR-GCN, we also performed the grid search, and the models with the best performance were used for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Comparisons between baseline methods and our proposed AGMN for coronary artery semantic labeling using our ICA dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' The means and standard deviations of the accuracy, precision, recall, and F1-scores among the five folds are presented.' metadata={'source': 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+page_content='0424 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content='7569±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content='0508 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content='8262±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content='0301 According to Table 4, our AGMN achieved the highest average accuracy of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content='8264, average precision of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content='8276, average recall of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content='8264, and average F1-score of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content='8262 among all types of coronary arteries, which outperformed other baseline models significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' All models achieved high performance on main branches prediction, except for DTU net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' The UTD and BiTreeLSTM performed the perfect classification on LMA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' the accuracy was 100%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Because LMA was required to be manually assigned to build the tree-structured arteries, it did not reflect its actual performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' The input sequence of UTD and BiTreeLSTM stems from LMA to the side branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' For the LAD branch, the BiTreeLSTM achieved a higher precision and F1-score than the proposed AGMN due to the correct classification of LMA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Since LAD is connected to LMA directly, so the classification task is relaxed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' For the LCX branch, the BiTreeLSTM achieved a higher recall than the proposed AGMN because of the prior knowledge of the LMA branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Our AGMN outperformed other baselines with a large margin for the side branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' By comparing the similarities of side branches between different individual graphs, the AGMN can differentiate D branches and OM branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' In addition, the D branches are connected with LAD, and OM branches are connected with LCX;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' the performance of side branch classification is guaranteed since the AGMN has achieved a high classification performance of LAD and LCX branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Feature importance In this study, we designed 121 hand-craft features, including pixel features, positional features, and topological features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' A leave-one-out technique [40] is adopted to identify the feature significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' A feature is significant if the performance of semantic labeling decreases significantly when this feature is replaced by zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' By ranking the accuracy drops, the importance of the feature is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' We set the same hyperparameters demonstrated in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' We compared the averaged the accuracy changes among five- folds between using the original dataset and the corrupted datasets, as shown in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Feature importance ranking for classifying coronary arterial segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Feature significance was determined by the accuracy drops between using raw features and zero-filled features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' The vertical axis indicates the feature names, while the horizontal axis indicates the drops in accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' The top 15 features with the highest accuracy changes are shown in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Among these 15 features, 2 are topological features, 2 are pixel-wise features and the other 11 are positional features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' For the topological features, p1 degree and p2 degree indicate the degrees of the two endpoints of an artery segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' If we set these two features as zeros, then the accuracy of coronary artery semantic labeling dropped about 30%, indicating the convincible importance of these two features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' For the pixel features, original_glcm_ldmn and original_glcm_lmc2 are the features calculated by the Gray Level Co-occurrence Matrix, which describes the second-order joint probability function of the region of the arterial segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' The remaining 11 features belong to hand-craft positional features, representing the weighted or absolute coordinates of the center pixels within the arterial segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Due to the anatomical structure of the coronary artery, the position of the artery is important for classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=" Data attack The proposed AGMN was trained and evaluated based only on the 'ideal' individual graphs." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' However, even though our previous coronary artery binary segmentation model [12] has achieved the Dice similarity coefficient of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content='8899, we cannot guarantee that it would generate satisfactory arterial contours for all ICAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' To test the robustness of the designed model, we created the corrupted datasets by randomly removing parts of arterial segments from the ICAs in the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' The removed arterial segment must contain one endpoint to generate a connected graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Otherwise, if the arterial segment contains 2 bifurcation points and is removed, the individual arterial graph would be split into two individual graphs as well as the vascular tree would be separated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' In this situation, human intervention is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Using the corrupted dataset, we compared the AGMN with the graph- or tree-based baseline models, including BiTreeLSTM and CPR-GCN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' We compared the performance drops using the corrupted dataset by randomly removing 5%, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content='5%, 10%, 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content='5%, 15%, 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content='5%, and 20% arterial segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' The ACC, PREC, REC, and F1 and their changes using different corrupted datasets are shown in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' p2_degree pl_degree weighted_y_center y_center Feature Name x_center weighted_x_center pl_x_center p2_abs_y_center p2_abs_y_weighted_center pl_x_weighted center original_glcm_Idmn p2_x_center original_glcm_Imc2 pl_abs_y_center pl_abs_x_weighted_center 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content='30 △ Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' The achieved ACC, PREC, REC, and F1 of the proposed AGMN, CPR-GCN, and BiTreeLSTM using different corrupted datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' The horizontal axis indicates the probability of deleting an artery segment randomly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' The results demonstrated that our AGMN was robust since the accuracy was above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content='79, even using the corrupted datasets with 20% of the arterial segments removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' However, the accuracy changes of AGMN were more significant than those of BiTreeLSTM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' According to Table 4, we observed that the BiTreeLSTM achieved a low performance on side branch classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=" When generating the corrupted datasets, parts of the side branches were removed, which didn't affect the overall performance significantly." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=" We didn't remove the LMA branches when testing the BiTreeLSTM using the corrupted datasets." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' The LMA is the Model Names AGMN CPR-GCN BiTreeLSTM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content='9 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content='5% 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' NO 20root of the arterial tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' If we removed the LMA branch, the arterial tree would be separated into two sub- trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' The tree-based model, BiTreeLSTM, failed to perform artery semantic labeling when using the data without LMA, which is the root of the vascular tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' In addition, our proposed AGMN achieved the highest ACCs, PRECs, RECs, and F1s using different corrupted datasets than other baseline models, indicating its robustness and powerful performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Limitations and future work The limitation of the proposed AGMN is that during the prediction, the graph matching procedures are required to be performed between the test subject and every subject in the template set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' If we employ fewer subjects in the template set, the prediction time would be reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' In the future, graph clustering will be used to select the most representative subjects in each cluster and then construct the template set to accelerate the prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' We use hand-crafted features as the pixel-level features to reduce the training time and model complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' However, the feature representation capability is limited compared to CNN-extracted features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' In the future, a light-weight deep learning-based method is recommended to automatically extract the pixel features for each segment rather than the hand-crafted radiomics features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Conclusion In this paper, we developed and validated a new algorithm for coronary artery semantic labeling on ICAs with high accuracy, interpretability, and robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' A new workflow for the individual graph generation and the association graph-based graph matching network was proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' The association graph-based approach, per-node, and per-edge feature representation learning network performed inexact graph matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' The experimental results showed that our AGMN achieved the best performance and significantly outperformed existing approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' By analyzing the feature importance, the interpretability of AGMN is guaranteed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Our AGMN still performed highly in the data attack experiments, even using corrupted datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Credit authorship contribution statement Chen Zhao: Conceptualization, methodology, coding, manuscript writing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Zhihui Xu: Data management and clinical validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Jingfeng Jiang: Methodology and manuscript writing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Michele Esposito: Clinical validation and manuscript writing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Drew Pienta: Methodology and manuscript writing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Guang-Uei Hung: Data management and manuscript writing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Weihua Zhou: Supervision, project administration, funding acquisition, manuscript writing, review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Declaration of Competing Interest The authors declare no conflicts of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Acknowledgment This research was supported by a new faculty startup grant from Michigan Technological University Institute of Computing and Cybersystems (PI: Weihua Zhou).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' It was also supported in part by a research seed fund from Michigan Technological University Health Research Institute and an NIH grant (U19AG055373).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Reference [1] Okrainec K, Banerjee DK, Eisenberg MJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Coronary artery disease in the developing world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' American heart journal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' Elsevier;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE3T4oBgHgl3EQf0Atf/content/2301.04733v1.pdf'} +page_content='148(1):7–15.' metadata={'source': 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sha256:8f69da958ac8487da16fc3fdecdbbe828d87bbf381a07bdc8cd6f3f420e96105 +size 131428 diff --git a/Y9FOT4oBgHgl3EQf-TSC/content/tmp_files/2301.12973v1.pdf.txt b/Y9FOT4oBgHgl3EQf-TSC/content/tmp_files/2301.12973v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..3242533a99a2127cded2397821c0364f003919a9 --- /dev/null +++ b/Y9FOT4oBgHgl3EQf-TSC/content/tmp_files/2301.12973v1.pdf.txt @@ -0,0 +1,1055 @@ +arXiv:2301.12973v1 [eess.SP] 30 Jan 2023 +Robust Precoding via Characteristic Functions for +VSAT to Multi-Satellite Uplink Transmission +Maik R¨oper∗, Bho Matthiesen +∗†, Dirk W¨ubben∗, Petar Popovski +‡,†, and Armin Dekorsy∗ +∗ Gauss-Olbers Center, c/o University of Bremen, Dept. of Communications Engineering, 28359 Bremen, Germany +† University of Bremen, U Bremen Excellence Chair, Dept. of Communications Engineering, 28359 Bremen, Germany +‡ Aalborg University, Department of Electronic Systems, 9220 Aalborg, Denmark +Email: {roeper, matthiesen, wuebben, dekorsy}@ant.uni-bremen.de, petarp@es.aau.dk +Abstract—The uplink from a very small aperture terminal +(VSAT) towards multiple satellites is considered, in this paper. +VSATs can be equipped with multiple antennas, allowing parallel +transmission to multiple satellites. A low-complexity precoder +based on imperfect positional information of the satellites is +presented. The probability distribution of the position uncertainty +and the statistics of the channel elements are related by the char- +acteristic function of the position uncertainty. This knowledge is +included in the precoder design to maximize the mean signal- +to-leakage-and-noise ratio (SLNR) at the satellites. Furthermore, +the performance w.r.t. the inter-satellite distance is numerically +evaluated. It is shown that the proposed approach achieves the +capacity for perfect position knowledge and sufficiently large +inter-satellite distances. In case of imperfect position knowledge, +the performance degradation of the robust precoder is relatively +small. +Index Terms—LEO satellites, 3D networks, massive MIMO, +beamforming, beamspace MIMO +I. INTRODUCTION +In future mobile networks, the terrestrial infrastructure will +be extended by non-terrestrial networks (NTNs), consisting +of unmanned aerial vehicles (UAVs), high altitude platform +stations (HAPSs) and communication satellites, forming a +global heterogeneous 3D network. An integral component to +these spatial networks are constellations of small satellites +in low Earth orbits (LEOs) [1], [2]. This is because, in +comparison to satellites in medium Earth orbit (MEO) and +geostationary orbit (GEO), these constellations have reduced +deployment costs and acceptable transmission delays [3], [4]. +Recent results examine the application of multiple-input- +multiple-output (MIMO) technologies to simultaneously com- +municate with multiple satellites [5]–[8]. In combination with +formation flying techniques, which allow multiple satellites to +move in clusters and act as a single entity [9]–[11], swarms +of small satellites can form huge virtual antenna arrays with +the promise of massive spectral efficiency improvements. +In [12], [13], optimal geometrical conditions for satellite +swarms with respect to (w. r. t.) the channel capactiy are de- +rived. Additionally, a low-complexity linear transceiver design +for the downlink from a satellite swarm towards a MIMO +ground terminal has been proposed for the considered scenario. +It has been observed that the derived transceiver achieves the +channel capacity for sufficiently large inter-satellite distances +and perfect position knowledge. +In case of imperfect position knowledge, the statistics of +the uncertainty can be included to design a robust precoder. +In [14] robust precoding under imperfect phase knowledge +for multi-user downlink scenario is presented. The precoder is +designed to achieve a given signal-to-interference-and-noise +ratio (SINR) per user with minimum transmit power. In order +to deal with the phase uncertainty, the SINR constraint is +relaxed to be satisfied either only in mean, or with a given +probabilistic. Similarly, in [15], [16] robust precoder with a +relaxed average SINR constraint in the presence of phase +uncertainties are presented. The precoder in [15] allows a +trade-off between spectral and energy efficiency, while in [16] +the energy efficiency for a hybrid precoder is maximized. A +precoder under the strict constraint to ensure an SINR above +a given threshold in case of bounded position uncertainty +is given in [17]. There, different objective functions are +numerically optimized, while the uncertainty region is sampled +to obtain a finite set of constraints. In [18] another robust +precoder for the multi-user downlink is presented to maximize +the signal power of a user while keeping the interference +leakage to other users within a bounded uncertainty region +below a given threshold. A different approach to deal with +position uncertainties is been presented in [13]. There, it is +shown that the characteristic function (CF) of the position error +is related to the autocorrelation matrix of the channel. +In this paper, we utilize the CF of the position uncertainty +to present a novel robust precoder to maximize the mean +signal-to-leakage-and-noise ratio (SLNR) [19] for very small +aperture terminal (VSAT) to multi-satellite uplink applications. +Therefore, different to [17], [18], our precoder is suitable +for arbitrary probability distributions of the receiver positions. +Furthermore, we obtain an analytic solution, and therefore the +computational complexity is relatively low, as the precoder +only requires an eigendecomposition. Furthermore, we show +numerically that the proposed precoder achieves the capacity +for sufficiently large inter-satellite distances and perfect posi- +tion knowledge of the satellites at the VSAT. +The rest of the paper is organized as follows. Next, in +Section II, the general system model and the performance +benchmark for perfect channel state information (CSI) is +presented. In Section III the proposed robust precoder is +presented and numerically evaluated in Section IV. Finally, +Section V concludes the paper. + +VSAT array +y +x +z +dℓ +di +Satellite ℓ +Satellite i +θel +ℓ +θaz +ℓ +Fig. 1: Geometric relation between VSAT and satellite ℓ +II. SYSTEM MODEL AND PERFORMANCE BOUNDS +We consider the uplink from a single VSAT to a group of NS +single antenna satellites. The VSAT is equipped with a uniform +rectangular array (URA) consisting of NT = N x +TN y +T ≥ NS +antennas. Coordinates are stated in a VSAT-centric reference +frame as depicted in Fig. 1. In particular, the z-axis points +toward the VSAT’s zenith, and the x- and y-axes are aligned +with the VSAT’s URA. Thus, N x +T and N y +T are the number +of antennas of the VSAT in x- and y-direction, respectively. +The position of satellite ℓ ∈ {1, ..., NS} is given by the triplet +(dℓ, θel +ℓ , θaz +ℓ ), where dℓ is the distance between satellite ℓ and +the VSAT and θel +ℓ ∈ [θel +ℓ,min, π/2] and θaz +ℓ +∈ [0, 2π] are the +elevation and azimuth angles, respectively, as shown in Fig. 1. +The VSAT transmits NS independent Gaussian data streams +s ∼ CN(0, INS), where INS is the identity matrix of di- +mension NS. These streams are linearly precoded with G = +[g1, ..., gNS] ∈ CNT×NS to obtain the transmit signal x = Gs. +The VSAT is subject to an average power constraint PTx, i.e., +E +� +xHx +� += tr +� +GGH� +≤ PTx . +(1) +The received signal yℓ at satellite ℓ is yℓ = hH +ℓ x + nℓ, where +hℓ ∈ CNT is the channel from the VSAT to satellite ℓ and nℓ +is circularly-symmetric complex white Gaussian noise with +power σ2 +n. +A. Channel and Error Model +We assume a line-of-sight (LOS) path between the VSAT +and satellite ℓ and the received power from the non-LOS +(NLOS) paths to be negligible small. Then, the elements of +hℓ differ only in their phase, but have the same magnitude +[12], [20], [21]. Thus, we can define the channel vector hℓ +as a multiplication of a scalar factor αℓ ∈ C and a steering +vector aℓ ∈ CNT, whose elements have magnitude one, i.e., +hℓ = αℓaℓ. +(2) +The complex channel gain αℓ includes the signals attenuation +and phase rotation due to the free space propagation and the +atmospheric effects. The steering vector aℓ depends on the +satellite position. Due to the regular structure of the URA, +it is helpful to separate aℓ into two steering vectors ax +ℓ = +[ax +ℓ,1, ..., ax +ℓ,N x +T]T and ay +ℓ = [ay +ℓ,1, ..., ay +ℓ,N y +T]T , for the x- and +y-direction, respectively, such that +aℓ = ax +ℓ ⊗ ay +ℓ , +(3) +where ⊗ denotes the Kronecker product. With the angles of +departure (AoDs) as defined in Fig. 1 and given that the +distance between the transmit antennas DA at the VSAT is +much smaller than the distance between the VSAT and satellite +ℓ, i.e., DA ≪ dℓ, the elements of the steering vectors are [22] +ax +ℓ,m = e−jνDA(m−1) cos(θel +ℓ ) cos(θaz +ℓ ) , +m = 1, ..., N x +T , (4a) +ay +ℓ,n = e−jνDA(n−1) cos(θel +ℓ ) sin(θaz +ℓ ) , +n = 1, ..., N y +T , +(4b) +where ν is the wavenumber of the carrier wave . Furthermore, +we define the space angles φx +ℓ = cos +� +θel +ℓ +� +cos (θaz +ℓ ) and +φy +ℓ = cos +� +θel +ℓ +� +sin (θaz +ℓ ). While the AoDs θel +ℓ +and θaz +ℓ +are +related to the spherical coordinates, the space angles φx +ℓ and +φy +ℓ are related to the Cartesian x- and y- coordinates of satellite +ℓ, respectively. Note that in practical systems only imperfect +knowledge of the satellites positions can be assumed. Let ξx +ℓ +and ξy +ℓ be the estimation errors for the space angles in x- and +y-direction, respectively, the estimated space angles are +ˆφx +ℓ = φx +ℓ + ξx +ℓ, +ˆφy +ℓ = φy +ℓ + ξy +ℓ . +(5) +Then, the k = n+(m−1)N y +Tth element of the channel vector +hℓ can be written as +hℓ,k = αℓax +ℓ,may +ℓ,n += αℓe−jνDA((m−1)( ˆφx +ℓ−ξx +ℓ)+(n−1)( ˆφy +ℓ−ξy +ℓ)) , +(6) +which includes three statistically independent random vari- +ables from the transmitters perspective, i.e., αℓ, ξx +ℓ and ξy +ℓ. The +complex channel gain αℓ is a circularly-symmetric random +variable with variance E{|αℓ|2} = σ2 +αℓ. Note that the variance +σ2 +αℓ can be different for each satellite ℓ, due to the different +path losses. For the estimation errors of the space angles ξx +ℓ +and ξy +ℓ, we assume the same probability distribution for every +satellite ℓ and for both directions. Their probability distribution +is described by the characteristic function ϕξ(t) [23]. +B. Upper bound on the channel capacity +To obtain an upper bound on the throughput performance, +perfect CSI at the transmitter and receiver as well as instanta- +neous and error-free inter-satellite communication is assumed. +Hence, perfect coordination between satellites is possible and +the system can be modelled as an NT × NS MIMO system. +The joint receive signal of all satellites is y = Hx + n, +with the channel matrix H = [h1, ..., hNS]H and the noise +vector n = [n1, ..., nNS]T , where the components of n are + +mutually independent. Let λµ be the µth eigenvalue of HHH, +the capacity of this channel is [24] +C = +NS +� +µ=1 +log2 +� +1 + λµ +pµ +σ2n +� +, +(7) +where and pµ is the optimal power allocated to the µth stream. +Given the power constraint � +µ pµ = PTx, the optimal power +allocation is obtained from the water-filling algorithm [24]. +III. PROPOSED UPLINK APPROACH +In this section, we present a novel low-complexity robust +precoder for uplink transmission from a VSAT to multiple +satellites based on imperfect knowledge of the satellites po- +sitions. For sufficiently large distances between the satellites, +the channel vectors become nearly orthogonal, i.e., aH +i aℓ ≈ 0. +In this case, the channel capacity is maximized, because the +eigenvalues of HHH are almost equal [12]. Given orthog- +onal channels, the capacity can be achieved by transmitting +independent data streams [24].1 Thus, we assign one stream +per satellite and then design the precoders. Given independent +data streams, no inter-satellite communication is required for +estimation and decoding because satellite ℓ can estimate and +decode the transmitted symbol sℓ independently from the +other satellites. Numerical results in Section IV-A show that +this approach is, despite its simplicity, capacity achieving +for sufficiently large orbital separations and perfect position +knowledge of the satellites. The received signal at satellite ℓ +is yℓ = hH +ℓ gℓsℓ + � +i̸=ℓ hH +ℓ gisi + nℓ. Correspondingly, the +instantaneous SINR Γℓ at satellite ℓ is given as +Γℓ = +��hH +ℓ gℓ +��2 +� +i̸=ℓ |hH +ℓ gi|2 + σ2n +(8) +and the achievable rate R is equal to the sum rate +R = +NS +� +ℓ=1 +log2 (1 + Γℓ) . +(9) +The goal is to obtain precoders that maximize this rate. +However, the exact values of hℓ are not known but only +their estimates based on ˆφx +ℓ and ˆφy +ℓ. Furthermore, directly +maximizing this sum rate is NP-hard and has, to the best of +our knowledge, no easy analytical solution. In the following, +we solve a slightly simplified version of this problem that, as +will be seen later in Sec. IV, will result in a solution that is +often close to the optimal throughput performance. +A. Problem Formulation +Observe that the SINRs Γℓ in (8) are coupled through their +denominator. Substituting Γℓ in (9) with the instantaneous +SLNR for satellite ℓ defined as +γℓ = +|hH +ℓ gℓ|2 +� +i̸=ℓ |hH +i gℓ|2 + σ2n +(10) +1Observe that this does not imply the transmission of several independent +messages. See, e.g., [25] or [26, Sec. 5.8.1]. +and assuming equal power allocation among streams, leads to +the substitute optimization problem +max +gℓ +|hH +ℓ gℓ|2 +� +i̸=ℓ |hH +i gℓ|2 + |σ2n +s.t. +gH +ℓ gℓ ≤ PTx +NS +, +(11) +which has to be solved for each ℓ ∈ {1, ..., NS}. However, this +problem still relies on the exact knowledge of the channels. +Our goal is to design robust precoders that take the statistics +of the estimation error into account. This can be achieved by +optimizing over the mean SLNR ¯γ = E {γ}, i.e., +max +gℓ E +� +|hH +ℓ gℓ|2 +� +i̸=ℓ |hH +i gℓ|2 + σ2n +� +s.t. +gH +ℓ gℓ ≤ PTx +NS +, (12) +where the expectation is taken w. r. t. the channel vectors +h1, . . . , hNS. Then, the robust precoders grob +ℓ +are a solution +of (12). +B. Robust Precoding via CF +The solution of (12) requires a closed-form expression of +the autocorrelation matrix of the channel E +� +hℓhH +ℓ +� +, which +is closely related to the autocorrelation matrix of the steering +vectors. In particular, let Raℓ = E +� +aℓaH +ℓ +� +be the autocorre- +lation matrix of the steering vector aℓ. With (3) and due to the +statistical independence of the position uncertainties ξx +ℓ and ξy +ℓ, +we can write +Raℓ = E +� +aℓaH +ℓ +� += E +� +(ax +ℓ ⊗ ay +ℓ) (ax +ℓ ⊗ ay +ℓ)H� +(13a) += E +� +ax +ℓax +ℓ +H� +⊗ E +� +ay +ℓay +ℓ +H� += Rax +ℓ ⊗ Ray +ℓ. +(13b) +The random variables in Rax +ℓ and Ray +ℓ are only the estimation +errors ξx +ℓ and ξy +ℓ, respectively. The characteristic function ϕξ(t) +of these random variables is defined as [23] +ϕξ(t) = E +� +ejtξx +ℓ +� += E +� +ejtξy +ℓ +� +. +(14) +Thus, the (m, m′)th element of the autocorrelation matrix Rax +ℓ +is given by +� +Rax +ℓ +� +m,m′ = E +� +e−jνDA(m−m′)( ˆφx +ℓ−ξx +ℓ)� +(15a) += e−jνDA(m−m′) ˆφℓϕξ(νDA(m′ − m)) . +(15b) +The elements of Ray +ℓ are given, analogously. +Now, we can formulate the optimal precoder, as stated in +the following theorem. +Theorem 1. The robust precoder, i.e., the solution to (12), is +grob +ℓ += PTx +NS +ψℓ,max , +(16) +where ψℓ,max is the eigenvector corresponding to the largest +eigenvalue of (� +i̸=ℓ σ2 +αiRai + NSσ2 +n/PTxINT)−1σ2 +αℓRaℓ. +Proof: Note that αℓ, ξx +ℓ and ξy +ℓ are statistically inde- +pendent of αi, ξx +i and ξy +i for i ̸= ℓ. Therefore, hℓ and hi +are statistically independent, too. Thus, we can rewrite the +objective function (12) as +¯γℓ = E +� +|hH +ℓ gℓ|2 +� +i̸=ℓ |hH +i gℓ|2 + σ2n +� +(17a) + += +gH +ℓ E +� +hℓhH +ℓ +� +gℓ +gH +ℓ +�� +i̸=ℓ E +� +hihH +i +� ++ σ2n +pℓ INT +� +gℓ +(17b) +where pℓ = gH +ℓ gℓ is the transmit power of the ℓth stream. +With (2) and (13), the expected value in (17b) can be +written as E +� +hℓhH +ℓ +� += σ2 +αℓRaℓ. Furthermore, the term in +(17b) is monotonically increasing with the transmit power pℓ. +Therefore, at the optimum solution the constraint must be +active, i.e., pℓ = PTx/NS. Correspondingly, the optimization +problem (12) is equivalent to +max +gℓ +σ2 +αℓgH +ℓ Raℓgℓ +gH +ℓ +�� +i̸=ℓ σ2αiRai + NSσ2 +n +PTx I +� +gℓ +(18a) +s.t. +gH +ℓ gℓ = PTx +NS +(18b) +The objective function (18a) is a generalized Rayleigh +quotient. Consequently, the maximum is achieved by the +scaled eigenvector corresponding to the largest eigenvalue of +(� +i̸=ℓ σ2 +αiRai+NSσ2 +n/PTxINT)−1Raℓ [19], i.e., the precoder +grob +ℓ +must be proportional to ψℓ,max. Given the constraint +(18b), the only solution is given by (16). +The precoder in Theorem 1 optimizes (12) for any prob- +ability distribution of ξx +ℓ and ξy +ℓ. For the special case of +perfect position knowledge, we obtain a closed-form solution, +as stated in the following corollary. +Corollary 1. For perfect knowledge of the steering vectors, +the optimal precoder for (12) is +gper +ℓ += β +� NS +� +i=1 +σ2 +αiaiaH +i + NSσ2 +n +PTx +INT +�−1 +aℓ +(19) +where the normalization coefficient β is chosen such that +tr +� +GℓGH +ℓ +� += PTx/NS. +Proof: For perfect position knowledge, i.e., ξx +ℓ = ξy +ℓ = 0, +the characteristic function ϕξ(t) becomes one. Thus, the +objective function is +¯γ|ϕξ(t)=1 = +σ2 +αℓgH +ℓ aℓaH +ℓ gℓ +gH +ℓ +�� +i̸=ℓ σ2αiaiaH +i + NSσ2n +PTx I +� +gℓ +. +(20) +Consequently, +the +optimal +precoder +with +perfect +position +knowledge +must +be +a +scaled +eigenvector +of +�� +i̸=ℓ σ2 +αiaiaH +i + NSσ2 +n/PTxINT +�−1 +aℓaH +ℓ . +Such +an +eigenvector is given by (19) [27] +Furthermore, given perfect position knowledge and orthog- +onal channels, the following corollary holds as well. +Corollary 2. If ∀i ̸= ℓ, the steering vectors ai and aℓ are +orthogonal and the path losses |αi|2 and |αℓ|2 are the same, +i.e., aH +i aℓ = 0 and |αi|2 = |αℓ|2, respectively, the precoder +with perfect position knowledge (19) is capacity achieving. +Proof: Note that the precoder (19) is capacity achieving, +if its columns are given by the right singular vectors of H +and the norm of each column is equal to the optimal power +allocation, obtained via the water-filling algorithm [24]. Let +A = [a1, ..., aNS], with AHA = NTINS, and |αi|2 = |αℓ|2 = +|α|2, for all i and ℓ, the precoder matrix Gper = [gper +1 , ..., gper +NS ] +can be written as +Gper = β +� +|α|2AAH + NSσ2 +n +PTx +INT +�−1 +A +(21a) += βA +� +|α|2AHA + NSσ2 +n +PTx +INS +�−1 +(21b) += +� +PTx/NTA , +(21c) +where (21b) follows from the the matrix inversion lemma [28] +and (21c) is obtained due to the orthogonal steering vectors. +Now, given the channel model (2), the matrix H can be +factorized via the singular value decomposition (SVD) as +H = (diag (α1, ..., αNS) A)H = UΣVH , +(22) +where +U += +1/|α| diag (α1, ..., αNS)H +and +V += +1/√NT[A, vNS+1, ..., vNT] are unitary matrices. The vectors +{vNS+1, ..., vNT} are the right singular vectors belonging +to the nullspace of the channel matrix H. Furthermore, +Σ = [|α|√NTINS, 0NS×(NT−NS)], where 0NS×(NT−NS) is an +all zero matrix of dimension NS×(NT −NS), is a rectangular +diagonal matrix with the singular values of H on its diagonal. +Thus, the precoder matrix (21c) is proportional to the right +singular vectors of H, which do not belong to the nullspace. +Finally, as there is only a single non-zero singular value with +multiplicty NS, the capacity is achieved by allocating the +same power to each right singular vector. This is given by +the precoder Gper, which concludes the proof. +Note that in Corollary 2, the assumptions on the channel +are very strict. In the next section, we show numerically that +the precoder also achieves the capacity in more general cases. +IV. NUMERICAL EVALUATIONS +To evaluate the proposed precoder approach, we consider a +satellite swarm in triangle formation with a fixed inter-satellite +distance DS between any of these NS = 3 satellites. The +VSAT is equipped with a 32 × 32 URA with an antenna +spacing of DA = 2.5 cm. The antenna gain at the VSAT +and the satellites are ζTx,dB = 43.2 dBi − 10 log10(NT) ≈ +13.1 dBi and ζRx,dB = 30.5 dBi − 10 log10(NS) ≈ 25.7 dBi, +respectively, to match the 3GPP recommendation [29]. The +carrier frequency and the noise power are fc = 30 GHz and +PN = −120 dBW, respectively. The channel is modeled as a +pure LOS channel. Thus, the scaling factor is αℓ = 1/Lℓejφℓ,0, +where φℓ,0 ∼ U(0, 2π) is a random phase rotation and Lℓ is the +path loss, including the antenna gains ζTx,dB and ζRx,dB, free +space path loss, shadow fading, clutter loss [30], atmospheric +gas absorption [31] and tropospheric scintillation [32], [33]. +Furthermore, the altitude of the satellites is d0 = 600 km and +the minimum elevation angle is θ +el +ℓ,min = 30◦. + +10−1 +100 +101 +102 +103 +104 +2 +4 +6 +8 +10 +DS,opt +Inter-satellite distance DS km +Achievable Rate [bps/Hz] +C +R +Fig. 2: Achievable rate for different inter-satellite distances DS and perfect +position knowledge +A. Optimal Inter-Satellite Distance +In [12], the optimal inter-satellite distance for a simplified +downlink scenario has been derived. In this subsection, we +evaluate the performance w. r. t. the inter-satellite distance for +an uplink scenario and a fixed transmit power of PTx = +5 dBW. Given the altitude d0 += 600 km, the minimum +elevation angle θ +el +ℓ,min = 30◦, N x +T = 32 antennas along +the x-axis and νDA = 5π, the analytic solution for the +optimum inter-satellite distance is DS,opt ≈ 40 km [12]. In +Fig. 2, the channel capacity C and the sum rate R with the +proposed precoder and perfect position knowledge are shown. +It can be observed that both rates increases with increasing +inter-satellite distances DS up to a certain distance and then +slightly decrease. The inter-satellite distance, where both rates +achieve their maximum matches with the analytic solution +DS,opt from [12]. For DS < DS,opt, the sum rate is significantly +smaller than the channel capacity, due to the relatively big +difference between the maximum and minimum eigenvalue. +If DS ≥ DS,opt, all eigenvalues are approximately the same +because aH +i aℓ ≈ 0, for i ̸= ℓ. Therefore, the difference +between the sum rate and the capacity is negligible small. The +performance degradation for very large inter-satellite distances +is due to the increased path loss averaged over the satellites. +B. Robust Precoding +Now, we evaluate the performance of the proposed robust +precoder (16) for a constant inter-satellite distance DS = +40 km. Furthermore, we assume two different error distri- +butions of the position uncertainty. The robust precoder re- +quires knowledge about the CF of the probability distribution. +Therefore, the CFs of both probability distribution are given +in the appendix. For comparison, a heuristic precoder gheu +ℓ +for imperfect knowledge is obtained by substituting the true +steering vectors by the estimated ones in (19), i.e., +gheu +ℓ += β + +� +i̸=ℓ +σ2 +αiˆaiˆaH +i + NSσ2 +n +PTx +INT + + +−1 +ˆaℓ. +(23) +Note that for perfect position knowledge, the heuristic pre- +coder (23) as well as the robust precoder (16) are the same. +−5 +0 +5 +10 +15 +20 +25 +30 +0 +10 +20 +30 +Transmit power PTx [dBW] +Achievable Rate [bps/Hz] +C +robust, ξmax = +1 +128 +heuristic, ξmax = +1 +128 +robust, ξmax = +1 +64 +heuristic, ξmax = +1 +64 +Fig. 3: Achievable rate of robust and heuristic precoder in case of uniformly +distributed position uncertainty +−5 +0 +5 +10 +15 +20 +25 +30 +0 +10 +20 +30 +Transmit power PTx [dBW] +Achievable Rate [bps/Hz] +C +robust, σ2 +ξ = 2 · 10−5 +heuristic, σ2 +ξ = 2 · 10−5 +robust, σ2 +ξ = 8 · 10−5 +heuristic, σ2 +ξ = 8 · 10−5 +Fig. 4: Achievable rate of robust and heuristic precoder in case of gaussian +distributed position uncertainty +In Fig. 3, the achievable rates for the proposed robust +and heuristic precoder (16) and (23), respectively, are shown +for a uniformly distributed position uncertainty. Thus, the +error of the space angles are uniformly distributed, i.e., +ξx +ℓ, ξy +ℓ ∼ U(−ξmax, ξmax). It can be seen, that the sum rate with +the heuristic precoder degrades, especially for high transmit +powers. For the robust precoder, the sum rate is almost parallel +to the channel capacity. Thus, the performance degradation is +less severe and the robust precoder clearly outperforms the +heuristic precoder. +In Fig. 4, the corresponding performance for Gaussian +distribution, i.e., ξx +ℓ, ξy +ℓ ∼ N(0, σ2 +ξ), is shown. The variance σ2 +ξ +is chosen such that it is almost the same as for the uniformly +distributed error in Fig. 3. It can be seen that the performance +gain of the robust precoder compared to the heuristic precoder +is smaller than for uniformly distributed position uncertainty. +On the one hand, the slope for the robust precoder is not +parallel to the channel capacity anymore. Instead, the sum +rate degrades stronger for high transmit powers. On the other +hand, the impact of gaussian distributed position uncertainty +seem to be less severe for the heuristic precoder. +V. CONCLUSION +In this paper, a novel robust precoder for LOS communica- +tion is presented. Instead of full CSI, the proposed precoder + +is based on the second order statistics of the channel, which +include imperfect position knowledge of the receivers, as well +as statistical knowledge of the position uncertainty and the +long term fading of the channel. The position uncertainty +of the receivers induce a correlated phase error among the +transmit antennas. It has been shown that the resulting statistic +of the phase error and the channel statistics are connected +via the CF of the phase error distribution. Furthermore, the +proposed appraoch to transmit data from a VSAT to multiple +satellite has low complexity and is capacity achieving for +perfect position knowledge of the satellites and sufficiently +large distances between them. +ACKNOWLEDGMENT +This research was supported in part by the German Fed- +eral Ministry of Education and Research (BMBF) within +the project Open6GHub under grant number 16KISK016A, +the German Research Foundation (DFG) under Germany’s +Excellence Strategy (EXC 2077 at University of Bremen, +University Allowance) and by the European Space Agency +(ESA) within the SatNEx V activity WI Y2.2-A. +APPENDIX A +CHARACTERISTIC FUNCTION +a) Uniform Distribution: Let ξuni ∼ U(−υmax, υmax) be +uniformly distributed in the interval [−ξmax, ξmax]. 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Dietrich, Robust Signal Processing for Wireless Communications, ser. +Foundations in Signal Processing, Communications and Networking. +Springer-Verlag Berlin Heidelberg, 2008. +[29] 3GPP, “Solutions for NR to support non-terrestrial networks (NTN),” +TR 38.821 V16.0.0, Dec. 2019. +[30] ——, “Study on new radio (NR) to support non-terrestrial networks,” +TR 38.811 V15.4.0, Sep. 2020. +[31] ITU-R, “Attenuation by atmospheric gases and related effects,” ITU-R +P.676-12, Aug. 2019. +[32] ——, “Propagation data and prediction method required for the design +of Earth-space telecommunication systems,” ITU-R P.618-13, Dec. 2017. +[33] ——, “Inospheric propagation data and prediction methods required for +the design of satellite networks and systems,” ITU-R P.531-14, Aug. +2019. + diff --git a/Y9FOT4oBgHgl3EQf-TSC/content/tmp_files/load_file.txt b/Y9FOT4oBgHgl3EQf-TSC/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..0d6fe75aa9218019b005002bfef8c7cfa8ef4a4f --- /dev/null +++ b/Y9FOT4oBgHgl3EQf-TSC/content/tmp_files/load_file.txt @@ -0,0 +1,574 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf,len=573 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content='12973v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content='SP] 30 Jan 2023 Robust Precoding via Characteristic Functions for VSAT to Multi-Satellite Uplink Transmission Maik R¨oper∗, Bho Matthiesen ∗†, Dirk W¨ubben∗, Petar Popovski ‡,†, and Armin Dekorsy∗ ∗ Gauss-Olbers Center, c/o University of Bremen, Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' of Communications Engineering, 28359 Bremen, Germany † University of Bremen, U Bremen Excellence Chair, Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' of Communications Engineering, 28359 Bremen, Germany ‡ Aalborg University, Department of Electronic Systems, 9220 Aalborg, Denmark Email: {roeper, matthiesen, wuebben, dekorsy}@ant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content='uni-bremen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content='de, petarp@es.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content='aau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content='dk Abstract—The uplink from a very small aperture terminal (VSAT) towards multiple satellites is considered, in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' VSATs can be equipped with multiple antennas, allowing parallel transmission to multiple satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' A low-complexity precoder based on imperfect positional information of the satellites is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' The probability distribution of the position uncertainty and the statistics of the channel elements are related by the char- acteristic function of the position uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' This knowledge is included in the precoder design to maximize the mean signal- to-leakage-and-noise ratio (SLNR) at the satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' Furthermore, the performance w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' the inter-satellite distance is numerically evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' It is shown that the proposed approach achieves the capacity for perfect position knowledge and sufficiently large inter-satellite distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' In case of imperfect position knowledge, the performance degradation of the robust precoder is relatively small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' Index Terms—LEO satellites, 3D networks, massive MIMO, beamforming, beamspace MIMO I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' INTRODUCTION In future mobile networks, the terrestrial infrastructure will be extended by non-terrestrial networks (NTNs), consisting of unmanned aerial vehicles (UAVs), high altitude platform stations (HAPSs) and communication satellites, forming a global heterogeneous 3D network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' An integral component to these spatial networks are constellations of small satellites in low Earth orbits (LEOs) [1], [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' This is because, in comparison to satellites in medium Earth orbit (MEO) and geostationary orbit (GEO), these constellations have reduced deployment costs and acceptable transmission delays [3], [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' Recent results examine the application of multiple-input- multiple-output (MIMO) technologies to simultaneously com- municate with multiple satellites [5]–[8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' In combination with formation flying techniques, which allow multiple satellites to move in clusters and act as a single entity [9]–[11], swarms of small satellites can form huge virtual antenna arrays with the promise of massive spectral efficiency improvements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' In [12], [13], optimal geometrical conditions for satellite swarms with respect to (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=') the channel capactiy are de- rived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' Additionally, a low-complexity linear transceiver design for the downlink from a satellite swarm towards a MIMO ground terminal has been proposed for the considered scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' It has been observed that the derived transceiver achieves the channel capacity for sufficiently large inter-satellite distances and perfect position knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' In case of imperfect position knowledge, the statistics of the uncertainty can be included to design a robust precoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' In [14] robust precoding under imperfect phase knowledge for multi-user downlink scenario is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' The precoder is designed to achieve a given signal-to-interference-and-noise ratio (SINR) per user with minimum transmit power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' In order to deal with the phase uncertainty, the SINR constraint is relaxed to be satisfied either only in mean, or with a given probabilistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' Similarly, in [15], [16] robust precoder with a relaxed average SINR constraint in the presence of phase uncertainties are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' The precoder in [15] allows a trade-off between spectral and energy efficiency, while in [16] the energy efficiency for a hybrid precoder is maximized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' A precoder under the strict constraint to ensure an SINR above a given threshold in case of bounded position uncertainty is given in [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' There, different objective functions are numerically optimized, while the uncertainty region is sampled to obtain a finite set of constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' In [18] another robust precoder for the multi-user downlink is presented to maximize the signal power of a user while keeping the interference leakage to other users within a bounded uncertainty region below a given threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' A different approach to deal with position uncertainties is been presented in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' There, it is shown that the characteristic function (CF) of the position error is related to the autocorrelation matrix of the channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' In this paper, we utilize the CF of the position uncertainty to present a novel robust precoder to maximize the mean signal-to-leakage-and-noise ratio (SLNR) [19] for very small aperture terminal (VSAT) to multi-satellite uplink applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' Therefore, different to [17], [18], our precoder is suitable for arbitrary probability distributions of the receiver positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' Furthermore, we obtain an analytic solution, and therefore the computational complexity is relatively low, as the precoder only requires an eigendecomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' Furthermore, we show numerically that the proposed precoder achieves the capacity for sufficiently large inter-satellite distances and perfect posi- tion knowledge of the satellites at the VSAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' Next, in Section II, the general system model and the performance benchmark for perfect channel state information (CSI) is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' In Section III the proposed robust precoder is presented and numerically evaluated in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' Finally, Section V concludes the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' VSAT array y x z dℓ di Satellite ℓ Satellite i θel ℓ θaz ℓ Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' 1: Geometric relation between VSAT and satellite ℓ II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' SYSTEM MODEL AND PERFORMANCE BOUNDS We consider the uplink from a single VSAT to a group of NS single antenna satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' The VSAT is equipped with a uniform rectangular array (URA) consisting of NT = N x TN y T ≥ NS antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' Coordinates are stated in a VSAT-centric reference frame as depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' In particular, the z-axis points toward the VSAT’s zenith, and the x- and y-axes are aligned with the VSAT’s URA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' Thus, N x T and N y T are the number of antennas of the VSAT in x- and y-direction, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' The position of satellite ℓ ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=', NS} is given by the triplet (dℓ, θel ℓ , θaz ℓ ), where dℓ is the distance between satellite ℓ and the VSAT and θel ℓ ∈ [θel ℓ,min, π/2] and θaz ℓ ∈ [0, 2π] are the elevation and azimuth angles, respectively, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' The VSAT transmits NS independent Gaussian data streams s ∼ CN(0, INS), where INS is the identity matrix of di- mension NS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' These streams are linearly precoded with G = [g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=', gNS] ∈ CNT×NS to obtain the transmit signal x = Gs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' The VSAT is subject to an average power constraint PTx, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=', E � xHx � = tr � GGH� ≤ PTx .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' (1) The received signal yℓ at satellite ℓ is yℓ = hH ℓ x + nℓ, where hℓ ∈ CNT is the channel from the VSAT to satellite ℓ and nℓ is circularly-symmetric complex white Gaussian noise with power σ2 n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' Channel and Error Model We assume a line-of-sight (LOS) path between the VSAT and satellite ℓ and the received power from the non-LOS (NLOS) paths to be negligible small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' Then, the elements of hℓ differ only in their phase, but have the same magnitude [12], [20], [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' Thus, we can define the channel vector hℓ as a multiplication of a scalar factor αℓ ∈ C and a steering vector aℓ ∈ CNT, whose elements have magnitude one, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=', hℓ = αℓaℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' (2) The complex channel gain αℓ includes the signals attenuation and phase rotation due to the free space propagation and the atmospheric effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' The steering vector aℓ depends on the satellite position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' Due to the regular structure of the URA, it is helpful to separate aℓ into two steering vectors ax ℓ = [ax ℓ,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=', ax ℓ,N x T]T and ay ℓ = [ay ℓ,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=', ay ℓ,N y T]T , for the x- and y-direction, respectively, such that aℓ = ax ℓ ⊗ ay ℓ , (3) where ⊗ denotes the Kronecker product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' With the angles of departure (AoDs) as defined in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' 1 and given that the distance between the transmit antennas DA at the VSAT is much smaller than the distance between the VSAT and satellite ℓ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=', DA ≪ dℓ, the elements of the steering vectors are [22] ax ℓ,m = e−jνDA(m−1) cos(θel ℓ ) cos(θaz ℓ ) , m = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=', N x T , (4a) ay ℓ,n = e−jνDA(n−1) cos(θel ℓ ) sin(θaz ℓ ) , n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=', N y T , (4b) where ν is the wavenumber of the carrier wave .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' Furthermore, we define the space angles φx ℓ = cos � θel ℓ � cos (θaz ℓ ) and φy ℓ = cos � θel ℓ � sin (θaz ℓ ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' While the AoDs θel ℓ and θaz ℓ are related to the spherical coordinates, the space angles φx ℓ and φy ℓ are related to the Cartesian x- and y- coordinates of satellite ℓ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' Note that in practical systems only imperfect knowledge of the satellites positions can be assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' Let ξx ℓ and ξy ℓ be the estimation errors for the space angles in x- and y-direction, respectively, the estimated space angles are ˆφx ℓ = φx ℓ + ξx ℓ, ˆφy ℓ = φy ℓ + ξy ℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' (5) Then, the k = n+(m−1)N y Tth element of the channel vector hℓ can be written as hℓ,k = αℓax ℓ,may ℓ,n = αℓe−jνDA((m−1)( ˆφx ℓ−ξx ℓ)+(n−1)( ˆφy ℓ−ξy ℓ)) , (6) which includes three statistically independent random vari- ables from the transmitters perspective, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=', αℓ, ξx ℓ and ξy ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' The complex channel gain αℓ is a circularly-symmetric random variable with variance E{|αℓ|2} = σ2 αℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' Note that the variance σ2 αℓ can be different for each satellite ℓ, due to the different path losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' For the estimation errors of the space angles ξx ℓ and ξy ℓ, we assume the same probability distribution for every satellite ℓ and for both directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' Their probability distribution is described by the characteristic function ϕξ(t) [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' Upper bound on the channel capacity To obtain an upper bound on the throughput performance, perfect CSI at the transmitter and receiver as well as instanta- neous and error-free inter-satellite communication is assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' Hence, perfect coordination between satellites is possible and the system can be modelled as an NT × NS MIMO system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' The joint receive signal of all satellites is y = Hx + n, with the channel matrix H = [h1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=', hNS]H and the noise vector n = [n1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=', nNS]T , where the components of n are mutually independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' Let λµ be the µth eigenvalue of HHH, the capacity of this channel is [24] C = NS � µ=1 log2 � 1 + λµ pµ σ2n � , (7) where and pµ is the optimal power allocated to the µth stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' Given the power constraint � µ pµ = PTx, the optimal power allocation is obtained from the water-filling algorithm [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' PROPOSED UPLINK APPROACH In this section, we present a novel low-complexity robust precoder for uplink transmission from a VSAT to multiple satellites based on imperfect knowledge of the satellites po- sitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' For sufficiently large distances between the satellites, the channel vectors become nearly orthogonal, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=', aH i aℓ ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' In this case, the channel capacity is maximized, because the eigenvalues of HHH are almost equal [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' Given orthog- onal channels, the capacity can be achieved by transmitting independent data streams [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content='1 Thus, we assign one stream per satellite and then design the precoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' Given independent data streams, no inter-satellite communication is required for estimation and decoding because satellite ℓ can estimate and decode the transmitted symbol sℓ independently from the other satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' Numerical results in Section IV-A show that this approach is, despite its simplicity, capacity achieving for sufficiently large orbital separations and perfect position knowledge of the satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' The received signal at satellite ℓ is yℓ = hH ℓ gℓsℓ + � i̸=ℓ hH ℓ gisi + nℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' Correspondingly, the instantaneous SINR Γℓ at satellite ℓ is given as Γℓ = ��hH ℓ gℓ ��2 � i̸=ℓ |hH ℓ gi|2 + σ2n (8) and the achievable rate R is equal to the sum rate R = NS � ℓ=1 log2 (1 + Γℓ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' (9) The goal is to obtain precoders that maximize this rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' However, the exact values of hℓ are not known but only their estimates based on ˆφx ℓ and ˆφy ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' Furthermore, directly maximizing this sum rate is NP-hard and has, to the best of our knowledge, no easy analytical solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' In the following, we solve a slightly simplified version of this problem that, as will be seen later in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' IV, will result in a solution that is often close to the optimal throughput performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' Problem Formulation Observe that the SINRs Γℓ in (8) are coupled through their denominator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' Substituting Γℓ in (9) with the instantaneous SLNR for satellite ℓ defined as γℓ = |hH ℓ gℓ|2 � i̸=ℓ |hH i gℓ|2 + σ2n (10) 1Observe that this does not imply the transmission of several independent messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' See, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=', [25] or [26, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' and assuming equal power allocation among streams, leads to the substitute optimization problem max gℓ |hH ℓ gℓ|2 � i̸=ℓ |hH i gℓ|2 + |σ2n s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' gH ℓ gℓ ≤ PTx NS , (11) which has to be solved for each ℓ ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=', NS}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' However, this problem still relies on the exact knowledge of the channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' Our goal is to design robust precoders that take the statistics of the estimation error into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' This can be achieved by optimizing over the mean SLNR ¯γ = E {γ}, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=', max gℓ E � |hH ℓ gℓ|2 � i̸=ℓ |hH i gℓ|2 + σ2n � s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' gH ℓ gℓ ≤ PTx NS , (12) where the expectation is taken w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' the channel vectors h1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' , hNS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' Then, the robust precoders grob ℓ are a solution of (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' Robust Precoding via CF The solution of (12) requires a closed-form expression of the autocorrelation matrix of the channel E � hℓhH ℓ � , which is closely related to the autocorrelation matrix of the steering vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' In particular, let Raℓ = E � aℓaH ℓ � be the autocorre- lation matrix of the steering vector aℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' With (3) and due to the statistical independence of the position uncertainties ξx ℓ and ξy ℓ, we can write Raℓ = E � aℓaH ℓ � = E � (ax ℓ ⊗ ay ℓ) (ax ℓ ⊗ ay ℓ)H� (13a) = E � ax ℓax ℓ H� ⊗ E � ay ℓay ℓ H� = Rax ℓ ⊗ Ray ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' (13b) The random variables in Rax ℓ and Ray ℓ are only the estimation errors ξx ℓ and ξy ℓ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' The characteristic function ϕξ(t) of these random variables is defined as [23] ϕξ(t) = E � ejtξx ℓ � = E � ejtξy ℓ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' (14) Thus, the (m, m′)th element of the autocorrelation matrix Rax ℓ is given by � Rax ℓ � m,m′ = E � e−jνDA(m−m′)( ˆφx ℓ−ξx ℓ)� (15a) = e−jνDA(m−m′) ˆφℓϕξ(νDA(m′ − m)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' (15b) The elements of Ray ℓ are given, analogously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' Now, we can formulate the optimal precoder, as stated in the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' The robust precoder, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=', the solution to (12), is grob ℓ = PTx NS ψℓ,max , (16) where ψℓ,max is the eigenvector corresponding to the largest eigenvalue of (� i̸=ℓ σ2 αiRai + NSσ2 n/PTxINT)−1σ2 αℓRaℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' Proof: Note that αℓ, ξx ℓ and ξy ℓ are statistically inde- pendent of αi, ξx i and ξy i for i ̸= ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' Therefore, hℓ and hi are statistically independent, too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' Thus, we can rewrite the objective function (12) as ¯γℓ = E � |hH ℓ gℓ|2 � i̸=ℓ |hH i gℓ|2 + σ2n � (17a) = gH ℓ E � hℓhH ℓ � gℓ gH ℓ �� i̸=ℓ E � hihH i � + σ2n pℓ INT � gℓ (17b) where pℓ = gH ℓ gℓ is the transmit power of the ℓth stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' With (2) and (13), the expected value in (17b) can be written as E � hℓhH ℓ � = σ2 αℓRaℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' Furthermore, the term in (17b) is monotonically increasing with the transmit power pℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' Therefore, at the optimum solution the constraint must be active, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=', pℓ = PTx/NS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' Correspondingly, the optimization problem (12) is equivalent to max gℓ σ2 αℓgH ℓ Raℓgℓ gH ℓ �� i̸=ℓ σ2αiRai + NSσ2 n PTx I � gℓ (18a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' gH ℓ gℓ = PTx NS (18b) The objective function (18a) is a generalized Rayleigh quotient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' Consequently, the maximum is achieved by the scaled eigenvector corresponding to the largest eigenvalue of (� i̸=ℓ σ2 αiRai+NSσ2 n/PTxINT)−1Raℓ [19], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=', the precoder grob ℓ must be proportional to ψℓ,max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' Given the constraint (18b), the only solution is given by (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' The precoder in Theorem 1 optimizes (12) for any prob- ability distribution of ξx ℓ and ξy ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' For the special case of perfect position knowledge, we obtain a closed-form solution, as stated in the following corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' For perfect knowledge of the steering vectors, the optimal precoder for (12) is gper ℓ = β � NS � i=1 σ2 αiaiaH i + NSσ2 n PTx INT �−1 aℓ (19) where the normalization coefficient β is chosen such that tr � GℓGH ℓ � = PTx/NS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' Proof: For perfect position knowledge, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=', ξx ℓ = ξy ℓ = 0, the characteristic function ϕξ(t) becomes one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' Thus, the objective function is ¯γ|ϕξ(t)=1 = σ2 αℓgH ℓ aℓaH ℓ gℓ gH ℓ �� i̸=ℓ σ2αiaiaH i + NSσ2n PTx I � gℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' (20) Consequently, the optimal precoder with perfect position knowledge must be a scaled eigenvector of �� i̸=ℓ σ2 αiaiaH i + NSσ2 n/PTxINT �−1 aℓaH ℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' Such an eigenvector is given by (19) [27] Furthermore, given perfect position knowledge and orthog- onal channels, the following corollary holds as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' If ∀i ̸= ℓ, the steering vectors ai and aℓ are orthogonal and the path losses |αi|2 and |αℓ|2 are the same, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=', aH i aℓ = 0 and |αi|2 = |αℓ|2, respectively, the precoder with perfect position knowledge (19) is capacity achieving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' Proof: Note that the precoder (19) is capacity achieving, if its columns are given by the right singular vectors of H and the norm of each column is equal to the optimal power allocation, obtained via the water-filling algorithm [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' Let A = [a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=', aNS], with AHA = NTINS, and |αi|2 = |αℓ|2 = |α|2, for all i and ℓ, the precoder matrix Gper = [gper 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=', gper NS ] can be written as Gper = β � |α|2AAH + NSσ2 n PTx INT �−1 A (21a) = βA � |α|2AHA + NSσ2 n PTx INS �−1 (21b) = � PTx/NTA , (21c) where (21b) follows from the the matrix inversion lemma [28] and (21c) is obtained due to the orthogonal steering vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' Now, given the channel model (2), the matrix H can be factorized via the singular value decomposition (SVD) as H = (diag (α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=', αNS) A)H = UΣVH , (22) where U = 1/|α| diag (α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=', αNS)H and V = 1/√NT[A, vNS+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=', vNT] are unitary matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' The vectors {vNS+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=', vNT} are the right singular vectors belonging to the nullspace of the channel matrix H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' Furthermore, Σ = [|α|√NTINS, 0NS×(NT−NS)], where 0NS×(NT−NS) is an all zero matrix of dimension NS×(NT −NS), is a rectangular diagonal matrix with the singular values of H on its diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' Thus, the precoder matrix (21c) is proportional to the right singular vectors of H, which do not belong to the nullspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' Finally, as there is only a single non-zero singular value with multiplicty NS, the capacity is achieved by allocating the same power to each right singular vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' This is given by the precoder Gper, which concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' Note that in Corollary 2, the assumptions on the channel are very strict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' In the next section, we show numerically that the precoder also achieves the capacity in more general cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' NUMERICAL EVALUATIONS To evaluate the proposed precoder approach, we consider a satellite swarm in triangle formation with a fixed inter-satellite distance DS between any of these NS = 3 satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' The VSAT is equipped with a 32 × 32 URA with an antenna spacing of DA = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content='5 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' The antenna gain at the VSAT and the satellites are ζTx,dB = 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content='2 dBi − 10 log10(NT) ≈ 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content='1 dBi and ζRx,dB = 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content='5 dBi − 10 log10(NS) ≈ 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content='7 dBi, respectively, to match the 3GPP recommendation [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' The carrier frequency and the noise power are fc = 30 GHz and PN = −120 dBW, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' The channel is modeled as a pure LOS channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' Thus, the scaling factor is αℓ = 1/Lℓejφℓ,0, where φℓ,0 ∼ U(0, 2π) is a random phase rotation and Lℓ is the path loss, including the antenna gains ζTx,dB and ζRx,dB, free space path loss, shadow fading, clutter loss [30], atmospheric gas absorption [31] and tropospheric scintillation [32], [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' Furthermore, the altitude of the satellites is d0 = 600 km and the minimum elevation angle is θ el ℓ,min = 30◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' 10−1 100 101 102 103 104 2 4 6 8 10 DS,opt Inter-satellite distance DS km Achievable Rate [bps/Hz] C R Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' 2: Achievable rate for different inter-satellite distances DS and perfect position knowledge A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' Optimal Inter-Satellite Distance In [12], the optimal inter-satellite distance for a simplified downlink scenario has been derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' In this subsection, we evaluate the performance w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' the inter-satellite distance for an uplink scenario and a fixed transmit power of PTx = 5 dBW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' Given the altitude d0 = 600 km, the minimum elevation angle θ el ℓ,min = 30◦, N x T = 32 antennas along the x-axis and νDA = 5π, the analytic solution for the optimum inter-satellite distance is DS,opt ≈ 40 km [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' 2, the channel capacity C and the sum rate R with the proposed precoder and perfect position knowledge are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' It can be observed that both rates increases with increasing inter-satellite distances DS up to a certain distance and then slightly decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' The inter-satellite distance, where both rates achieve their maximum matches with the analytic solution DS,opt from [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' For DS < DS,opt, the sum rate is significantly smaller than the channel capacity, due to the relatively big difference between the maximum and minimum eigenvalue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' If DS ≥ DS,opt, all eigenvalues are approximately the same because aH i aℓ ≈ 0, for i ̸= ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' Therefore, the difference between the sum rate and the capacity is negligible small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' The performance degradation for very large inter-satellite distances is due to the increased path loss averaged over the satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' Robust Precoding Now, we evaluate the performance of the proposed robust precoder (16) for a constant inter-satellite distance DS = 40 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' Furthermore, we assume two different error distri- butions of the position uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' The robust precoder re- quires knowledge about the CF of the probability distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' Therefore, the CFs of both probability distribution are given in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' For comparison, a heuristic precoder gheu ℓ for imperfect knowledge is obtained by substituting the true steering vectors by the estimated ones in (19), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=', gheu ℓ = β \uf8eb \uf8ed� i̸=ℓ σ2 αiˆaiˆaH i + NSσ2 n PTx INT \uf8f6 \uf8f8 −1 ˆaℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' (23) Note that for perfect position knowledge, the heuristic pre- coder (23) as well as the robust precoder (16) are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' −5 0 5 10 15 20 25 30 0 10 20 30 Transmit power PTx [dBW] Achievable Rate [bps/Hz] C robust, ξmax = 1 128 heuristic, ξmax = 1 128 robust, ξmax = 1 64 heuristic, ξmax = 1 64 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' 3: Achievable rate of robust and heuristic precoder in case of uniformly distributed position uncertainty −5 0 5 10 15 20 25 30 0 10 20 30 Transmit power PTx [dBW] Achievable Rate [bps/Hz] C robust, σ2 ξ = 2 · 10−5 heuristic, σ2 ξ = 2 · 10−5 robust, σ2 ξ = 8 · 10−5 heuristic, σ2 ξ = 8 · 10−5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' 4: Achievable rate of robust and heuristic precoder in case of gaussian distributed position uncertainty In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' 3, the achievable rates for the proposed robust and heuristic precoder (16) and (23), respectively, are shown for a uniformly distributed position uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' Thus, the error of the space angles are uniformly distributed, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=', ξx ℓ, ξy ℓ ∼ U(−ξmax, ξmax).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' It can be seen, that the sum rate with the heuristic precoder degrades, especially for high transmit powers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' For the robust precoder, the sum rate is almost parallel to the channel capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' Thus, the performance degradation is less severe and the robust precoder clearly outperforms the heuristic precoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' 4, the corresponding performance for Gaussian distribution, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=', ξx ℓ, ξy ℓ ∼ N(0, σ2 ξ), is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' The variance σ2 ξ is chosen such that it is almost the same as for the uniformly distributed error in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' It can be seen that the performance gain of the robust precoder compared to the heuristic precoder is smaller than for uniformly distributed position uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' On the one hand, the slope for the robust precoder is not parallel to the channel capacity anymore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' Instead, the sum rate degrades stronger for high transmit powers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' On the other hand, the impact of gaussian distributed position uncertainty seem to be less severe for the heuristic precoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' CONCLUSION In this paper, a novel robust precoder for LOS communica- tion is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' Instead of full CSI, the proposed precoder is based on the second order statistics of the channel, which include imperfect position knowledge of the receivers, as well as statistical knowledge of the position uncertainty and the long term fading of the channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' The position uncertainty of the receivers induce a correlated phase error among the transmit antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' It has been shown that the resulting statistic of the phase error and the channel statistics are connected via the CF of the phase error distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' Furthermore, the proposed appraoch to transmit data from a VSAT to multiple satellite has low complexity and is capacity achieving for perfect position knowledge of the satellites and sufficiently large distances between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' ACKNOWLEDGMENT This research was supported in part by the German Fed- eral Ministry of Education and Research (BMBF) within the project Open6GHub under grant number 16KISK016A, the German Research Foundation (DFG) under Germany’s Excellence Strategy (EXC 2077 at University of Bremen, University Allowance) and by the European Space Agency (ESA) within the SatNEx V activity WI Y2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content='2-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' APPENDIX A CHARACTERISTIC FUNCTION a) Uniform Distribution: Let ξuni ∼ U(−υmax, υmax) be uniformly distributed in the interval [−ξmax, ξmax].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' Then, the characteristic function ϕU is a sinc-function, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=', ϕU(t) = � ξmax −ξmax 1 2ξmax ejξtdξ (24a) = sin (tξmax) tξmax = sinc (tξmax) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' (24b) b) Gaussian Distribution: Let ξgau ∼ N(0, σ2 ξ) be gaus- sian distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' Then the characteristic function ϕG follows also a gaussian function [23, Example 8.' metadata={'source': 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methods required for the design of satellite networks and systems,” ITU-R P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content='531-14, Aug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FOT4oBgHgl3EQf-TSC/content/2301.12973v1.pdf'} diff --git a/YNE5T4oBgHgl3EQfCw5O/content/tmp_files/2301.05398v1.pdf.txt b/YNE5T4oBgHgl3EQfCw5O/content/tmp_files/2301.05398v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a10076447b0c09fc5e20f3f6336f90144a98ded0 --- /dev/null +++ b/YNE5T4oBgHgl3EQfCw5O/content/tmp_files/2301.05398v1.pdf.txt @@ -0,0 +1,1300 @@ +Draft version January 16, 2023 +Typeset using LATEX twocolumn style in AASTeX631 +Multiwavelength Analysis of a Nearby Heavily Obscured AGN in NGC 449 +Xiaotong Guo (郭晓通) +,1 Qiusheng Gu (顾秋生) +,2, 3 Jun Xu (徐骏) +,1 Guanwen Fang (方官文) +,1 +Xue Ge (葛雪) +,4 Yongyun Chen (陈永云) +,5 Xiaoling Yu (俞效龄) +,5 and Nan Ding (丁楠) +6 +1Institute of Astronomy and Astrophysics, Anqing Normal University, Anqing, Anhui 246133, China +2School of Astronomy and Space Science, Nanjing University, Nanjing, Jiangsu 210093, China +3Key Laboratory of Modern Astronomy and Astrophysics (Nanjing University), Ministry of Education, Nanjing 210093, China +4School of Physics and Electronic Engineering, Jiangsu Second Normal University, Nanjing, Jiangsu 211200, China +5College of Physics and Electronic Engineering, Qujing Normal University, Qujing 655011, China +6School of Physical Science and Technology, Kunming University, Kunming 650214, China +ABSTRACT +We presented the multiwavelength analysis of a heavily obscured active galactic nucleus (AGN) in +NGC 449. We first constructed a broadband X-ray spectrum using the latest NuSTAR and XMM- +Newton data. Its column density (≃ 1024cm−2) and photon index (Γ ≃ 2.4) were reliably obtained +by analyzing the broadband X-ray spectrum. However, the scattering fraction and the intrinsic X-ray +luminosity could not be well constrained. Combined with the information obtained from the mid- +infrared (mid-IR) spectrum and spectral energy distribution (SED) fitting, we derived its intrinsic +X-ray luminosity (≃ 8.54 × 1042 erg s−1) and scattering fraction (fscat ≃ 0.26%). In addition, we also +derived the following results: (1). The mass accretion rate of central AGN is about 2.54×10−2M⊙ yr−1, +and the Eddington ratio is 8.39 × 10−2; (2). +The torus of this AGN has a high gas-to-dust ratio +(NH/AV = 8.40 × 1022 cm−2 mag−1); (3). The host galaxy and the central AGN are both in the early +stage of co-evolution. +Keywords: Active galactic nuclei(16) — Seyfert galaxies(1447) — X-ray active galactic nuclei(2035) +— AGN host galaxies(2017) +1. INTRODUCTION +It is well known that active galactic nuclei (AGNs) are +powered by the accretion of surrounding matter by su- +permassive black holes (SMBHs). The emission of AGNs +covers almost the whole electromagnetic band. +The +AGNs’ radiations at different wavelengths arise from +their diverse structures. +For example, the significant +emission of the accretion disk is in the ultraviolet (UV) +to the optical band (e.g., Richstone & Schmidt 1980), +while the emission of the torus is in the mid-infrared +(mid-IR) band (e.g., Antonucci 1993). The diverse in- +formation can be derived from the emission of AGNs at +different wavelengths. To fully understand an AGN, it +is often necessary to study with the multi-bands. +Corresponding author: Qiusheng Gu +qsgu@nju.edu.cn +The classification of AGNs usually utilizes the obser- +vation features of different wavelengths. For instance, +based on the characteristics of the optical emission +lines, AGNs can be classified into type 1 and type 2 +(e.g., Khachikian & Weedman 1974). The AGN unified +model proposes that different AGN types are caused +by different viewing angles with an obscuring torus +(e.g., Antonucci 1993; Urry & Padovani 1995; Netzer +2015). +Since the X-ray emission has a strong pierce, +the X-ray emission of AGNs is usually subject to lit- +tle absorption. Based on the column density (NH) of +the X-ray absorption, AGNs can be classified into four +categories: unobscured (NH < 1022 cm−2), obscured +(1022 cm−2 < NH < 1023 cm−2), heavily obscured +(1023 cm−2 < NH < 1.5 × 1024 cm−2), and Compton- +thick (NH ⩾ 1.5 × 1024 cm−2, e.g., Comastri 2004). +Many studies have also regarded the source with a col- +umn density of ∼ 1024 cm−2 as Compton-thick AGN +(CT-AGN, e.g., Risaliti et al. 1999; Kammoun et al. +2019). +arXiv:2301.05398v1 [astro-ph.GA] 13 Jan 2023 + +ID2 +Guo et al. +Some studies have pointed out that most AGNs were +heavily obscured (e.g., Hickox & Alexander 2018). The +heavily obscured AGNs are believed to be in an early +phase of the evolutionary scenario of AGNs, which is a +rapid growth state of central SMBH (e.g., Goulding et al. +2011). The host galaxies of heavily obscured AGNs are +also in an early phase of the evolutionary scenario of +galaxies. For instance, Kocevski et al. (2015) pointed +out that heavily obscured AGNs were twice as likely to +be hosted by late-type galaxies relative to unobscured +AGNs. As a class of the most luminous and persistent +X-ray sources in the universe, AGNs contribute the vast +majority of Cosmic X-ray Background (CXB, e.g., Ueda +et al. 2014) in the 1 to ∼200 – 300 keV. The heavily ob- +scured AGNs and CT-AGNs are significant contributors +to the hump of the CXB emission at 30 keV (e.g., Mar- +shall et al. 1980; Ajello et al. 2008). +The single-band analysis for AGNs can only derive a +part of their properties. For example, we can derive the +column density in the line-of-sight (LOS) direction by +fitting X-ray spectra. Using the spectral energy distri- +butions (SEDs) from the UV to the far-IR, we can de- +rive the star formation rates (SFRs) of the host galaxies, +stellar masses of the host galaxies, and AGN luminosi- +ties. To fully understand heavily obscured AGNs, their +host galaxies, and their co-evolution, we first derive their +properties through multi-band analysis. +The heavily obscured AGNs usually present a complex +spectral shape in their X-ray waveband. Theoretically, +the soft X-ray emission (E ≲ 2 keV) of heavily obscured +AGNs will be heavily (entirely) absorbed. Even in many +CT AGNs, the soft X-ray is not entirely absorbed, be- +cause the emission from the AGNs’ scattered and the +host galaxy can be observed (e.g., Turner et al. 1997; +LaMassa et al. 2012). As the energy increases, the ab- +sorption of X-ray emission (E>10 keV) is gradually re- +duced or cut-off. The hard X-ray of heavily obscured +AGN is dominated by the reflected component, the so- +called Compton reflection continuum characterized by a +broad bump peaking around 20–30 keV (e.g. Comastri +2004). One of the most prominent features in the X-ray +spectrum of heavily obscured AGN is the neutral iron +Kα emission line at 6.4 keV. However, the X-ray spec- +tra of some heavily obscured AGNs show a weak (or +absent) iron Kα emission line (Boorman et al. 2018), +especially the heavily obscured AGNs beyond the local +universe. To accurately determine the column density of +the gas and recover the intrinsic X-ray flux of AGNs, we +require proper modeling of the physical processes in the +AGNs that self-consistently account for the transmitted +emission, Compton-scattered reflected component and +the fluorescent line flux. Such modeling may also con- +strain the geometry of the torus and view angle. Over +the past several years, a series of X-ray spectral mod- +els (e.g., Murphy & Yaqoob 2009; Balokovi´c et al. 2018) +are created with assumed different geometries (spheri- +cal, toroidal, or clumpy). Many recent studies use these +models to constrain the parameters of AGNs (Gandhi +et al. 2017; Kammoun et al. 2019; LaMassa et al. 2019; +Toba et al. 2020). +From the UV to the optical band, the continua of +the heavily obscured AGNs are masked by the emis- +sion of the host galaxies, so we can only observe their +high-ionized emission lines and obtain little information +about them. It is difficult to constrain the gas column +density along LOS for the AGNs with low-quality X-ray +spectra. However, we can estimate the gas column den- +sity using the luminosity ratio of X-ray to high-ionized +emission lines (e.g., Maiolino et al. 1998; Cappi et al. +2006; Gilli et al. 2010; Lanzuisi et al. 2015). The mid-IR +emission of the AGNs is usually less suppressed due to +the low optical depth. The contribution of AGNs can be +easily decomposed in the mid-IR spectra. In addition, +for heavily obscured AGNs, we can also observe a strong +silicon absorption line at 9.7 µm. The dust in the torus +can be derived using silicon absorption strength (e.g., +Xu et al. 2020). By the SED fitting, we can accurately +obtain the contribution of the AGNs, the AGN lumi- +nosities, and some parameters about their host galaxies. +So the multi-band data can provide us with more com- +prehensive information on heavily obscured AGNs. +NGC 449 (also known as Mrk 1) is a nearby +(z=0.01595, D∼67 Mpc) Seyfert 2 galaxy, and its cen- +ter hosts a heavily obscured AGN. It was observed +by the XMM-Newton X-ray telescope (Jansen et al. +2001) in 2004. Guainazzi et al. (2005) fitted the X-ray +spectrum and constrained the column density (NH > +1.1 × 1024 cm−2) of this AGN. Moreover, some stud- +ies also derived its column density by using the ratio +of X-ray to high-ionized emission lines ([O III]5007, [Ne +V]3426) luminosity (Heckman et al. 2005; Gilli et al. +2010). These results indicated that the X-ray is severely +absorbed and that even the AGN may be a CT-AGN. +However, these results did not constrain its column den- +sity well. To better constrain its column density and +other properties, the spectrum above 10 keV is neces- +sary. Fortunately, this source was observed by the Nu- +clear Spectroscopic Telescope Array (NuSTAR, Harri- +son et al. 2013) in 2017. The NuSTAR data has still not +been analyzed (or presented in previous work). More- +over, there are abundant data about NGC 449 in the +archives, which can also help us to obtain more proper- +ties. So the motivation of our work is to constrain the + +Multiwavelength Analysis of NGC 449 +3 +properties of this heavily obscured AGN by multiwave- +length analysis. +The structure of the paper is as follows. +Section 2 +describes the multiwavelength data and processing for +NGC 449. +In Section 3, we analyzed the multiwave- +length data of this source in detail. +Then, we dis- +cuss the implications of our results in Section 4. +Fi- +nally, we present a summary of this work in Sec- +tion 5. We adopt a concordance flat Λ-cosmology with +H0 = 67.4 km s−1 Mpc−1, Ωm = 0.315, and ΩΛ = 0.685 +(Planck Collaboration et al. 2020). +2. MULTIWAVELENGTH DATA +2.1. X-Ray Observations and Data Reduction +A log of the XMM-Newton and NuSTAR observations +analyzed herein is presented in Table 1, and the individ- +ual data sets are described in this section. +2.1.1. XMM-Newton Observations +NGC 449 was observed by the XMM-Newton X-ray +telescope (Jansen et al. 2001) for about 11.9 ks of expo- +sure on January 9, 2004 (PI: Matteo Guainazzi, ObsID: +0200430301). The data is reduced in a standard man- +ner using the XMM-Newton Science Analysis System +(SAS, Gabriel et al. 2004) v17.0.0 and Current Cali- +bration Files (CCF) of June 22, 2018. The spectra are +extracted from a circular region (30 arcsec radius for PN +and 25 arcsec radius for MOS) around the source, and +the background spectra are taken from a nearby source- +free circular region (80 arcsec radius for PN and 100 +arcsec radius for MOS). The spectra are binned to have +minimum counts of 10 per energy bin. +2.1.2. NuSTAR Observations +NGC 449 was also observed by the NuSTAR (Harrison +et al. 2013) for about 32.7 ks of exposure on December 8, +2017 (ObsID: 60360002002). The data is processed us- +ing the NuSTAR data analysis software nustardas v1.9.5 +available in heasoft v6.27.2 and CALDB released on Au- +gust 13, 2020. The nupipeline script is used to produce +calibrated and clean event files. We extract source spec- +tra using the nuproducts task. The spectra are extracted +from a 45 arcsec radius aperture centered on the source, +while the background is extracted from a circular region +with a 100 arcsec radius from a source-free region on the +detector. To improve the spectral quality of NuSTAR, +we try to combine the two spectra of FPMA and FPMB. +The combined spectrum is also binned to have minimum +counts of 10 per energy bin. +2.2. Mid-IR spectral Observations and Data Reduction +We search the archives for its mid-IR data. There is +good mid-IR spectral data, covering wavelengths from +5 to 37 µm, observed by Spitzer’s IrsStare on February +5, 2009 (PI: Levenson, Nancy, ObsID: 25408000). The +mid-IR spectrum was processed using the Spitzer data +analysis software irs merge v2.1 with the pipeline script +(version: S18.18.0) on December 18, 2011. For more de- +tailed processing of the mid-IR spectrum of this source, +please refer to Section 2 of Lebouteiller et al. (2011). +2.3. Multiwavelength photometric data +To construct the SED for NGC 449, we search for +available multi-band photometric data from different +surveys or the archives of telescopes, i.e., the Sloan Dig- +ital Sky Survey (SDSS) Data Release 7 (Abazajian et al. +2009), the Two Micron All-Sky Survey (2MASS, Skrut- +skie et al. 2006), the Wide-field Infrared Survey Explorer +(WISE, +Wright et al. 2010), the AKARI and the In- +frared Astronomical Satellite (IRAS). In total, we collect +the photometric data for 17 filters. Among them, only +the filter SDSS-u is in the UV band. There are three +filters of the optical band, including SDSS-g, SDSS-r, +and SDSS-i. The other filters are in the IR band, in- +cluding four near-IR filters (SDSS-z, 2MASS-J, 2MASS- +H, 2MASS-K), seven mid-IR filters (W1, W2, W3, W4, +IRAS-PSC 25, AKARI-PSC 09, AKARI-PSC 18) and +two far-IR filters (IRAS-PSC 60, IRAS-PSC 100). The +details for photometric data and filters are listed in Ta- +ble 2. +3. ANALYSIS OF MULTIWAVELENGTH DATA +3.1. X-Ray Spectral Analysis +In this section, we analyze the X-ray spectra of NGC +449 in Xspec v.12.11.0 (Arnaud 1996). +First, we use +phenomenological modeling to determine whether there +is variation in the spectra for the same energy range of +XMM-Newton and NuSTAR. We then analyzed which +components contributed to the X-ray spectra of this +source. Based on the analysis of the X-ray components, +we finally use the self-consistent and physical models to +fit the X-ray spectra. +All errors represent the 68.0% +confidence intervals unless otherwise stated. +3.1.1. Testing for Variability Among Observations +To test whether there is significant variation in the +spectra of two observations within the same energy +range (3–10 keV), we need to compare their spectral +shapes and fluxes. We can obtain them by fitting the +X-ray spectra of this source. +However, the data for +this source is poor. To obtain more reliable properties, + +4 +Guo et al. +Table 1. Observation log. +ObsID +Observation date +Mission +Instrument(s) +Net Exposure +Net Counts +[ks] +(1) +(2) +(3) +(4) +(5) +(6) +MOS1 +11.4 +249.90 +0200430301 +2004-01-09 +Newton +MOS2 +11.5 +222.31 +PN +8.6 +735.63 +60360002002 +2017-12-23 +NuSTAR +FPMA +32.7 +130.66 +FPMB +32.6 +126.67 +Table 2. Photometry of NGC 449. +Band +Wavelength +Flux +References +[µm] +[mJy] +(1) +(2) +(3) +(4) +SDSS-u +0.3547 +1.5473 ± 0.0109 +(a) +SDSS-g +0.4767 +5.1128 ± 0.0104 +SDSS-r +0.6226 +8.5429 ± 0.0161 +SDSS-i +0.7615 +10.024 ± 0.0196 +SDSS-z +0.9123 +12.909 ± 0.0368 +2MASS-J +1.234 +19.8 ± 0.704 +(b) +2MASS-H +1.661 +23.4 ± 1.08 +2MASS-K +2.157 +22.6 ± 1.11 +WISE-W1 +3.4 +13.035 ± 0.2084 +(c) +WISE-W2 +4.6 +20.144 ± 0.3220 +WISE-W3 +12 +116.39 ± 1.8618 +WISE-W4 +22 +551.74 ± 1.9548 +IRAS-PSC 25 +25 +801.20 ± 128.19 +(d) +IRAS-PSC 60 +60 +2302.0 ± 299.26 +IRAS-PSC 100 +100 +2851.0 ± 342.12 +AKARI-PSC 09 +9 +101.14 ± 1.4838 +(e) +AKARI-PSC 18 +18 +531.34 ± 25.134 +Note—(a) Abazajian et al. (2009), (b) Skrutskie et al. (2006), (c) +Wright et al. (2010), (d) Moshir & et al. (1990), (e) Terashima +et al. (2015). +we use a phenomenological model to fit the spectra of +XMM-Newton1 and NuSTAR within the same energy +range. The phenomenological model is written (in the +1 The spectra of MOS1, MOS2 and PN are simultaneously fitted. +Table 3. +Variability among observations within the +same energy range. +Mission +log(f3−10 keV) +Γ3−10 keV +χ2/dof +[erg · s−1 · cm−2] +(1) +(2) +(3) +(4) +Newton +−12.85 ± 0.13 +−0.86 ± 0.86 +5.02/5 +NuSTAR +−13.06 ± 1.58 +0.70 ± 0.59 +31/17 +Xspec terminology) as follows: +model = phabs[1] ∗ cflux[2] ∗ powerlaw[3], +where phabs[1] accounts for the Galactic absorption +which is fixed at a column density of 5.16 × 1020 cm−2. +Table 3 lists some properties of these two spectra +within the same energy range. Their fluxes are the same +within the error range. The spectral shape of XMM- +Newton’s spectrum within the same energy range indi- +cates that the X-ray emission of NGC 449 is seriously +absorbed. +Similarly, the NuSTAR observation of this +source also presents a flat spectrum. The data in the +same energy range are so poor that their photon indices +cannot be well constrained. In short, these two observa- +tions have similar fluxes within the same energy range. +Therefore, there is no significant variation in the X-ray +emission for NGC 449 among these two observations. +For subsequent X-ray spectrum analysis, we consti- +tute a broadband X-ray spectrum using the X-ray spec- +tra obtained with XMM-Newton and NuSTAR, which +covers the 0.35–30 keV band (PN for XMM-Newton : +0.35–10.0 keV, NuSTAR : 3.0–30.0 keV). +3.1.2. Analysis of the basic components + +Multiwavelength Analysis of NGC 449 +5 +In this section, our goal is to analyze which compo- +nents contribute to the X-ray spectra of this source, pro- +viding support for subsequent model selection. +First, We use a powerlaw model (phabs ∗ powerlaw) +with a photon index of 2.17 to fit the broadband spec- +trum. +Of course, the goodness of fit is poor (see the +panel b of Figure 1). There is significant excess in the +spectrum above 10 keV, which is a flat spectrum. So the +broadband spectrum is again fitted by a broken power- +law model (phabs ∗ bknpower). Figure 1(c) shows the +residuals of the broken powerlaw model. The best-fitting +break energy is 3.93 keV. The low energy band is a steep +spectrum (Γ = 2.36), and the high energy band is a flat +spectrum (Γ = 0.51). +Although its goodness of fit is +significantly improved, there is still excess near 6.5 keV +and 1 keV. Therefore, we also need to add two more +models. Among them, a Gaussian model is used to fit +the iron emission line near 6.5 keV, and the equivalent +width (EW) of the iron emission line is 1.49 keV which +is consistent with previous work (EW<2 keV, Guainazzi +et al. 2005). The other model (diffuse thermal emission) +is used to match the excess near 1.0 keV. Finally, we +obtain the best-fitting of the spectrum, as shown in Fig- +ure 1(a). The fitted parameters are listed in Table 4. +We find a flat spectrum above 3.71 keV by fitting +the broadband spectrum, indicating that the emission +of this source is seriously absorbed. Therefore, the re- +processed emission is dominant in hard X-rays. +The- +oretically, there should be a flat spectrum at energies +below 3.71 keV. In fact, there is a steep spectrum below +3.71 keV with a photon index of 2.38. The power-law +emission below 3.71 keV is most likely contributed by +the scattered component of primary X-ray emission. Be- +sides, the diffuse thermal emission and the iron emission +line are required. +3.1.3. Pexmon model +According to the analysis in Section 3.1.2, we fit the +spectrum using the neutral reflection model Pexmon +(Nandra et al. 2007). The detailed model can be written +as follows: +modelP exmon =phabs[1] ∗ (zphabs[2] ∗ powerlaw[3]+ +constant[4] ∗ powerlaw[5] + mekal[6]+ +pexmon[7]). +In this model, the phabs[1] component stands for the +Galactic absorption is fixed at a column density of +5.16 × 1020cm−2, powerlaw[3] represents the primary +X-ray emission of the AGN, which is absorbed by ob- +scured material (zphabs[2]). The primary X-ray emis- +sion is scattered by ionized gas in the polar regions, +while the scattering component is usually little or not +absorbed. The soft scattering component into our LOS +is represented by constant[4] ∗ powerlaw[5]. Thus the +parameters of powerlaw[5] are tied to the parameters of +powerlaw[3]. The diffuse thermal radiation is denoted +by mekal[6], similar to Section 3.1.2. We use pexmon[7] +to represent the reprocessed emission. +The Pexmon model assumes a slab obscurer/reflector +with an infinite optical depth, as may be expected in a +standard geometrically thin accretion disc. The incident +source in X-rays is the primary emission (powerlaw) +from a hot electron corona. Therefore, the photon index +and normalization of pexmon[7] are tied to the identi- +cal parameters of powerlaw[3]. +The cutoff energy of +pexmon[7] is fixed as 500 keV. We are not able to con- +strain the Fe abundance (tied to the elemental abun- +dance), so we fix it to its best-fit value. The reflection +fraction was fixed to -1 in the Pexmon fit. +This model fits the data well (χ2/dof=101.38/105). +The best-fitting and residuals are shown in Figure 2. +The Pexmon model implies that the X-ray emission of +central AGN is absorbed by Compton-thick material +with NH = (1.27 ± 0.26) × 1024 cm−2. The intrinsic 2– +10 keV luminosity is L2−10 = (0.64±0.16)×1042 erg s−1. +The photon index is 2.44, suggesting that the spectrum +of central AGN is very soft. We note that the scatter- +ing component is found to be fscat = (3.22 ± 1.57)% of +the primary emission, and that the 2–10 keV luminosity +of scattering component is (2.05 ± 1.09) × 1040 erg s−1. +The diffuse thermal gas model (mekal[6]) adequately +describes the soft X-rays with a temperature kT = +0.82 keV. More best-fit parameters for the model are +listed in Table 5. +3.1.4. Borus model +Similarly, according to the analysis in Section 3.1.2, +we refit the spectrum of the reprocessed emission using +the Borus model (Balokovi´c et al. 2018). Borus model is +defined with the following command sequence in Xspec: +modelBorus = phabs[1] ∗ (zphabs[2] ∗ cabs[3]∗ +powerlaw[4] + constant[5] ∗ powerlaw[6] ++ mekal[7] + Borus02[8]). +The phabs[1], constant[5] ∗ powerlaw[6], and mekal[7] +components are equivalent to the ones in the Pexmon +fit. The zphabs[2]∗cabs[3] represents LOS absorption at +the redshift of the X-ray source (generally independent +from the average column density of the torus), including +Compton scattering losses out of the line of sight. The +photon index and normalization of Borus02[8] are also + +6 +Guo et al. +Table 4. Analysis of the basic components. +Parameter +Power-law +Broken power-law +Broken+Gauss+Diffuse +(1) +(2) +(3) +(4) +(5) +powerlaw +Γ +2.17 ± 0.06 +. . . +. . . +Γ1 +. . . +2.36 ± 0.08 +2.38 ± 0.12 +bknpower +EBreak (keV) +. . . +3.93 ± 0.50 +3.71 ± 0.60 +Γ2 +. . . +0.51 ± 0.18 +0.36 ± 0.19 +zgauss +lineE(keV) +. . . +. . . +6.56 ± 0.09 +σ(keV) +. . . +. . . +0.16 ± 0.15 +EW(keV) +. . . +. . . +1.49 +mekal +kT(keV) +. . . +. . . +0.82 ± 0.06 +χ2/dof +224.61/110 +150.52/108 +100.52/103 +0.5 +1.0 +2.0 +5.0 +10.0 +20.0 +Energy [keV] +10 +6 +10 +5 +10 +4 +Photons cm +2 s +1 keV +1 +(a) +XMM-Newton +NuSTAR +10 +20 +Ratio +(b) +Power-law +2/dof=2.04 +2 +4 +Ratio +(c) +Broken power-law +2/dof=1.39 +0.5 +1.0 +2.0 +5.0 +10.0 +20.0 +Energy [keV] +0 +2 +4 +Ratio +(d) +Broken+Gauss+Diffuse +2/dof=0.98 +Figure 1. XMM-Newton PN (black) and NuSTAR (red) spectra of NGC 449. The black line in panel (a) represents the +best-fit model (broken + gauss + diffuse). Panel (b) shows the ratio (data/model) by fitting the X-ray spectra using a simple +power-law model. Panels (c) and (d) show the residuals obtained by fitting broken power-law and broken + gauss + diffuse +models, respectively. +tied to the powerlaw[3], and the cutoff energy is fixed +as 500 keV. Some parameters can not be constrained, +so we fix them to their best-fit values, i.e., half-opening +angle, inclination angle, and iron abundance. +This model fits the data well (χ2/dof=101.35/105). +The best-fitting and residuals are shown in Figure 3. +The column density of LOS is NH = (0.97 ± 0.16) × +1024 cm−2 , and the average column density of the torus +is about NH,Torus = 1.23 × 1024 cm−2. The intrinsic 2– +10 keV luminosity is L2−10 = (0.93±0.24)×1042 erg s−1. +More best-fit parameters for the model are listed in Ta- +ble 5. +Through the above analysis and the broadband X-ray +spectrum fitting of two physical models (Pexmon and +Borus), we have obtained some parameters about the +AGN in NGC 449. +Most of the parameters obtained +by the two models are the identical within the error +range, i.e., column density of LOS, the photon index, the +luminosity of scattering component, the temperature of +diffuse thermal gas, and the absorbed luminosity. These +suggest that these parameters can be well constrained in +the X-ray band. However, the fraction of the scattering +component and the intrinsic luminosity can not be well +constrained. Therefore, we need information from other +bands to constrain these parameters. +3.2. Mid-IR spectrum Analysis + +Multiwavelength Analysis of NGC 449 +7 +10 +5 +10 +4 +10 +3 +keV2 [Photons cm +2 s +1 keV +1] +XMM-Newton +NuSTAR +0.5 +1.0 +2.0 +5.0 +10.0 +20.0 +Energy [keV] +0 +2 +4 +Ratio +Figure 2. +Pexmon model fit to the XMM-Newton PN +(black) and NuSTAR (red) data. The top panel represents +the best-fitting. The bottom panel shows the residuals. +Table 5. Best-fit parameters for Pexmon and Borus. +Parameter +Pexmon +Borus +(1) +(2) +(3) +NH,LOS (1024 cm−2) +1.27 ± 0.26 +0.97 ± 0.16 +log NH,Torus (cm−2) +· · · +24.09 ± 0.13 +Γ +2.44 ± 0.13 +2.38 ± 0.10 +fscat(%) +3.22 ± 1.52 +2.50 ± 1.04 +f2−10,scat(10−14 erg s−1 cm−2) +3.81 ± 2.04 +4.28 ± 2.10 +L2−10,scat(1040 erg s−1) +2.05 ± 1.09 +2.30 ± 1.13 +kT(keV) +0.82 ± 0.06 +0.82 ± 0.06 +θtorus(◦) +. . . +25.0 +θinc(◦) +76.23 ± 9.44 +84.4 +f2−10,abs(10−12 erg s−1 cm−2) +0.104+0.002 +−0.034 +0.104+0.005 +−0.018 +L2−10,abs(1042 erg s−1) +0.063+0.002 +−0.013 +0.063+0.005 +−0.013 +f2−10,int(10−12 erg s−1 cm−2) +1.18 ± 0.30 +1.72 ± 0.44 +L2−10,int(1042 erg s−1) +0.64 ± 0.16 +0.93 ± 0.24 +χ2/dof +101.38/105 +101.35/105 +Note—NH,LOS: the column density of LOS. NH,Torus : the av- +erage column density of torus. +Γ: the photon index of the +X-ray spectrum. +fscat: the fraction of the scattering com- +ponent. f2−10,scat: the flux density of the scattering compo- +nent. L2−10,scat: the luminosity of the scattering component. +kT: the temperature of diffuse thermal gas. θtorus: the half- +opening angle of torus. θinc: the inclination angle. f2−10,abs: +the absorbed flux density. L2−10,abs: the absorbed luminos- +ity. f2−10,int: the intrinsic flux density. L2−10,int: the intrinsic +luminosity. +10 +5 +10 +4 +10 +3 +keV2 [Photons cm +2 s +1 keV +1] +XMM-Newton +NuSTAR +0.5 +1.0 +2.0 +5.0 +10.0 +20.0 +Energy [keV] +0 +2 +4 +Ratio +Figure 3. Borus model fit to the XMM-Newton PN (black) +and NuSTAR (red) data. The top panel represents the best- +fitting. The bottom panel shows the residuals. +6 × 100 +101 +102 +103 +Flux density [mJy] +Best-fitting model +AGN +PAH +Stellar +Spectrum Data +6.0 +10.0 +16.0 +Wavelength [ m] +0 +1 +2 +Ratio +Figure 4. The best-fitting of the mid-IR spectrum. The +black line indicates the best-fit model. The blue, red, and +orange dash lines represent stellar, PAH, and AGN emission, +respectively. The bottom panel shows the residuals. +In +this +section, +we +use +DeblendIRS2 +(Hern´an- +Caballero et al. 2015) to analyze the mid-IR spectrum of +NGC 449. DeblendIRS is an IDL package that fits the +mid-IR spectra with a linear combination of three spec- +tral templates, i.e., a “pure” AGN template, a “pure” +stellar template, and a “pure” Polycyclic Aromatic Hy- +drocarbon (PAH, which accounts for the interstellar +emission) template. The mid-IR spectra of the sources +can be well decomposed into the contributions of dif- +ferent components using this package. Thus we obtain +2 For +more +details, +refer +to +http://www.denebola.org/ahc/ +deblendIRS/ + +8 +Guo et al. +100 +101 +102 +103 +104 +S (mJy) +Stellar attenuated +Stellar unattenuated +Dust emission +AGN emission +Model spectrum +Model fluxes +Observed fluxes +100 +101 +102 +Observed ( m) +1 +0 +1 +Relative +residual +(Obs-Mod)/Obs +Best model for NGC449 + (z=0.0159, reduced ²=1.4) +Figure 5. The best-fit SED for NGC 449. The black line in- +dicates the best-fit model. The blue, yellow, red, and orange +lines represent unattenuated stellar, attenuated stellar, dust, +and AGN emission, respectively. The lower panel indicates +residual of the best fitting. +some physical properties of the AGNs and their host +galaxies, such as the silicate strength3 (SSi), spectral in- +dex (α), AGN emission at rest-frame 6 µm, and 12 µm. +We use this package to fit the mid-IR spectrum (5 – +16 µm) and provide the best-fitting results. Figure 4 +shows the best-fitting of the mid-IR spectrum of NGC +449. From this fitting, we derive a silicate strength of +−0.71, a luminosity at 6 µm of 7.65 × 1042erg s−1, and +a luminosity at 12 µm of 1.84 × 1043erg s−1. +3.3. Analysis of Multi-band SED +In this section, we analyze the Multi-band SED of +this source using Code Investigating GALaxy Emission +(CIGALE 2020.0, Boquien et al. 2019). CIGALE 2020.0 +is an open python code designed to estimate the physical +properties (i.e., SFR, stellar mass, AGN luminosity) of +galaxies and AGNs. We fit the SED of NGC 449 with +the same modules and parameters as Guo et al. (2020). +For details, please refer to Section 3 of Guo et al. (2020). +Figure 5 presents the best-fit SED for this source. In +addition, we also obtain some physical parameters of +3 The silicate strength is defined as +SSi = ln F(λp) +FC(λp) , +where F(λp)and FC(λp) stand for the maximum flux density of +the silicate line profile near 9.7 µm and the corresponding flux +density of the underlying continuum profile,respectively. SSi is +a negative value, which means that the central radiation of the +AGN is absorbed by the torus.The optical depth of the silicate +absorption τ9.7 = −SSi. +Table 6. Some parameters are obtained/derived by +the SED fitting. +Parameter +Value +(1) +(2) +M∗ (M⊙) +(5.28 ± 0.29) × 109 +SFRSED (M⊙ yr−1) +2.02 ± 0.25 +AGN Luminosity (erg s−1) +(5.35 ± 0.58) × 1043 +χ2/dof +1.40 +LUV( L⊙) +8.22 × 108 +LIR( L⊙) +2.61 × 1010 +SFRUV+IR (M⊙ yr−1) +3.12 +NGC 449 by the SED fitting, such as SFR, stellar mass, +and AGN luminosity. Table 6 lists several parameters +that may be used in this work. The SFR estimate by +the SED fitting is more dependent on the star formation +history model, so we also use the calibration from Bell +et al. (2005) to estimate SFR by the UV and IR lumi- +nosities, scaled to Chabrier (2003) initial mass function: +SFR(M⊙/yr) = 1.09 × 10−9(3.3LUV + LIR) +(1) +where LUV = νLν is an estimation of the integrated 1216 +– 3000˚A rest-frame UV luminosity, and LIR is the 8 – +1000 µm rest-frame IR luminosity. Both LUV and LIR +(listed in Table 6) are in units of L⊙. The SFR of NGC +449 is 3.12 M⊙/yr which is estimated by the UV and IR +luminosities. +4. RESULTS AND DISCUSSION +We have obtained some properties of NGC 449 +through individual band analyses. However, some prop- +erties are not well-constrained. +Combined with the +multiband results, the properties of this source will be +better constrained. In addition, we will also derive some +other properties of this source. +4.1. The properties of the AGN central engine +The AGN at the center of NGC 449 is an optical +type-2 AGN, so we cannot directly observe its cen- +tral emission (e.g., accretion disk, corona). +It is dif- +ficultly to understand the coronal (intrinsic) emission +only by the X-ray spectrum fitting. +For heavily ob- +scured AGNs, the estimation of intrinsic luminosities +(flux densities) is very dependent on the X-ray model +employed. Although Table 5 presents the intrinsic 2–10 +keV luminosities (flux densities) obtained by the Pex- + +Multiwavelength Analysis of NGC 449 +9 +mon and Borus models, they are unreliable. +To de- +rive a reliable intrinsic luminosity, we had better es- +timate it in other ways. +Fortunately, there is a very +close correlation between 2–10 keV and mid-IR 12 µm +luminosities of AGNs (e.g., Asmus et al. 2015). In Sec- +tion 3.2, we have obtained the mid-IR 12 µm luminosity +(νLν(12 µm) = 1.84 × 1043erg s−1) of central AGN. We +use Equation 2 of Asmus et al. (2015) to estimate intrin- +sic 2–10 keV luminosity of (8.54 ± 0.75) × 1042 erg s−1. +Moreover, the fraction of the scattering component is +also not well constrained. +However, the Pexmon and +Borus models have well-constrained the scattering lumi- +nosity, whose average value is (2.18±1.11)×1040 erg s−1. +The relationship between the intrinsic luminosity and +scattering luminosity is fscat = Lscat/LInt. So the scat- +tering fraction of the X-ray emission of central AGN is +about (0.26 ± 0.13)%. +Since the UV-to-optical continuum of this source is +mainly contributed by the host galaxy, we cannot ob- +tain the bolometric emission of the accretion disk by +SED fitting. However, the coronal and torus emissions +can well track the thermal radiation of the accretion +disk. +So they can be used to estimate the bolomet- +ric luminosity of type-2 AGNs. Using the intrinsic 2– +10 keV luminosity, we solve the Equation 21 of Mar- +coni et al. (2004) to estimate the bolometric luminos- +ity Lbol = 1.44 × 1044 erg s−1 (bolometric correction +kbol ≃ 16.85). This bolometric luminosity is consistent +with that provided by Woo & Urry (2002). +The mass accretion rate is a measurement of SMBH +mass growth and is related to bolometric luminosity. +The relationship is written as ˙macc = Lbol/ηc2, where +η is the efficiency that converts the rest-mass energy +of accreted material into radiation. Assuming η = 0.1 +(Frank et al. 2002), we estimate the mass accretion +rate of NGC449, which is 2.54 × 10−2M⊙ yr−1. +The +Eddington ratio is a measurement of the SMBH ac- +cretion efficiency and is defined as λEdd = Lbol/LEdd, +where Lbol is the bolometric luminosity of the AGN, +LEdd is the Eddington luminosity of the AGN (1.3 × +1038 MBH/M⊙ erg s−1). Using the SMBH mass (1.32 × +107 M⊙) given in Woo & Urry (2002), we estimate the +Eddington Luminosity to be LEdd ≃ 1.72×1045 erg s−1. +The Eddington ratio is about λEdd = 8.39 × 10−2. The +Eddington ratio of most CT-AGNs in the local universe +is similar to our result (e.g., Tanimoto et al. 2022). +4.2. The properties of the torus +Previous studies have attempted to constrain the col- +umn density of NGC449 using XMM-Newton data or the +ratio of X-ray to high-ionized emission lines ([O III]5007, +[Ne V]3426) luminosities (Guainazzi et al. 2005; Heck- +man et al. 2005; Gilli et al. 2010). Those results sug- +gested that the center of NGC449 hosted a CT-AGN. +We use the latest X-ray data above 10 keV to con- +struct a broadband X-ray spectrum. +Compared with +the previous studies, we can better constrain the pa- +rameters of the torus in Section 3.1. +In the Pexmon +model, the column density of the torus is obtained as +(1.27 ± 0.26) × 1024 cm−2, indicating that it may be a +CT-AGN. In Section 3.1.4, we re-fit the broadband X- +ray spectrum using the Borus model and obtain a series +of parameters about torus. Among these parameters, +the column density of LOS is (0.97 ± 0.16) × 1024 cm−2, +and the average column density of the torus is about +1.23 × 1024 cm−2, which is close to Compton-thick. In +conclusion, the column density of this source we obtain is +similar to that estimated by previous works (Guainazzi +et al. 2005; Heckman et al. 2005; Gilli et al. 2010), but +it is well-constrained. +In Section 3.2, we decomposed the contribution of +AGN by the mid-IR spectrum fitting, and derived the +silicate strength of the AGN component. Using the sili- +cate strength, we can derive the optical depth of silicate +absorption as τ9.7 = 0.71. +The optical depth of sili- +cate absorption is a good way to track the dust in the +torus, i.e., the V-band extinction is AV = 19 × A9.7 = +19 × 1.086τ9.7 = 14.65 mag (Roche & Aitken 1985). +Therefore, the gas-to-dust ratio of the torus is NH/AV = +8.40×1022 cm−2 mag−1 (NH is the average column den- +sity of torus). This gas-to-dust ratio is about 45 times +that of the Galaxy (NH/AV = 1.87 × 1021 cm−2 mag−1, +Draine 2003). Even this gas-to-dust ratio is higher than +that of nearby CT-AGN in the Circinus galaxy (Tani- +moto et al. 2019), which is the most obscured AGN in +the nearby universe. Such a high gas-to-dust ratio of +this source means that the radiation of the central AGN +may have destroyed dust in the torus. +4.3. The evolutionary stage of the source +Many studies have suggested a co-evolution scheme +between host galaxies and AGNs (e.g., Kormendy & Ho +2013), such as MBH −σ∗ relation (e.g., Ferrarese & Mer- +ritt 2000), AGN feedback (e.g., Bower et al. 2006). Some +studies have suggested that the host galaxies of heavily +obscured AGNs appear to be at the stage of intense +star formation (e.g., Hopkins et al. 2006). These intense +star-forming activities may be associated with AGNs, +i.e. AGNs trigger star formation of host galaxies. +NGC 449 is a late-type galaxy, meaning it is in an early +stage of galaxy evolution. Its center hosts a heavily ob- +scured AGN, which is considered to be at the early stage +of AGN evolution. These suggest that the host galaxy +and central AGN of NGC 449 are at the early stage of + +10 +Guo et al. +co-evolution. In Section 3.3, we derive some parameters +about the host galaxy, including SFR and stellar mass. +The specific star formation rate (sSFR = SFR/M∗) of +this source is approximately 0.59 Gyr−1. This value is +consistent with other star formation galaxies at similar +redshifts (Salim et al. 2007). This result suggests that +the central AGN does not appear to trigger intense star +formation in its host galaxy. +5. SUMMARY +We studied a heavily obscured AGN in NGC 449 +using the multiwavelength data, which included NuS- +TAR, XMM-Newton, Spitzer, and multi-band photo- +metric data. +After, we analyzed its X-ray spectrum +and obtained the column density (NH ≃ 1024cm−2), +the photon index (Γ ≃ 2.4), the luminosity (flux den- +sity) of the scattering component, and the temperature +(kT ≃ 0.82 keV) of diffuse thermal gas. However, some +parameters could not be well constrained, i.e., the frac- +tion of the scattering component, the intrinsic flux den- +sity, and luminosity. We fitted its mid-IR spectrum and +decomposed the AGN contribution to derive some AGN +parameters, and used its SED to obtain some parame- +ters. Combining the information from the mid-IR spec- +trum and SED, we first derived the intrinsic X-ray lumi- +nosity (≃ 8.54×1042 erg s−1) and the scattering fraction +of primary X-ray emission (≃ 0.26%). In addition, we +also derived the following results: +1. The bolometric luminosity of the AGN is about +1.44 × 1044 erg s−1. The mass accretion rate of +central AGN is about 2.54 × 10−2M⊙ yr−1, and +the Eddington ratio is 8.39 × 10−2. +2. The torus of this AGN has a high gas-to-dust ratio +(NH/AV = 8.40 × 1022 cm−2 mag−1), which is +about 45 times that of the Galaxy. Such a high +gas-to-dust ratio means that the radiation of the +central AGN may have destroyed dust in the torus. +3. The host galaxy and the central AGN are both in +the early stage of co-evolution. +We sincerely thank the anonymous referee for useful sug- +gestions. +We acknowledge financial support from the +National Key Research and Development Program of +China grant (No. 2017YFA0402703) and National Nat- +ural Science Foundation of China grant (NSFC; grant +Nos.11733002). +We acknowledge the science research +grants from the China Manned Space Project with +NO. CMS-CSST-2021-A07. Yongyun Chen is grateful- +for financial support from the National Natural Science +Foundation of China (No. +12203028). +This work is +supported by the youth project of Yunnan Provincial +Science and Technology Department (202101AU070146, +2103010006). Yongyun Chen is grateful for funding for +the training Program for talents in Xingdian, Yunnan +Province. +Nan Ding thanks for the financial support +from the National Natural Science Foundation of China +(No. 12103022) and the Special Basic Cooperative Re- +search Programs of Yunnan Provincial Undergraduate +Universities’Association (No. 202101BA070001-043). +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +13 +14 +15 +16 +17 +18 +19 +Software: +numpy, pandas, matplotlib (Hunter +2007), astropy(Astropy Collaboration et al. 2013) +REFERENCES +Abazajian, K. N., Adelman-McCarthy, J. K., Ag¨ueros, +M. 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K., et al. +2010, AJ, 140, 1868, doi: 10.1088/0004-6256/140/6/1868 +Xu, J., Sun, M.-Y., Xue, Y.-Q., Li, J.-Y., & He, Z.-C. 2020, +Research in Astronomy and Astrophysics, 20, 147, +doi: 10.1088/1674-4527/20/9/147 + diff --git a/YNE5T4oBgHgl3EQfCw5O/content/tmp_files/load_file.txt b/YNE5T4oBgHgl3EQfCw5O/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..457e52ac981961350be657c73c00ce8a4601a315 --- /dev/null +++ b/YNE5T4oBgHgl3EQfCw5O/content/tmp_files/load_file.txt @@ -0,0 +1,1270 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf,len=1269 +page_content='Draft version January 16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' 2023 Typeset using LATEX twocolumn style in AASTeX631 Multiwavelength Analysis of a Nearby Heavily Obscured AGN in NGC 449 Xiaotong Guo (郭晓通) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='1 Qiusheng Gu (顾秋生) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' 3 Jun Xu (徐骏) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='1 Guanwen Fang (方官文) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='1 Xue Ge (葛雪) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='4 Yongyun Chen (陈永云) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='5 Xiaoling Yu (俞效龄) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='5 and Nan Ding (丁楠) 6 1Institute of Astronomy and Astrophysics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Anqing Normal University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Anqing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Anhui 246133,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' China 2School of Astronomy and Space Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Nanjing University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Nanjing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Jiangsu 210093,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' China 3Key Laboratory of Modern Astronomy and Astrophysics (Nanjing University),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Ministry of Education,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Nanjing 210093,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' China 4School of Physics and Electronic Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Jiangsu Second Normal University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Nanjing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Jiangsu 211200,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' China 5College of Physics and Electronic Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Qujing Normal University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Qujing 655011,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' China 6School of Physical Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Kunming University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Kunming 650214,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' China ABSTRACT We presented the multiwavelength analysis of a heavily obscured active galactic nucleus (AGN) in NGC 449.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' We first constructed a broadband X-ray spectrum using the latest NuSTAR and XMM- Newton data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Its column density (≃ 1024cm−2) and photon index (Γ ≃ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='4) were reliably obtained by analyzing the broadband X-ray spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' However, the scattering fraction and the intrinsic X-ray luminosity could not be well constrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Combined with the information obtained from the mid- infrared (mid-IR) spectrum and spectral energy distribution (SED) fitting, we derived its intrinsic X-ray luminosity (≃ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='54 × 1042 erg s−1) and scattering fraction (fscat ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='26%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' In addition, we also derived the following results: (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' The mass accretion rate of central AGN is about 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='54×10−2M⊙ yr−1, and the Eddington ratio is 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='39 × 10−2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' The torus of this AGN has a high gas-to-dust ratio (NH/AV = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='40 × 1022 cm−2 mag−1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' The host galaxy and the central AGN are both in the early stage of co-evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Keywords: Active galactic nuclei(16) — Seyfert galaxies(1447) — X-ray active galactic nuclei(2035) — AGN host galaxies(2017) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' INTRODUCTION It is well known that active galactic nuclei (AGNs) are powered by the accretion of surrounding matter by su- permassive black holes (SMBHs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' The emission of AGNs covers almost the whole electromagnetic band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' The AGNs’ radiations at different wavelengths arise from their diverse structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' For example, the significant emission of the accretion disk is in the ultraviolet (UV) to the optical band (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=', Richstone & Schmidt 1980), while the emission of the torus is in the mid-infrared (mid-IR) band (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=', Antonucci 1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' The diverse in- formation can be derived from the emission of AGNs at different wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' To fully understand an AGN, it is often necessary to study with the multi-bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Corresponding author: Qiusheng Gu qsgu@nju.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='cn The classification of AGNs usually utilizes the obser- vation features of different wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' For instance, based on the characteristics of the optical emission lines, AGNs can be classified into type 1 and type 2 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=', Khachikian & Weedman 1974).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' The AGN unified model proposes that different AGN types are caused by different viewing angles with an obscuring torus (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=', Antonucci 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Urry & Padovani 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Netzer 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Since the X-ray emission has a strong pierce, the X-ray emission of AGNs is usually subject to lit- tle absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Based on the column density (NH) of the X-ray absorption, AGNs can be classified into four categories: unobscured (NH < 1022 cm−2), obscured (1022 cm−2 < NH < 1023 cm−2), heavily obscured (1023 cm−2 < NH < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='5 × 1024 cm−2), and Compton- thick (NH ⩾ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='5 × 1024 cm−2, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=', Comastri 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Many studies have also regarded the source with a col- umn density of ∼ 1024 cm−2 as Compton-thick AGN (CT-AGN, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=', Risaliti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Kammoun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='05398v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='GA] 13 Jan 2023 ID2 Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Some studies have pointed out that most AGNs were heavily obscured (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=', Hickox & Alexander 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' The heavily obscured AGNs are believed to be in an early phase of the evolutionary scenario of AGNs, which is a rapid growth state of central SMBH (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=', Goulding et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' The host galaxies of heavily obscured AGNs are also in an early phase of the evolutionary scenario of galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' For instance, Kocevski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' (2015) pointed out that heavily obscured AGNs were twice as likely to be hosted by late-type galaxies relative to unobscured AGNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' As a class of the most luminous and persistent X-ray sources in the universe, AGNs contribute the vast majority of Cosmic X-ray Background (CXB, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=', Ueda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' 2014) in the 1 to ∼200 – 300 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' The heavily ob- scured AGNs and CT-AGNs are significant contributors to the hump of the CXB emission at 30 keV (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=', Mar- shall et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' 1980;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Ajello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' The single-band analysis for AGNs can only derive a part of their properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' For example, we can derive the column density in the line-of-sight (LOS) direction by fitting X-ray spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Using the spectral energy distri- butions (SEDs) from the UV to the far-IR, we can de- rive the star formation rates (SFRs) of the host galaxies, stellar masses of the host galaxies, and AGN luminosi- ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' To fully understand heavily obscured AGNs, their host galaxies, and their co-evolution, we first derive their properties through multi-band analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' The heavily obscured AGNs usually present a complex spectral shape in their X-ray waveband.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Theoretically, the soft X-ray emission (E ≲ 2 keV) of heavily obscured AGNs will be heavily (entirely) absorbed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Even in many CT AGNs, the soft X-ray is not entirely absorbed, be- cause the emission from the AGNs’ scattered and the host galaxy can be observed (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=', Turner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' LaMassa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' As the energy increases, the ab- sorption of X-ray emission (E>10 keV) is gradually re- duced or cut-off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' The hard X-ray of heavily obscured AGN is dominated by the reflected component, the so- called Compton reflection continuum characterized by a broad bump peaking around 20–30 keV (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Comastri 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' One of the most prominent features in the X-ray spectrum of heavily obscured AGN is the neutral iron Kα emission line at 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='4 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' However, the X-ray spec- tra of some heavily obscured AGNs show a weak (or absent) iron Kα emission line (Boorman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' 2018), especially the heavily obscured AGNs beyond the local universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' To accurately determine the column density of the gas and recover the intrinsic X-ray flux of AGNs, we require proper modeling of the physical processes in the AGNs that self-consistently account for the transmitted emission, Compton-scattered reflected component and the fluorescent line flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Such modeling may also con- strain the geometry of the torus and view angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Over the past several years, a series of X-ray spectral mod- els (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=', Murphy & Yaqoob 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Balokovi´c et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' 2018) are created with assumed different geometries (spheri- cal, toroidal, or clumpy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Many recent studies use these models to constrain the parameters of AGNs (Gandhi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Kammoun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' LaMassa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Toba et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' From the UV to the optical band, the continua of the heavily obscured AGNs are masked by the emis- sion of the host galaxies, so we can only observe their high-ionized emission lines and obtain little information about them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' It is difficult to constrain the gas column density along LOS for the AGNs with low-quality X-ray spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' However, we can estimate the gas column den- sity using the luminosity ratio of X-ray to high-ionized emission lines (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=', Maiolino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Cappi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Gilli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Lanzuisi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' The mid-IR emission of the AGNs is usually less suppressed due to the low optical depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' The contribution of AGNs can be easily decomposed in the mid-IR spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' In addition, for heavily obscured AGNs, we can also observe a strong silicon absorption line at 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='7 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' The dust in the torus can be derived using silicon absorption strength (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=', Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' By the SED fitting, we can accurately obtain the contribution of the AGNs, the AGN lumi- nosities, and some parameters about their host galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' So the multi-band data can provide us with more com- prehensive information on heavily obscured AGNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' NGC 449 (also known as Mrk 1) is a nearby (z=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='01595, D∼67 Mpc) Seyfert 2 galaxy, and its cen- ter hosts a heavily obscured AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' It was observed by the XMM-Newton X-ray telescope (Jansen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' 2001) in 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Guainazzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' (2005) fitted the X-ray spectrum and constrained the column density (NH > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='1 × 1024 cm−2) of this AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Moreover, some stud- ies also derived its column density by using the ratio of X-ray to high-ionized emission lines ([O III]5007, [Ne V]3426) luminosity (Heckman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Gilli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' These results indicated that the X-ray is severely absorbed and that even the AGN may be a CT-AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' However, these results did not constrain its column den- sity well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' To better constrain its column density and other properties, the spectrum above 10 keV is neces- sary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Fortunately, this source was observed by the Nu- clear Spectroscopic Telescope Array (NuSTAR, Harri- son et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' 2013) in 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' The NuSTAR data has still not been analyzed (or presented in previous work).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' More- over, there are abundant data about NGC 449 in the archives, which can also help us to obtain more proper- ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' So the motivation of our work is to constrain the Multiwavelength Analysis of NGC 449 3 properties of this heavily obscured AGN by multiwave- length analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' The structure of the paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Section 2 describes the multiwavelength data and processing for NGC 449.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' In Section 3, we analyzed the multiwave- length data of this source in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Then, we dis- cuss the implications of our results in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Fi- nally, we present a summary of this work in Sec- tion 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' We adopt a concordance flat Λ-cosmology with H0 = 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='4 km s−1 Mpc−1, Ωm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='315, and ΩΛ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='685 (Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' MULTIWAVELENGTH DATA 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' X-Ray Observations and Data Reduction A log of the XMM-Newton and NuSTAR observations analyzed herein is presented in Table 1, and the individ- ual data sets are described in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' XMM-Newton Observations NGC 449 was observed by the XMM-Newton X-ray telescope (Jansen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' 2001) for about 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='9 ks of expo- sure on January 9, 2004 (PI: Matteo Guainazzi, ObsID: 0200430301).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' The data is reduced in a standard man- ner using the XMM-Newton Science Analysis System (SAS, Gabriel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' 2004) v17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='0 and Current Cali- bration Files (CCF) of June 22, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' The spectra are extracted from a circular region (30 arcsec radius for PN and 25 arcsec radius for MOS) around the source, and the background spectra are taken from a nearby source- free circular region (80 arcsec radius for PN and 100 arcsec radius for MOS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' The spectra are binned to have minimum counts of 10 per energy bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' NuSTAR Observations NGC 449 was also observed by the NuSTAR (Harrison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' 2013) for about 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='7 ks of exposure on December 8, 2017 (ObsID: 60360002002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' The data is processed us- ing the NuSTAR data analysis software nustardas v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='5 available in heasoft v6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='2 and CALDB released on Au- gust 13, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' The nupipeline script is used to produce calibrated and clean event files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' We extract source spec- tra using the nuproducts task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' The spectra are extracted from a 45 arcsec radius aperture centered on the source, while the background is extracted from a circular region with a 100 arcsec radius from a source-free region on the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' To improve the spectral quality of NuSTAR, we try to combine the two spectra of FPMA and FPMB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' The combined spectrum is also binned to have minimum counts of 10 per energy bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Mid-IR spectral Observations and Data Reduction We search the archives for its mid-IR data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' There is good mid-IR spectral data, covering wavelengths from 5 to 37 µm, observed by Spitzer’s IrsStare on February 5, 2009 (PI: Levenson, Nancy, ObsID: 25408000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' The mid-IR spectrum was processed using the Spitzer data analysis software irs merge v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='1 with the pipeline script (version: S18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='0) on December 18, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' For more de- tailed processing of the mid-IR spectrum of this source, please refer to Section 2 of Lebouteiller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Multiwavelength photometric data To construct the SED for NGC 449, we search for available multi-band photometric data from different surveys or the archives of telescopes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=', the Sloan Dig- ital Sky Survey (SDSS) Data Release 7 (Abazajian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' 2009), the Two Micron All-Sky Survey (2MASS, Skrut- skie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' 2006), the Wide-field Infrared Survey Explorer (WISE, Wright et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' 2010), the AKARI and the In- frared Astronomical Satellite (IRAS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' In total, we collect the photometric data for 17 filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Among them, only the filter SDSS-u is in the UV band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' There are three filters of the optical band, including SDSS-g, SDSS-r, and SDSS-i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' The other filters are in the IR band, in- cluding four near-IR filters (SDSS-z, 2MASS-J, 2MASS- H, 2MASS-K), seven mid-IR filters (W1, W2, W3, W4, IRAS-PSC 25, AKARI-PSC 09, AKARI-PSC 18) and two far-IR filters (IRAS-PSC 60, IRAS-PSC 100).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' The details for photometric data and filters are listed in Ta- ble 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' ANALYSIS OF MULTIWAVELENGTH DATA 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' X-Ray Spectral Analysis In this section, we analyze the X-ray spectra of NGC 449 in Xspec v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='0 (Arnaud 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' First, we use phenomenological modeling to determine whether there is variation in the spectra for the same energy range of XMM-Newton and NuSTAR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' We then analyzed which components contributed to the X-ray spectra of this source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Based on the analysis of the X-ray components, we finally use the self-consistent and physical models to fit the X-ray spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' All errors represent the 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='0% confidence intervals unless otherwise stated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Testing for Variability Among Observations To test whether there is significant variation in the spectra of two observations within the same energy range (3–10 keV), we need to compare their spectral shapes and fluxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' We can obtain them by fitting the X-ray spectra of this source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' However, the data for this source is poor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' To obtain more reliable properties, 4 Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Observation log.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' ObsID Observation date Mission Instrument(s) Net Exposure Net Counts [ks] (1) (2) (3) (4) (5) (6) MOS1 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='4 249.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='90 0200430301 2004-01-09 Newton MOS2 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='5 222.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='31 PN 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='6 735.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='63 60360002002 2017-12-23 NuSTAR FPMA 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='7 130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='66 FPMB 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='6 126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='67 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Photometry of NGC 449.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Band Wavelength Flux References [µm] [mJy] (1) (2) (3) (4) SDSS-u 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='3547 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='5473 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='0109 (a) SDSS-g 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='4767 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='1128 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='0104 SDSS-r 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='6226 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='5429 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='0161 SDSS-i 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='7615 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='024 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='0196 SDSS-z 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='9123 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='909 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='0368 2MASS-J 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='234 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='704 (b) 2MASS-H 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='661 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='4 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='08 2MASS-K 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='157 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='6 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='11 WISE-W1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='4 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='035 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='2084 (c) WISE-W2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='6 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='144 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='3220 WISE-W3 12 116.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='39 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='8618 WISE-W4 22 551.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='74 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='9548 IRAS-PSC 25 25 801.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='20 ± 128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='19 (d) IRAS-PSC 60 60 2302.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='0 ± 299.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='26 IRAS-PSC 100 100 2851.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='0 ± 342.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='12 AKARI-PSC 09 9 101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='14 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='4838 (e) AKARI-PSC 18 18 531.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='34 ± 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='134 Note—(a) Abazajian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' (2009), (b) Skrutskie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' (2006), (c) Wright et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' (2010), (d) Moshir & et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' (1990), (e) Terashima et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' we use a phenomenological model to fit the spectra of XMM-Newton1 and NuSTAR within the same energy range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' The phenomenological model is written (in the 1 The spectra of MOS1, MOS2 and PN are simultaneously fitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Variability among observations within the same energy range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Mission log(f3−10 keV) Γ3−10 keV χ2/dof [erg · s−1 · cm−2] (1) (2) (3) (4) Newton −12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='85 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='13 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='86 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='86 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='02/5 NuSTAR −13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='06 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='58 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='70 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='59 31/17 Xspec terminology) as follows: model = phabs[1] ∗ cflux[2] ∗ powerlaw[3], where phabs[1] accounts for the Galactic absorption which is fixed at a column density of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='16 × 1020 cm−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Table 3 lists some properties of these two spectra within the same energy range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Their fluxes are the same within the error range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' The spectral shape of XMM- Newton’s spectrum within the same energy range indi- cates that the X-ray emission of NGC 449 is seriously absorbed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Similarly, the NuSTAR observation of this source also presents a flat spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' The data in the same energy range are so poor that their photon indices cannot be well constrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' In short, these two observa- tions have similar fluxes within the same energy range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Therefore, there is no significant variation in the X-ray emission for NGC 449 among these two observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' For subsequent X-ray spectrum analysis, we consti- tute a broadband X-ray spectrum using the X-ray spec- tra obtained with XMM-Newton and NuSTAR, which covers the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='35–30 keV band (PN for XMM-Newton : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='35–10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='0 keV, NuSTAR : 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='0–30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='0 keV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Analysis of the basic components Multiwavelength Analysis of NGC 449 5 In this section, our goal is to analyze which compo- nents contribute to the X-ray spectra of this source, pro- viding support for subsequent model selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' First, We use a powerlaw model (phabs ∗ powerlaw) with a photon index of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='17 to fit the broadband spec- trum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Of course, the goodness of fit is poor (see the panel b of Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' There is significant excess in the spectrum above 10 keV, which is a flat spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' So the broadband spectrum is again fitted by a broken power- law model (phabs ∗ bknpower).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Figure 1(c) shows the residuals of the broken powerlaw model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' The best-fitting break energy is 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='93 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' The low energy band is a steep spectrum (Γ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='36), and the high energy band is a flat spectrum (Γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='51).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Although its goodness of fit is significantly improved, there is still excess near 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='5 keV and 1 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Therefore, we also need to add two more models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Among them, a Gaussian model is used to fit the iron emission line near 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='5 keV, and the equivalent width (EW) of the iron emission line is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='49 keV which is consistent with previous work (EW<2 keV, Guainazzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' The other model (diffuse thermal emission) is used to match the excess near 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='0 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Finally, we obtain the best-fitting of the spectrum, as shown in Fig- ure 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' The fitted parameters are listed in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' We find a flat spectrum above 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='71 keV by fitting the broadband spectrum, indicating that the emission of this source is seriously absorbed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Therefore, the re- processed emission is dominant in hard X-rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' The- oretically, there should be a flat spectrum at energies below 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='71 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' In fact, there is a steep spectrum below 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='71 keV with a photon index of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' The power-law emission below 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='71 keV is most likely contributed by the scattered component of primary X-ray emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Be- sides, the diffuse thermal emission and the iron emission line are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Pexmon model According to the analysis in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='2, we fit the spectrum using the neutral reflection model Pexmon (Nandra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' The detailed model can be written as follows: modelP exmon =phabs[1] ∗ (zphabs[2] ∗ powerlaw[3]+ constant[4] ∗ powerlaw[5] + mekal[6]+ pexmon[7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' In this model, the phabs[1] component stands for the Galactic absorption is fixed at a column density of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='16 × 1020cm−2, powerlaw[3] represents the primary X-ray emission of the AGN, which is absorbed by ob- scured material (zphabs[2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' The primary X-ray emis- sion is scattered by ionized gas in the polar regions, while the scattering component is usually little or not absorbed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' The soft scattering component into our LOS is represented by constant[4] ∗ powerlaw[5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Thus the parameters of powerlaw[5] are tied to the parameters of powerlaw[3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' The diffuse thermal radiation is denoted by mekal[6], similar to Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' We use pexmon[7] to represent the reprocessed emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' The Pexmon model assumes a slab obscurer/reflector with an infinite optical depth, as may be expected in a standard geometrically thin accretion disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' The incident source in X-rays is the primary emission (powerlaw) from a hot electron corona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Therefore, the photon index and normalization of pexmon[7] are tied to the identi- cal parameters of powerlaw[3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' The cutoff energy of pexmon[7] is fixed as 500 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' We are not able to con- strain the Fe abundance (tied to the elemental abun- dance), so we fix it to its best-fit value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' The reflection fraction was fixed to -1 in the Pexmon fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' This model fits the data well (χ2/dof=101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='38/105).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' The best-fitting and residuals are shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' The Pexmon model implies that the X-ray emission of central AGN is absorbed by Compton-thick material with NH = (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='27 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='26) × 1024 cm−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' The intrinsic 2– 10 keV luminosity is L2−10 = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='64±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='16)×1042 erg s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' The photon index is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='44, suggesting that the spectrum of central AGN is very soft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' We note that the scatter- ing component is found to be fscat = (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='22 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='57)% of the primary emission, and that the 2–10 keV luminosity of scattering component is (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='05 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='09) × 1040 erg s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' The diffuse thermal gas model (mekal[6]) adequately describes the soft X-rays with a temperature kT = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='82 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' More best-fit parameters for the model are listed in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Borus model Similarly, according to the analysis in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='2, we refit the spectrum of the reprocessed emission using the Borus model (Balokovi´c et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Borus model is defined with the following command sequence in Xspec: modelBorus = phabs[1] ∗ (zphabs[2] ∗ cabs[3]∗ powerlaw[4] + constant[5] ∗ powerlaw[6] + mekal[7] + Borus02[8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' The phabs[1], constant[5] ∗ powerlaw[6], and mekal[7] components are equivalent to the ones in the Pexmon fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' The zphabs[2]∗cabs[3] represents LOS absorption at the redshift of the X-ray source (generally independent from the average column density of the torus), including Compton scattering losses out of the line of sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' The photon index and normalization of Borus02[8] are also 6 Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Analysis of the basic components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Parameter Power-law Broken power-law Broken+Gauss+Diffuse (1) (2) (3) (4) (5) powerlaw Γ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='17 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='06 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Γ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='36 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='08 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='38 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='12 bknpower EBreak (keV) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='93 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='50 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='71 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='60 Γ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='51 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='36 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='19 zgauss lineE(keV) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='56 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='09 σ(keV) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='16 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='15 EW(keV) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='49 mekal kT(keV) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='82 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='06 χ2/dof 224.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='61/110 150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='52/108 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='52/103 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='0 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='0 Energy [keV] 10 6 10 5 10 4 Photons cm 2 s 1 keV 1 (a) XMM-Newton NuSTAR 10 20 Ratio (b) Power-law 2/dof=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='04 2 4 Ratio (c) Broken power-law 2/dof=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='0 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='0 Energy [keV] 0 2 4 Ratio (d) Broken+Gauss+Diffuse 2/dof=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='98 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' XMM-Newton PN (black) and NuSTAR (red) spectra of NGC 449.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' The black line in panel (a) represents the best-fit model (broken + gauss + diffuse).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Panel (b) shows the ratio (data/model) by fitting the X-ray spectra using a simple power-law model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Panels (c) and (d) show the residuals obtained by fitting broken power-law and broken + gauss + diffuse models, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' tied to the powerlaw[3], and the cutoff energy is fixed as 500 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Some parameters can not be constrained, so we fix them to their best-fit values, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=', half-opening angle, inclination angle, and iron abundance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' This model fits the data well (χ2/dof=101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='35/105).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' The best-fitting and residuals are shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' The column density of LOS is NH = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='97 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='16) × 1024 cm−2 , and the average column density of the torus is about NH,Torus = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='23 × 1024 cm−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' The intrinsic 2– 10 keV luminosity is L2−10 = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='93±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='24)×1042 erg s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' More best-fit parameters for the model are listed in Ta- ble 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Through the above analysis and the broadband X-ray spectrum fitting of two physical models (Pexmon and Borus), we have obtained some parameters about the AGN in NGC 449.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Most of the parameters obtained by the two models are the identical within the error range, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=', column density of LOS, the photon index, the luminosity of scattering component, the temperature of diffuse thermal gas, and the absorbed luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' These suggest that these parameters can be well constrained in the X-ray band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' However, the fraction of the scattering component and the intrinsic luminosity can not be well constrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Therefore, we need information from other bands to constrain these parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Mid-IR spectrum Analysis Multiwavelength Analysis of NGC 449 7 10 5 10 4 10 3 keV2 [Photons cm 2 s 1 keV 1] XMM-Newton NuSTAR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='0 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='0 Energy [keV] 0 2 4 Ratio Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Pexmon model fit to the XMM-Newton PN (black) and NuSTAR (red) data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' The top panel represents the best-fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' The bottom panel shows the residuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Best-fit parameters for Pexmon and Borus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Parameter Pexmon Borus (1) (2) (3) NH,LOS (1024 cm−2) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='27 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='97 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='16 log NH,Torus (cm−2) · · 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='09 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='13 Γ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='44 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='13 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='38 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='10 fscat(%) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='22 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='52 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='50 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='04 f2−10,scat(10−14 erg s−1 cm−2) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='81 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='04 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='28 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='10 L2−10,scat(1040 erg s−1) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='05 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='09 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='30 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='13 kT(keV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='82 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='82 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='06 θtorus(◦) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='0 θinc(◦) 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='23 ± 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='44 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='4 f2−10,abs(10−12 erg s−1 cm−2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='104+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='002 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='034 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='104+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='005 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='018 L2−10,abs(1042 erg s−1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='063+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='002 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='013 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='063+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='005 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='013 f2−10,int(10−12 erg s−1 cm−2) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='18 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='30 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='72 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='44 L2−10,int(1042 erg s−1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='64 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='93 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='24 χ2/dof 101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='38/105 101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='35/105 Note—NH,LOS: the column density of LOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' NH,Torus : the av- erage column density of torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Γ: the photon index of the X-ray spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' fscat: the fraction of the scattering com- ponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' f2−10,scat: the flux density of the scattering compo- nent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' L2−10,scat: the luminosity of the scattering component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' kT: the temperature of diffuse thermal gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' θtorus: the half- opening angle of torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' θinc: the inclination angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' f2−10,abs: the absorbed flux density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' L2−10,abs: the absorbed luminos- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' f2−10,int: the intrinsic flux density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' L2−10,int: the intrinsic luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' 10 5 10 4 10 3 keV2 [Photons cm 2 s 1 keV 1] XMM-Newton NuSTAR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='0 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='0 Energy [keV] 0 2 4 Ratio Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Borus model fit to the XMM-Newton PN (black) and NuSTAR (red) data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' The top panel represents the best- fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' The bottom panel shows the residuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' 6 × 100 101 102 103 Flux density [mJy] Best-fitting model AGN PAH Stellar Spectrum Data 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='0 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='0 Wavelength [ m] 0 1 2 Ratio Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' The best-fitting of the mid-IR spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' The black line indicates the best-fit model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' The blue, red, and orange dash lines represent stellar, PAH, and AGN emission, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' The bottom panel shows the residuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' In this section, we use DeblendIRS2 (Hern´an- Caballero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' 2015) to analyze the mid-IR spectrum of NGC 449.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' DeblendIRS is an IDL package that fits the mid-IR spectra with a linear combination of three spec- tral templates, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=', a “pure” AGN template, a “pure” stellar template, and a “pure” Polycyclic Aromatic Hy- drocarbon (PAH, which accounts for the interstellar emission) template.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' The mid-IR spectra of the sources can be well decomposed into the contributions of dif- ferent components using this package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Thus we obtain 2 For more details, refer to http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='denebola.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='org/ahc/ deblendIRS/ 8 Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' 100 101 102 103 104 S (mJy) Stellar attenuated Stellar unattenuated Dust emission AGN emission Model spectrum Model fluxes Observed fluxes 100 101 102 Observed ( m) 1 0 1 Relative residual (Obs-Mod)/Obs Best model for NGC449 (z=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='0159, reduced ²=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='4) Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' The best-fit SED for NGC 449.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' The black line in- dicates the best-fit model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' The blue, yellow, red, and orange lines represent unattenuated stellar, attenuated stellar, dust, and AGN emission, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' The lower panel indicates residual of the best fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' some physical properties of the AGNs and their host galaxies, such as the silicate strength3 (SSi), spectral in- dex (α), AGN emission at rest-frame 6 µm, and 12 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' We use this package to fit the mid-IR spectrum (5 – 16 µm) and provide the best-fitting results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Figure 4 shows the best-fitting of the mid-IR spectrum of NGC 449.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' From this fitting, we derive a silicate strength of −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='71, a luminosity at 6 µm of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='65 × 1042erg s−1, and a luminosity at 12 µm of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='84 × 1043erg s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Analysis of Multi-band SED In this section, we analyze the Multi-band SED of this source using Code Investigating GALaxy Emission (CIGALE 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='0, Boquien et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' CIGALE 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='0 is an open python code designed to estimate the physical properties (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=', SFR, stellar mass, AGN luminosity) of galaxies and AGNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' We fit the SED of NGC 449 with the same modules and parameters as Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' For details, please refer to Section 3 of Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Figure 5 presents the best-fit SED for this source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' In addition, we also obtain some physical parameters of 3 The silicate strength is defined as SSi = ln F(λp) FC(λp) , where F(λp)and FC(λp) stand for the maximum flux density of the silicate line profile near 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='7 µm and the corresponding flux density of the underlying continuum profile,respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' SSi is a negative value, which means that the central radiation of the AGN is absorbed by the torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='The optical depth of the silicate absorption τ9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='7 = −SSi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Some parameters are obtained/derived by the SED fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Parameter Value (1) (2) M∗ (M⊙) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='28 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='29) × 109 SFRSED (M⊙ yr−1) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='02 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='25 AGN Luminosity (erg s−1) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='35 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='58) × 1043 χ2/dof 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='40 LUV( L⊙) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='22 × 108 LIR( L⊙) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='61 × 1010 SFRUV+IR (M⊙ yr−1) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='12 NGC 449 by the SED fitting, such as SFR, stellar mass, and AGN luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Table 6 lists several parameters that may be used in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' The SFR estimate by the SED fitting is more dependent on the star formation history model, so we also use the calibration from Bell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' (2005) to estimate SFR by the UV and IR lumi- nosities, scaled to Chabrier (2003) initial mass function: SFR(M⊙/yr) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='09 × 10−9(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='3LUV + LIR) (1) where LUV = νLν is an estimation of the integrated 1216 – 3000˚A rest-frame UV luminosity, and LIR is the 8 – 1000 µm rest-frame IR luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Both LUV and LIR (listed in Table 6) are in units of L⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' The SFR of NGC 449 is 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='12 M⊙/yr which is estimated by the UV and IR luminosities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' RESULTS AND DISCUSSION We have obtained some properties of NGC 449 through individual band analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' However, some prop- erties are not well-constrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Combined with the multiband results, the properties of this source will be better constrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' In addition, we will also derive some other properties of this source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' The properties of the AGN central engine The AGN at the center of NGC 449 is an optical type-2 AGN, so we cannot directly observe its cen- tral emission (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=', accretion disk, corona).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' It is dif- ficultly to understand the coronal (intrinsic) emission only by the X-ray spectrum fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' For heavily ob- scured AGNs, the estimation of intrinsic luminosities (flux densities) is very dependent on the X-ray model employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Although Table 5 presents the intrinsic 2–10 keV luminosities (flux densities) obtained by the Pex- Multiwavelength Analysis of NGC 449 9 mon and Borus models, they are unreliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' To de- rive a reliable intrinsic luminosity, we had better es- timate it in other ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Fortunately, there is a very close correlation between 2–10 keV and mid-IR 12 µm luminosities of AGNs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=', Asmus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' In Sec- tion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='2, we have obtained the mid-IR 12 µm luminosity (νLν(12 µm) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='84 × 1043erg s−1) of central AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' We use Equation 2 of Asmus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' (2015) to estimate intrin- sic 2–10 keV luminosity of (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='54 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='75) × 1042 erg s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Moreover, the fraction of the scattering component is also not well constrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' However, the Pexmon and Borus models have well-constrained the scattering lumi- nosity, whose average value is (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='18±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='11)×1040 erg s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' The relationship between the intrinsic luminosity and scattering luminosity is fscat = Lscat/LInt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' So the scat- tering fraction of the X-ray emission of central AGN is about (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='26 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='13)%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Since the UV-to-optical continuum of this source is mainly contributed by the host galaxy, we cannot ob- tain the bolometric emission of the accretion disk by SED fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' However, the coronal and torus emissions can well track the thermal radiation of the accretion disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' So they can be used to estimate the bolomet- ric luminosity of type-2 AGNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Using the intrinsic 2– 10 keV luminosity, we solve the Equation 21 of Mar- coni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' (2004) to estimate the bolometric luminos- ity Lbol = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='44 × 1044 erg s−1 (bolometric correction kbol ≃ 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='85).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' This bolometric luminosity is consistent with that provided by Woo & Urry (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' The mass accretion rate is a measurement of SMBH mass growth and is related to bolometric luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' The relationship is written as ˙macc = Lbol/ηc2, where η is the efficiency that converts the rest-mass energy of accreted material into radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Assuming η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='1 (Frank et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' 2002), we estimate the mass accretion rate of NGC449, which is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='54 × 10−2M⊙ yr−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' The Eddington ratio is a measurement of the SMBH ac- cretion efficiency and is defined as λEdd = Lbol/LEdd, where Lbol is the bolometric luminosity of the AGN, LEdd is the Eddington luminosity of the AGN (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='3 × 1038 MBH/M⊙ erg s−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Using the SMBH mass (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='32 × 107 M⊙) given in Woo & Urry (2002), we estimate the Eddington Luminosity to be LEdd ≃ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='72×1045 erg s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' The Eddington ratio is about λEdd = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='39 × 10−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' The Eddington ratio of most CT-AGNs in the local universe is similar to our result (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=', Tanimoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' The properties of the torus Previous studies have attempted to constrain the col- umn density of NGC449 using XMM-Newton data or the ratio of X-ray to high-ionized emission lines ([O III]5007, [Ne V]3426) luminosities (Guainazzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Heck- man et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Gilli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Those results sug- gested that the center of NGC449 hosted a CT-AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' We use the latest X-ray data above 10 keV to con- struct a broadband X-ray spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Compared with the previous studies, we can better constrain the pa- rameters of the torus in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' In the Pexmon model, the column density of the torus is obtained as (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='27 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='26) × 1024 cm−2, indicating that it may be a CT-AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' In Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='4, we re-fit the broadband X- ray spectrum using the Borus model and obtain a series of parameters about torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Among these parameters, the column density of LOS is (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='97 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='16) × 1024 cm−2, and the average column density of the torus is about 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='23 × 1024 cm−2, which is close to Compton-thick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' In conclusion, the column density of this source we obtain is similar to that estimated by previous works (Guainazzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Heckman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Gilli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' 2010), but it is well-constrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' In Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='2, we decomposed the contribution of AGN by the mid-IR spectrum fitting, and derived the silicate strength of the AGN component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Using the sili- cate strength, we can derive the optical depth of silicate absorption as τ9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='7 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' The optical depth of sili- cate absorption is a good way to track the dust in the torus, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=', the V-band extinction is AV = 19 × A9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='7 = 19 × 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='086τ9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='7 = 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='65 mag (Roche & Aitken 1985).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Therefore, the gas-to-dust ratio of the torus is NH/AV = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='40×1022 cm−2 mag−1 (NH is the average column den- sity of torus).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' This gas-to-dust ratio is about 45 times that of the Galaxy (NH/AV = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='87 × 1021 cm−2 mag−1, Draine 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Even this gas-to-dust ratio is higher than that of nearby CT-AGN in the Circinus galaxy (Tani- moto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' 2019), which is the most obscured AGN in the nearby universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Such a high gas-to-dust ratio of this source means that the radiation of the central AGN may have destroyed dust in the torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' The evolutionary stage of the source Many studies have suggested a co-evolution scheme between host galaxies and AGNs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=', Kormendy & Ho 2013), such as MBH −σ∗ relation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=', Ferrarese & Mer- ritt 2000), AGN feedback (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=', Bower et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Some studies have suggested that the host galaxies of heavily obscured AGNs appear to be at the stage of intense star formation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=', Hopkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' These intense star-forming activities may be associated with AGNs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' AGNs trigger star formation of host galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' NGC 449 is a late-type galaxy, meaning it is in an early stage of galaxy evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Its center hosts a heavily ob- scured AGN, which is considered to be at the early stage of AGN evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' These suggest that the host galaxy and central AGN of NGC 449 are at the early stage of 10 Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' co-evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' In Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='3, we derive some parameters about the host galaxy, including SFR and stellar mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' The specific star formation rate (sSFR = SFR/M∗) of this source is approximately 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='59 Gyr−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' This value is consistent with other star formation galaxies at similar redshifts (Salim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' This result suggests that the central AGN does not appear to trigger intense star formation in its host galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' SUMMARY We studied a heavily obscured AGN in NGC 449 using the multiwavelength data, which included NuS- TAR, XMM-Newton, Spitzer, and multi-band photo- metric data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' After, we analyzed its X-ray spectrum and obtained the column density (NH ≃ 1024cm−2), the photon index (Γ ≃ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='4), the luminosity (flux den- sity) of the scattering component, and the temperature (kT ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='82 keV) of diffuse thermal gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' However, some parameters could not be well constrained, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=', the frac- tion of the scattering component, the intrinsic flux den- sity, and luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' We fitted its mid-IR spectrum and decomposed the AGN contribution to derive some AGN parameters, and used its SED to obtain some parame- ters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Combining the information from the mid-IR spec- trum and SED, we first derived the intrinsic X-ray lumi- nosity (≃ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='54×1042 erg s−1) and the scattering fraction of primary X-ray emission (≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='26%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' In addition, we also derived the following results: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' The bolometric luminosity of the AGN is about 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='44 × 1044 erg s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' The mass accretion rate of central AGN is about 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='54 × 10−2M⊙ yr−1, and the Eddington ratio is 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='39 × 10−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' The torus of this AGN has a high gas-to-dust ratio (NH/AV = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='40 × 1022 cm−2 mag−1), which is about 45 times that of the Galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Such a high gas-to-dust ratio means that the radiation of the central AGN may have destroyed dust in the torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' The host galaxy and the central AGN are both in the early stage of co-evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' We sincerely thank the anonymous referee for useful sug- gestions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' We acknowledge financial support from the National Key Research and Development Program of China grant (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' 2017YFA0402703) and National Nat- ural Science Foundation of China grant (NSFC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' grant Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content='11733002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' We acknowledge the science research grants from the China Manned Space Project with NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' CMS-CSST-2021-A07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Yongyun Chen is grateful- for financial support from the National Natural Science Foundation of China (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' 12203028).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' This work is supported by the youth project of Yunnan Provincial Science and Technology Department (202101AU070146, 2103010006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Yongyun Chen is grateful for funding for the training Program for talents in Xingdian, Yunnan Province.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' Nan Ding thanks for the financial support from the National Natural Science Foundation of China (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE5T4oBgHgl3EQfCw5O/content/2301.05398v1.pdf'} +page_content=' 12103022) and the Special Basic Cooperative Re- search Programs of Yunnan Provincial Undergraduate Universities’Association (No.' metadata={'source': 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mode 100644 index 0000000000000000000000000000000000000000..aea50601add3a69e50ea95756eb99c386b676149 --- /dev/null +++ b/a9E4T4oBgHgl3EQfoQ0b/content/tmp_files/2301.05182v1.pdf.txt @@ -0,0 +1,3111 @@ +Thompson Sampling with Diffusion Generative Prior +Yu-Guan Hsieh ∗ +Université Grenoble Alpes +yu-guan.hsieh@univ-grenoble-alpes.fr +Shiva Kasiviswanathan +Amazon +kasivisw@gmail.com +Branislav Kveton +AWS AI Labs +bkveton@amazon.com +Patrick Blöbaum +Amazon +bloebp@amazon.com +Abstract +In this work, we initiate the idea of using denoising diffusion models to learn priors for online decision +making problems. Our special focus is on the meta-learning for bandit framework, with the goal of +learning a strategy that performs well across bandit tasks of a same class. To this end, we train a diffusion +model that learns the underlying task distribution and combine Thompson sampling with the learned +prior to deal with new tasks at test time. Our posterior sampling algorithm is designed to carefully +balance between the learned prior and the noisy observations that come from the learner’s interaction +with the environment. To capture realistic bandit scenarios, we also propose a novel diffusion model +training procedure that trains even from incomplete and/or noisy data, which could be of independent +interest. Finally, our extensive experimental evaluations clearly demonstrate the potential of the proposed +approach. +∗Work done during internship at Amazon. +1 +arXiv:2301.05182v1 [cs.LG] 12 Jan 2023 + +Contents +1 +Introduction +3 +2 +Preliminaries and Problem Description +4 +2.1 +Denoising Diffusion Probabilistic Model +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +4 +2.2 +Meta-Learning of Bandit Tasks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +5 +3 +Using Trained Diffusion Models in Thompson Sampling +6 +3.1 +Variance Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +6 +3.2 +Thompson Sampling with Diffusion Prior +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . +7 +4 +Training Diffusion Models from Imperfect Data +8 +4.1 +Training with Imperfect Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +9 +4.2 +Variance Calibration with Imperfect Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +12 +5 +Numerical Experiments +12 +6 +Concluding Remarks +14 +A Mathematics of Algorithm Design +19 +A.1 Recurrent Step in Posterior Sampling from Diffusion Prior . . . . . . . . . . . . . . . . . . . . +19 +A.2 On SURE-based Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +21 +B Missing Experimental Details +21 +B.1 +Construction of Bandit Instances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +21 +B.2 +Diffusion Models– Model Design +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +22 +B.3 +Diffusion Models– Training +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +23 +B.4 +Other Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +24 +C Ablation Study +25 +C.1 Predicted versus Sampled Noise in Posterior Sampling . . . . . . . . . . . . . . . . . . . . . . +25 +C.2 Importance of Variance Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +25 +C.3 Ablation Study for Training from Imperfect Data . . . . . . . . . . . . . . . . . . . . . . . . . +27 +D Additional Experiments +27 +D.1 Experimental Results with Different Assumed Noise Levels +. . . . . . . . . . . . . . . . . . . +29 +D.2 Comparison of Posterior Sampling Strategies on a Toy Problem . . . . . . . . . . . . . . . . . +29 +D.3 Training from Imperfect Image Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +29 +E Expected Reward Visualization +32 +2 + +1 +Introduction +Uncertainty quantification is an integral part of online decision making and forms the basis of various online +algorithms that trade-off exploration against exploitation. Among these methods, Bayesian approaches allow +us to quantify the uncertainty using probability distributions, with the help of the powerful tools of Bayesian +inference. Nonetheless, their performance is known to be sensitive to the choice of prior. +For concreteness, let us consider the problem of stochastic multi-armed bandits (MABs) [Bubeck and +Cesa-Bianchi, 2012, Lattimore and Szepesvári, 2020], in which a learner repeatedly pulls one of the K +arms from a given set A = {1, ..., K} and receives rewards that depend on the learner’s choices. More +precisely, when arm a is pulled at round t, the learner receives reward rt ∈ R drawn from an arm-dependent +distribution Pa. The goal of the learner is either to i) accumulate the highest possible reward over time (a.k.a. +regret-minimization) or to ii) find the arm with the highest expected reward within a prescribed number of +rounds (a.k.a. best-arm identification). +For both purposes, we need to have a reasonable estimate of the arms’ mean rewards µa = Era∼Pa[ra]. In +general, this would require us to pull each arm a certain number of times, which becomes inefficient when +K is large. While the no-free-lunch principle prevents us from improving upon this bottleneck in general +situations, it is worth noticing that the bandit instances (referred as tasks hereinafter) that we encounter in +most practical problems are far from arbitrary. To name a few examples, in recommendation systems, each +task corresponds to a user with certain underlying preferences that affect how much they like each item; in +online shortest path routing, we operate in real-world networks that feature specific characteristics. In this +regard, introducing such inductive bias to the learning algorithm would be beneficial. In Bayesian models, this +can be expressed through the choice of the prior distribution. Moreover, as suggested by the mete-learning +paradigm, the prior itself can also be learned from data, which often leads to superior performance [Rothfuss +et al., 2021, Hospedales et al., 2021]. This has led to the idea of meta-learning a prior for bandits [Peleg +et al., 2022, Cella et al., 2020, Basu et al., 2021]. +On the other hand, we have recently witnessed the success of deep generative modeling in producing +high-quality synthetic data across various modalities [Saharia et al., 2022, Wu et al., 2021, Brown et al., +2020]. The impressive results shows that these models come out as a powerful tool for modeling complex +distributions. While different models have their own strength and weakness, diffusion models [Sohl-Dickstein +et al., 2015, Ho et al., 2020] turn out to be particularly appealing for our use case as its iterative sampling +scheme makes it much more flexible to be applied on a downstream task. In this regard, this paper attempts +to answer the following question: +Can diffusion models provide better priors to address the exploration-exploitation trade-off in bandits? +Our Contributions. +In this work, we initiate the idea of using diffusion models to meta-learn a prior for +bandit problems. Working towards this direction, we make the following contributions: +• We propose a new Thompson sampling scheme that incorporates a prior represented by a diffusion +model. +The designed algorithm strikes a delicate balance between the learned prior and bandit +observations, bearing in mind the importance of having an accurate uncertainty estimate. In particular, +the deployment of the diffusion model begins with a variance calibration step. Then, in each round of +the interaction we summarize the interaction history by a masked vector of dimension K, and perform +posterior sampling with a modified iterative sampling process that makes use of this vector in each step. +• Standard diffusion model training assumes access to noise-free samples. This is however nearly impossible +in most bandit applications. To overcome this limitation, we propose a novel diffusion model training +procedure which only utilizes incomplete and/or noisy observations. Our method alternates between +sampling from the posterior distribution and minimizing a modified loss function that is suited to +imperfect data. We believe that this training procedure could be of interest beyond bandit setup for +example in deep generative modeling scenarios where noise-free training data are not accessible or +expensive to get. +3 + +• We perform extensive experimental evaluations on various synthetic and real datasets to demonstrate +the benefit of the considered approach against several baseline methods including Thomspon sampling +with Gaussian priors [Thompson, 1933], Thompson sampling with Gaussian mixture model (GMM) +priors [Hong et al., 2022b], and UCB [Auer, 2002]. The results confirm that the use of diffusion +prior always leads to improved performance and the improvement is particularly significant when the +underlying problem has a complex structure. +Related Work. +Prior to our work, the use of diffusion models in decision making has been explored by +Janner et al. [2022], Ajay et al. [2022], who used conditional diffusion models to synthesize trajectories +in offline decision making. Their approaches demonstrated good performance on various benchmarks. In +contrast, our focus is on online decision making, where exploration is crucial for the success of the algorithm. +Additionally, we use diffusion models to learn a task prior, rather than a distribution specific to a single task. +More generally, diffusion models have been used as priors in various areas, primarily for the goal of +inverse problem solving. Our posterior sampling algorithm shares some similarity with those presented in +previous studies by Sohl-Dickstein et al. [2015], Song et al. [2021], Chung et al. [2022a]. Essentially, these +algorithms combine each unconditional sampling step with a step that ensures coherence with the observation. +Alternatively, close form expression for the conditional score function and the conditional reverse step can +be derived if we assume the observed noise is carved from the noise of the diffusion process. This approach +was taken by Kawar et al. [2021, 2022]. Another solution is to approximate the posterior with a Gaussian +distribution, as proposed by Graikos et al. [2022]. In this case, samples are reconstructed by minimizing a +weighted sum of the denoising loss and a constraint loss, rather than using an iterative sampling scheme. +Regarding the algorithmic framework, we build upon the well-known Thompson sampling idea introduced +by Thompson [1933] nearly a century ago. It has reemerged as one of the most popular algorithms for bandit +problems in the last decade due to its simplicity and generality [Chapelle and Li, 2012, Russo and Van Roy, +2014, Russo et al., 2018]. Nonetheless, it is only until more recently that a series of work [Lu and Van Roy, +2019, Simchowitz et al., 2021] provides a through investigation into the influence of the algorithm’s prior, and +confirms the benefit of learning a meta-prior in bandits via both empirical and theoretical evidence [Cella +et al., 2020, Basu et al., 2021, Kveton et al., 2021, Peleg et al., 2022]. The main difference between our work +and the above is the use of a more complex prior, which also goes beyond the previously studied mixture +prior [Hong et al., 2022b] and multi-layered Gaussian prior [Hong et al., 2022a]. On a slightly different note, +a large corpus of work have investigated other ways to encode prior knowledge, including the use of arm +hierarchy [Sen et al., 2021], graphs [Valko et al., 2014], or more commonly a latent parameter shared by the +arms [Lattimore and Munos, 2014, Maillard and Mannor, 2014, Hong et al., 2020, Gupta et al., 2020]. +Notation. +All the variables are multi-dimensional unless otherwise specified. For a vector x, xa represents +the a-th coordinate of the vector, and x2 represents its coordinate-wise square. A sequence of vectors +(xl)l∈{l1,...,l2} is written as xl1:l2 for conciseness. [n] denotes the sequence of integers {1, ..., n}. To distinguish +random variables from their instantiation, we represent the former with capital letters and the latter with the +corresponding lowercase letters. Conditioning on X = x is then abbreviated as · | x. A Gaussian distribution +centered at µ ∈ Rd with covariance Σ ∈ Rd×d is written as N(X; µ, Σ) or simply N(µ, Σ) if the random +variable in question is clear from the context. +2 +Preliminaries and Problem Description +In this section, we briefly review denoising diffusion models and introduce our meta-learning for bandits +framework. +2.1 +Denoising Diffusion Probabilistic Model +First introduced by Sohl-Dickstein et al. [2015] and recently popularized by Ho et al. [2020] and Song and +Ermon [2019], denoising diffusion models (or the closely related score-based models) have demonstrated +4 + +state-of-the-art performance in various data generation tasks. A large number of variants of these models +have been proposed since then. In this paper, we primarily follow the notation and formulation of Ho et al. +[2020], with minor modifications to suit our purposes. +Intuitively speaking, diffusion models learn to approximate a distribution Q0 over Rd by training a series +of denoisers with samples drawn from this distribution. Writing q for the probability density function (assume +everything is Lebesgue measurable for simplicity) and X0 for the associated random variable, we define the +forward diffusion process with respect to a sequence of scale factors (αℓ) ∈ (0, 1)L by +q(x1:L | x0) = +L−1 +� +ℓ=0 +q(xℓ+1 | xℓ), +q(Xℓ+1 | xℓ) = N(Xℓ+1; √αℓ+1xℓ, (1 − αℓ+1)Id). +The first equality suggests that the forward process forms a Markov chain that starts at x0 ∈ Rd, while the +second equality implies that the transition kernel is Gaussian. Further denoting the product of the scale +factors by ¯αℓ = �ℓ +i=1 αi, we then have q(Xℓ | x0) = N(Xℓ; √¯αℓx0, (1 − ¯αℓ)Id). +The sequence (αℓ) ∈ (0, 1)L is chosen to be decreasing and such that ¯αL ≈ 0. We thus expect q(Xℓ) ≈ +N(0, Id). A denoising diffusion model learns to reverse the diffusion process by optimizing a certain parameter +θ that defines a distribution Pθ over random variables X′ +0:L. The hope is that the marginal distribution +Pθ(X′ +0) would be a good approximation of Q0. In practice, this is achieved by setting pθ(Xℓ) = N(0, Id), +enforcing the learned reverse process to be Markovian, and modeling pθ(Xℓ | xℓ+1) as a Gaussian parameterized +by1 +pθ(Xℓ | xℓ+1) = q(Xℓ | xℓ+1, X0 = hθ(xℓ+1, ℓ + 1)) +� +�� +� +ˆx0 += N +� +Xℓ; +√¯αℓ(1 − αℓ+1) +1 − ¯αℓ+1 +ˆx0 + +√αℓ+1(1 − ¯αℓ) +1 − ¯αℓ+1 +xℓ+1; 1 − ¯αℓ +1 − ¯αℓ+1 +(1 − αℓ+1)Id +� +(1) +In the above hθ is the learned denoiser and hθ(xℓ+1, ℓ + 1) is the predicted clean sample.2 +2.2 +Meta-Learning of Bandit Tasks +Our work focuses on meta-learning problems in which the tasks are bandit instances drawn from an underlying +distribution that we denote by T . As in standard meta-learning, the goal is to learn an inductive bias from +the meta training set that would improve the overall performance of an algorithm on new tasks drawn from +the same distribution. In the context of this paper, the inductive bias is encoded in the form of a prior +distribution that would be used by the Thompson sampling algorithm when the learner interacts with new +bandit instances. +For the sake of simplicity, we restrict our attention to the multi-armed bandit scenario presented in +Section 1, with the additional assumption that the noise in the rewards are Gaussian with known variance +σ2.3 The only unknown information is thus the vector of the mean rewards µ = (µa)a∈A. For this specific +situation, Thompson sampling takes as input a prior distribution over RK, samples a guess ˜µt of the mean +reward vector from the posterior distribution at each round, and pulls arm at ∈ arg maxa∈A ˜µa +t in that round. +The posterior distribution itself is determined by both the prior and the interaction history, i.e., the sequence +of the action-reward pairs (as, rs)s∈{1,...,t−1}. +As for the meta-training phase, we consider two situations that are distinguished by whether the learner +has access to perfect data or not. In the former case, the meta-training set is composed of the exact means +Dtr = {µB}B of training tasks B drawn from the distribution T , whereas in the latter case the training set is +1With a slight abuse of notation, we drop the prime from X′ +0:L in the remaining of the work, but one should keep in mind +that the distributions of X0:L induced by the forward process and of X′ +0:L modeled by the diffusion model are distinct. +2To obtain hθ we typically train a neural network with a U-Net architecture. In [Ho et al., 2020], this network is trained to +output the predicted noise ¯zℓ = (xℓ − √¯αℓhθ(xℓ, ℓ))/√1 − ¯αℓ. +3We make this assumption as we are using diffusion prior. As far as we are aware, all the existing diffusion model posterior +sampling algorithms for the case of Gaussian noise either rely on this assumption or circumvent it by adding some adjustable +hyperparameter. How to extend these algorithms to cope with unknown noise variance properly is an interesting open question. +5 + +Model +Training +Variance +Calibration +Bandit +Deployment +perfect / imperfect observations of + from different tasks +expected rewards +diffusion model +calibrated variances +diffusion prior +Task Distribution +Task +action +feedback +Figure 1: Overview of the meta-learning for bandits with diffusion prior framework. +composed of incomplete and/or noisy observations of these vectors (see Section 4 for details). We use the +term imperfect data to informally refer to the scenario where the data is incomplete and/or noisy. The entire +algorithm flow is summarized in Figure 1 and Algorithm 1, where both the model training and the variance +calibration blocks together define the diffusion prior that is used by Thompson sampling in the deployment +phase, as we will immediately see in Section 3. +Algorithm 1 Meta-learning for Bandits with Diffusion Models +1: Meta-Training Phase a): +Diffusion Model Training +2: Input: Training set containing reward observations from different tasks +3: Train a diffusion model hθ to model the distribution of the mean rewards (in case of imperfect data use +Algorithm 5) +4: Meta-Training Phase b): +Variance Calibration +5: Input: Diffusion model hθ and calibration set containing reward observations from different tasks +6: Use Algorithm 2 to estimate the mean squared reconstruction errors τ 1:L of the model hθ from different +diffusion steps to calibrate the variance of each reverse step (in case of imperfect data use Algorithm 6) +7: Meta-Deployment Phase +8: Input: Diffusion model hθ, reconstruction error τ 1:L, and assumed noise level ˆσ +9: For any new task, run Thompson sampling with diffusion prior (Algorithm 4) with provided parameters +3 +Using Trained Diffusion Models in Thompson Sampling +In this section, we describe how a learned diffusion model can be incorporated as a prior for Thompson +sampling. For sake of presenting the core ideas, we focus here on the case where clean datasets Dtr and +Dcal are used for training and calibration. The case where only imperfect datasets are available is addressed +in Section 4. With a clean dataset Dtr, diffusion model can be trained using well-known techniques, see +e.g., [Ho et al., 2020]. Then, as outlined in Algorithm 1, given a trained model, the two remaining steps are: +a) variance calibration, and b) Thompson sampling. +3.1 +Variance Calibration +While Ho et al. [2020] fixed the variance of pθ(Xℓ | xℓ+1) to that of q(Xℓ | xℓ+1, x0) as expressed by Eq. (1), it +was recently shown by Bao et al. [2021] that this choice was sub-optimal. This defect turns out to be critical +when we use diffusion model as prior in online decision problems, as it falls short in quantifying the right +level of uncertainty. To remedy this problem, we follow closely the approach of Bao et al. [2022] and calibrate +the variances of the reverse process with a separate calibration set Dcal. Precisely, we write +6 + +pθ,τ(Xℓ | xℓ+1) = +� +q(Xℓ | xℓ+1, x0)p′ +θ,τ(x0 | xℓ+1) dx0. +(2) +In the above, p′ +θ,τ(X0 | xℓ+1) is a Gaussian distribution centered at the denoiser output ˆx0 = hθ(xℓ+1, ℓ + 1) +and τ = τ 1:L is the optimized variance parameter. This is different from (1) where instead of p′ +θ,τ(x0 | xℓ+1)dx0 +we only have a Dirac concentrated at x0. The covariance of p′ +θ,τ(X0 | xℓ+1) is taken as the diagonal matrix +diag(τ 2 +ℓ+1). As for the variance parameter τ ℓ+1, it represents the (coordinate-wise) root mean squared +reconstruction error and is computed on the calibration set Dcal by constructing Dcal,ℓ of pairs (x0, xℓ) with +x0 ∈ Dcal and xℓ sampled from Xℓ | x0, and setting +τ a +ℓ = +� +� +x0,xℓ∈Dcal,ℓ +∥xa +0 − ha +θ(xℓ, ℓ)∥2/ card(Dcal,ℓ). +The above procedure is summarized in Algorithm 2. Intuitively, the calibration step automatically adjusts +how much we rely on the learned model in the upcoming tasks by taking the reconstruction error as a proxy +for the model’s quality. We opt for a simple model here in which the covariance matrix is the same at all +points, whereas Bao et al. [2022] fit a neural network to predict the mean squared residual at every xℓ. Once +the reconstruction errors are computed, the covariance of pθ,τ can be derived from (2). +Algorithm 2 Diffusion Model Variance Calibration +1: Input: Diffusion model hθ, calibration set Dcal = {xi,0}i, noise standard deviation σ +2: Output: Reconstructions errors τ 1:L +3: for ℓ = 1 . . . L do +4: +Construct Dcal,ℓ = {xi,0, xi,ℓ}i by sampling xi,ℓ from Xℓ | xi,0 +5: +for a = 1 . . . K do +6: +Set τ a +ℓ = +�� +x0,xℓ∈Dcal,ℓ∥xa +0 − ha +θ(xℓ, ℓ)∥2/ card(Dcal,ℓ) +7: +end for +8: end for +3.2 +Thompson Sampling with Diffusion Prior +We next proceed to discuss how to perform Thompson sampling with a diffusion model learned prior pθ,τ. +For this, we need to sample from the posterior when the prior is specified as such. Concretely, for a given +evidence y0 of x0 with known q(y0 | x0), we are interested in sampling from X0 | y0. While an exact solution +does not exist in general, we may look at this problem as sampling from the prior mixed with evidence y0. In +this regard, our algorithm gradually guides the sample towards the evidence during the sampling process. +This is achieved by conditioning the reverse Markovian process on Y 0 = y0 and seeks an approximation for +each conditional reverse step. +In the case of multi-armed bandits, y0 = (as, rs)s∈{1,...,t} is the interaction history and x0 = µ ∈ RK is +the mean reward vector of the task, and it holds that +q(y0 | x0) ∝ +t� +s=1 +q(rs | µ, as) = +t� +s=1 +N(rs; µas, σ2) +[as a function of x0]. +(3) +By the proportionality we hide all the randomness in the learner’s actions. This is legitimate because the +learner’s actions only depend on the mean reward vector via their interaction history with the environment, +i.e., q(as | a1, r1, . . . , as−1, rs−1, µ) = q(as | a1, r1, . . . , as−1, rs−1). The initialization and the recursive steps +of our conditional sampling scheme tailored to this situation is then provided below. Detailed derivation +behind the algorithm is provided in Appendix A.1. +Sampling from XL | y0. +For this part, we simply ignore y0 and sample from N(0, Id) as before. +7 + +... +... +... +Unconditional +generation +Conditional +generation +Result +Predicted noise +Diffused observation +Observation +Unknown latent +Figure 2: Illustration of the proposed posterior sampling +with diffusion prior algorithm (Algorithm 3). +Warm-up +Posterior +Sampling +Loss +Minimization +minimize +minimize +sample +sample +with current model +EM +Figure 3: Overview of the proposed training procedure +to deal with incomplete and/or noisy data. +Sampling from Xℓ | xℓ+1, y0 +We first create an unconditional latent variable x′ +ℓ by sampling from the +unconditional reverse process pθ,τ(Xℓ | xℓ+1). We then perform coordinate-wise operation by distinguishing +between the following two situations. +• Arm a has never been pulled in the first t rounds: In this case we just set xa +ℓ to be x′a +ℓ . +• Arm a has been pulled in the first t rounds: We denote by ˆµa +t = �t +s=1 rs 1{as = a}/N a +t as the empirical +mean and σa +t = σ/ +� +N a +t as the adjusted standard deviation, where N a +t is the number of times that arm +a has been pulled up to time t (included). We also write ζa +ℓ,1 for the standard deviation of pθ,τ(Xa +ℓ | xℓ+1) +and define the diffused observation +˜ya +ℓ = √¯αℓˆµa +t + +√ +1 − ¯αℓ¯za +ℓ+1 + ζa +ℓ,2˜za +ℓ+1 +(4) +that contains a predicted noise component ¯zℓ+1 satisfying xℓ+1 = √¯αℓ+1hθ(xℓ+1, ℓ+1)+√1 − ¯αℓ+1¯zℓ+1 +and an independent noise component with ˜za +ℓ+1 sampled from N(0, 1) and further multiplied by +ζa +ℓ,2 = +� +¯αℓ +� +(σa +t )2 + ¯αℓ+1(1 − ¯αℓ) +¯αℓ(1 − ¯αℓ+1)(τ a +ℓ+1)2 +� +. +(5) +The output of the conditional reverse step is then a weighted sum of the diffused observation and the +unconditional latent variable45 +xa +ℓ = +(ζa +ℓ,1)−2x′a +ℓ + (ζa +ℓ,2)−2˜ya +ℓ +(ζa +ℓ,1)−2 + (ζa +ℓ,2)−2 +. +As we see above, while the case of single observation with missing entries is clearly a special case of bandit +observations, in terms of algorithmic scheme, it becomes equivalent when we summarize the interaction +history of bandits with mean ˆµa +t and standard deviation σt. Therefore, we present the posterior sampling +algorithm for the former situation in Algorithm 3, and depict the induced Thompson sampling algorithm in +Algorithm 4. It is also important to note that while our algorithms shares similarity with existing posterior +sampling methods [Song et al., 2021, Chung et al., 2022b], we supplement our diffused observation ˜ya +ℓ with a +predicted noise component that improves the coherence between the observation and the generated sample. +4 +Training Diffusion Models from Imperfect Data +Standard training procedure of diffusion models require access to a dataset of clean samples Dtr = {xi,0}i∈[n]. +Nonetheless, in most bandit applications, it is nearly impossible to obtain such dataset as the exact mean +4We set xa +ℓ = ˜ya +ℓ if ζa +ℓ,2 = 0. +5The weighted average is also equivalent to sampling xa +ℓ from a certain Gaussian distribution; see Appendix A.1 for details. +8 + +Algorithm 3 Posterior Sampling with Diffusion Prior +1: Input: Observation y0 ∈ RK, noise standard deviation σ ∈ RK, binary mask m ∈ {0, 1}K, diffusion +model hθ and associated reconstruction errors τ 1:L +2: Output: Posterior sample x0 (resp. x0:L) approximately sampled from X0 | y0 (resp. X0:L | y0) +3: Sample xL ∼ N(0, Id) +4: for ℓ ∈ L − 1, . . . , 0 do +5: +Predict clean sample ˆx0 ← hθ(xℓ+1, ℓ + 1) and associated noise ¯zℓ+1 +6: +Sample unconditional latent variable xℓ from pθ,τ(Xℓ | xℓ+1) +7: +for a ∈ A such that ma = 1 do +8: +Sample ˜za +ℓ+1 ∼ N(0, 1) and compute ζa +ℓ,2 following (5) +9: +Compute diffused observation ˜ya +ℓ ← √¯αℓya +0 + √1 − ¯αℓ¯za +ℓ+1 + ζa +ℓ,2˜za +ℓ+1 +10: +Set xa +ℓ ← +(ζa +ℓ,1)−2xa +ℓ +(ζa +ℓ,2)−2˜ya +ℓ +(ζa +ℓ,1)−2+(ζa +ℓ,2)−2 +▷ ζa +ℓ,1 is the standard deviation of pθ,τ(Xa +ℓ | xℓ+1) +11: +end for +12: end for +Algorithm 4 Thompson Sampling with Diffusion Prior (DiffTS) +1: Input: Diffusion model hθ, reconstruction errors τ 1:L, assumed noise level ˆσ ∈ R +2: for t = 1, . . . do +3: +Sample ˜x0 using Algorithm 3 with y0 ← ˆµt−1, σ ← σt−1, m defined by ma = 1{N a +t−1 > 0} +4: +Pull arm at ∈ arg maxa∈A ˜xa +0 +5: +Update number of pulls N a +t , scaled standard deviation σa +t , and empirical reward ˆµa +t for a ∈ A +6: end for +reward vector µ of each single task is never directly observed. Instead, one can collect imperfect observations +of these vectors, either through previous bandit interactions or forced exploration. Taking this into account, in +this section, we build towards a systematic procedure to train (and calibrate) diffusion models from imperfect +(incomplete and/or noisy) data. It is worth noticing that the application scope of our methodology goes +beyond the bandit setup and covers any situation where imperfect data are available. As an example, we +apply our approach to train from imperfect images (corrupted MNIST and Fashion-MNIST [Xiao et al., 2017] +datasets) and obtain promising results (details are provided in Appendix D.3). +Setup. +For ease of exposition, we first focus on the case of uniform noise variance. Extension to deal with +non-uniform noise variance is later presented in Remark 1. When all the observed noises have the same +variance σ2 ∈ R, the samples of the imperfect dataset ˇDtr = {yi,0}i∈[n] can be written as yi,0 = mi ⊙(xi,0 +zi) +where mi ∈ {0, 1}K is binary mask, zi is a noise vector sampled from N(0, σ2Id), and ⊙ denotes element-wise +multiplication.6 In the considered bandit problem, such dataset can be obtained by randomly pulling a subset +of arms once for each arm. We also assume that the associated masks {mi}i∈[n] and the noise standard +deviation σ are known. We can thus rewrite the dataset as ˇDtr = {yi,0, mi}i∈[n]. +4.1 +Training with Imperfect Data +In presence of perfect data, diffusion model training optimizes the denoising objective +Eℓ∼Uniform({1,...,L}),x0∼Q0,xℓ∼Xℓ | x0[∥x0 − hθ(xℓ, ℓ)∥2]. +(6) +Nonetheless, neither x0 nor xℓ are available when we only have access to an imperfect dataset ˇDtr. To +tackle these challenges, we propose an expectation-maximization (EM)-type procedure where in the place of +the expectation step we perform sampling of latent variables and in the place of the maximization step we +6As we will see Remark 1, the masking of an entry can also be viewed as an observation with infinite variance. +9 + +minimize a tailored loss function. An indicative algorithmic scheme is summarized in Algorithm 5 (without +mini-batching, dataset shuffling, and the use of specific optimization algorithm). +Posterior Sampling. +Due to the absence of a clean observation of x0, it is impossible to sample xℓ via the +forward diffusion process. Nonetheless, we can perform posterior sampling with the current model as done in +several variants of stochastic EM [Fort and Moulines, 2003]. In fact, as explained in Section 2.1, a diffusion +model can be regarded as a probability model over the random variables X0:L. A typical expectation step in +EM for a given parameter θ′ requires us to compute the expected log likelihood function +Q(θ) = +n +� +i=1 +EXi,0:L | yi,0,mi,θ′ log pθ(Xi,0:L). +Nonetheless, this is intractable in general due to the use of neural network in the definition of pθ. To +circumvent this issue, we can instead sample ˜xi,0:L from the posterior with density pθ′(· | yi,0, mi) and use +stochastic gradient ascent to maximize the log likelihood function. +Loss Minimization. +Having obtained the posterior samples, we have the option to either maximize the +log-likelihood of �Dtr or minimize the denoising loss � +˜x0:L∈ � +Dtr +�L +ℓ=1∥˜x0 − hθ(˜xℓ, ℓ)∥2. However, both of these +approaches heavily rely on the samples generated in the posterior sampling step, which can bias the model +towards generating low-quality samples during early stages of training. To address this issue, we propose to +consider a loss function that utilizes the actual observation y0 and not the reconstructed sample ˜x0. Fix a +small value ε and a regularization parameter λ, the new loss function for a sample pair (y0, ˜xℓ) at diffusion +step ℓ with associated mask m is defined as +L(θ; y0, ˜xℓ, m, ℓ) = ∥m ⊙ y0 − m ⊙ hθ(˜xℓ, ℓ)∥2 + 2λ√¯αℓσ2 Eb∼N (0,I) b⊤ +�hθ(˜xℓ + εb, ℓ) − hθ(˜xℓ, ℓ) +ε +� +. +(7) +The above loss function is composed of two components that address respectively the incompleteness and +the noise in the observations. First, it handles incomplete (missing) data by only considering the observed +entries as determined by the element-wise product with the mask in the first term. Next, to account for +noise, we include a regularization term that penalizes the denoiser from varying too much when the input +changes. Overall, our denoising loss find its roots in a series of work [Metzler et al., 2018, Zhussip et al., +2019] that investigates the training of denoiser in the absence of ground-truth clean data. In particular, +the expectation here is an approximation of the divergence div˜xℓ(hθ(˜xℓ, ℓ)) that appears in Stein’s unbiased +risk estimate (SURE) [Stein, 1981, Eldar, 2008], an unbiased estimator of the mean squared error whose +computation only requires the use of noisy samples.7 +From a practical viewpoint, the regularization term provides a trade-off between the bias and the variance +of the learned model. When λ is set to 0, the model learns to generate noisy samples, which corresponds to a +flatter prior that encourages exploration. When λ gets larger, the model tries to denoise from the observed +noisy samples. This can however deviate the model from the correct prior and accordingly jeopardize the +online learning procedure. +Overall Procedure. +The complete algorithm for training from imperfect data is presented in Algorithm 5. +It alternates between the posterior sampling and the loss minimization steps. While any posterior sam- +pling algorithm can be used for the former, in our experiments we simply rely on the one presented in +Section 3.2 (note that we actually sample the entire chain ˜x0:L in the procedure of sampling ˜x0) to ac- +quire posterior samples �Dtr = {˜xi,0:L}i∈S⊆[n] for (a subset of) the dataset. Then, in the loss minimization +step we sample from the dataset �D +′ +tr = {˜xi,0:L, yi,0, mi}i∈S and use stochastic gradient descent to mini- +mize � +(˜x0:L,y0,m)∈�D +′ +tr +�L +ℓ=1 L(θ; y0, ˜xℓ, m, ℓ). Moreover, we begin with a warm-up phrase where we sample +7When λ = 1, xℓ = ˜xℓ = √¯αℓy0, m = 1 (i.e., all the entries are observed), and the expectation is replaced by the divergence, +we recover SURE up to additive constant −Kσ2. See Appendix A.2 for details. +10 + +Algorithm 5 Diffusion Model Training from Imperfect (Incomplete and/or Noisy) Data +1: Input: Training set ˇDtr = {yi,0, mi}i, calibration set ˇDcal, noise standard deviation σ, number of +warm-up, outer, and inner training steps S, J, and S′ +2: Output: Diffusion model hθ +3: Warm-up +4: for s = 1, . . . , S do +5: +Sample y0, m from ˇDtr +6: +Sample ℓ from the uniform distribution over {1, ..., L} +7: +Sample yℓ from Xℓ | X0 = y0 +8: +Take gradient step to minimize L(θ; y0, yℓ, m, ℓ) (Eq. (7)) +9: end for +10: Main Training Procedure +11: for j = 1, . . . , J do +12: +Posterior Sampling +13: +Compute reconstructions errors τ 1:L with Algorithm 6 using ˇDcal +14: +Construct �D +′ +tr = {˜xi,0:L, yi,0, mi}i with Algorithm 3 +15: +Loss Minimization +16: +for s = 1, . . . , S′ do +17: +Sample ˜x0:L, y0, m from ˇDtr +18: +Sample ℓ from the uniform distribution over {1, ..., L} +19: +Take gradient step to minimize L(θ; y0, ˜xℓ, m, ℓ) (Eq. (7)) +20: +end for +21: end for +yℓ = √¯αℓy0 + √1 − ¯αℓ˜zℓ with ˜zℓ sampled from N(0, Id) as in standard diffusion model training but replace +the mean squared error by the loss function L introduced in (7).8 This initial phase produces better training +samples than posterior sampling with randomly initialized model. +Remark 1 (Bandit observations / observations with varying variances). When the observations come from +bandit interactions and each arm can be pulled more than once, we can first summarize the interaction history +by the empirical mean and the vector of adjusted standard deviation as suggested in Section 3.2. Therefore, +it remains to address the case where the noise vector zi is sampled from N(0, diag(σi2)) for some vector +σi ∈ RK. As the design of our posterior sampling algorithm already takes this into account, the posterior +sampling steps of the algorithm remains unchanged. The only difference would thus lie in the definition of +the modified loss (7). Intuitively, we would like to give more weights to samples that are less uncertain. This +can be achieved by weighting the loss by the inverse of the variances, that is, we set +L′(θ; y0, ˜xℓ, m, σ, ℓ) = +K +� +a=1 +ma|ya +0 − ha +θ(˜xℓ, ℓ)| +(σa)2 ++ 2λ√¯αℓ Eb∼N (0,I) b⊤ +�hθ(˜xℓ + εb, ℓ) − hθ(˜xℓ, ℓ) +ε +� +. +(8) +To make sure the above loss is always well defined, we may further replace (σa)2 by (σa)2 + δ for some +small δ > 0. It is worth noticing that one way to interpret the absence of observation ma = 0 is to set the +corresponding variance to infinite, i.e., σa = +∞. In this case we see there is even no need of m anymore as the +coordinates with σa = +∞ would already be given 0 weight. Finally, to understand why we choose to weight +with the inverse of the variance, we consider a scalar x, and a set of noisy observations y1, . . . , yn respectively +drawn from N(x, σ2 +1), . . . , N(x, σ2 +n). Then, the maximum likelihood estimate of x is �n +i=1 σ2 +i yi/(�n +i=1 σ2 +i ). +8In our experiments, we impute the missing values of y0 by a non-zero constant. +11 + +4.2 +Variance Calibration with Imperfect Data +As mentioned in Section 3.1, a reliable variance estimate of the reverse process is essential for building a +good diffusion prior. This holds true not only for the online learning process at test phase, but also for the +posterior sampling step of our training procedure. The algorithm introduced in Section 3.1 calibrates the +variance through perfect data. In this part, we extend it to operate with imperfect data. +Let ˇDcal be a set of imperfect data constructed in the same way as ˇDtr. We write ˇD +a +cal = {(y0, m) ∈ ˇDcal : +ma = 1} as the subset of ˇDcal for which a noisy observation of the feature at position a is available. Our +algorithm (outlined in Algorithm 6) is inspired by the following two observations. First, if the entries are +missing completely at random, observed ya +0 of ˇD +a +cal and sampled xa +0 + za with x0 ∼ Q0 and z ∼ N(0, σ2I) +have the same distribution. Moreover, for any triple (x0, y0, xℓ) with y0 = x0 + z, xℓ = √¯αℓx0 + √1 − ¯αℓ ¯zℓ +and x0, z, and ¯zℓ sampled independently from Q0, N(0, σ2I), and N(0, I), it holds that +E[∥ya +0 − ha +θ(xℓ, ℓ)∥2] = E[∥xa +0 − ha +θ(xℓ, ℓ)∥2] + σ2. +We can thus estimate E[∥xa +0 − ha +θ(xℓ, ℓ)∥2] if we manage to pair each ya +0 ∈ ˇD +a +cal with a such xℓ. +We again resort to Algorithm 3 for the construction of xℓ (referred to as ˜xℓ in Algorithm 6 and hereinafter). +Unlike the training procedure, here we first construct ˜x0 and sample ˜xℓ from Xℓ | ˜x0 to decrease the mutual +information between ˜xℓ and y0. Nonetheless, the use of our posterior sampling algorithm itself requires +a prior with calibrated variance. To resolve the chicken-and-egg dilemma, we add a warm-up step where +we precompute the reconstruction errors with Algorithm 2 by treating ˇDcal as the perfect dataset. In our +experiments, we observe this step yields estimates of the right order of magnitude but not good enough to be +used with Thompson sampling, while the second step brings the relative error to as small as 5% compare to +the estimate obtained with perfect validation data using Algorithm 2. +Algorithm 6 Diffusion Model Variance Calibration from Imperfect (incomplete and/or noisy) Data +1: Input: Diffusion model hθ, calibration set ˇDcal = {yi,0, mi}i, noise standard deviation σ +2: Output: Reconstructions errors τ 1:L +3: Data Set Preprocessing +4: Precompute reconstructions errors τ 1:L with Algorithm 2 and Dcal ← ˇDcal (masks ignored) +5: Construct �Dcal = {˜xi,0, yi,0, mi}i with Algorithm 3 +6: Variance Calibration +7: for ℓ = 1 . . . L do +8: +Construct �Dcal,ℓ = {˜xi,ℓ, yi,0, mi}i by sampling ˜xi,ℓ from Xℓ | ˜xi,0 +9: +for a = 1 . . . K do +10: +Let �Da +cal,ℓ = {˜xℓ, y0 : (˜xℓ, y0, m) ∈ �Dcal,ℓ, ma = 1} +11: +Set τ a +ℓ = +� +(� +˜xℓ,y0∈ � +Da +cal,ℓ∥xa +0 − ha +θ(xℓ, ℓ)∥2/ card( �Da +cal,ℓ)) − σ2 +12: +end for +13: end for +5 +Numerical Experiments +Figure +4: +An +example task of +the 2D Maze prob- +lem presented be- +low. The red path +indicates the opti- +mal (super-)arm. +In this section, we illustrate the benefit of using diffusion prior +through numerical experiments on both real and synthetic data. +Missing experimental details, ablation studies, and additional +experiments are presented in Appendices B to D. +Problem Construction. +To demonstrate the wide applica- +bility of our technique, we consider here three bandit problems +respectively inspired by the applications in recommendation +12 + +0 +1000 +2000 +3000 +4000 +5000 +# Iterations +0 +50 +100 +150 +200 +Regret +DiffTS (ours) +UCB +GTS-diag +GTS-full +GMMTS-10 +0 +1000 +2000 +3000 +4000 +5000 +# Iterations +0 +50 +100 +150 +200 +Regret +DiffTS (ours) +UCB +GTS-diag +GTS-full +GMMTS-25 +0 +1000 +2000 +3000 +4000 +5000 +# Iterations +0 +200 +400 +600 +800 +1000 +1200 +Regret +DiffTS (ours) +UCB +GTS-diag +GTS-full +GMMTS-25 +0 +1000 +2000 +3000 +4000 +5000 +# Iterations +0 +50 +100 +150 +200 +Regret +DiffTS (Ours) +UCB +GTS-diag +GTS-full +(a) Popular and Niche +0 +1000 +2000 +3000 +4000 +5000 +# Iterations +0 +50 +100 +150 +200 +Regret +DiffTS (ours) +UCB +GTS-diag +GTS-full +(b) iPinYou Bidding +0 +1000 +2000 +3000 +4000 +5000 +# Iterations +0 +200 +400 +600 +800 +1000 +1200 +Regret +DiffTS (Ours) +UCB +GTS-diag +GTS-full +(c) 2D Maze +Figure 5: Regret performances on three different problems with priors fitted/trained on either exact expected rewards +(top) or partially observed noisy rewards (bottom). The results are averaged over tasks of a test set and shaded areas +represent standard errors. +system, online pricing, and online shortest path routing. Detailed description of the problems and some +visualization that help understand the problem structures are provided in Appendices B.1 and E. The first +and the third problems listed below rely on synthetic data, where we only specify the construction of the +means and the rewards are obtained by perturbing the means with Gaussian noise of standard deviation +σ = 0.1, and for the second problem we use the iPinYou dataset [Liao et al., 2014]. +1. Popular and Niche Problem. We consider here the problem of choosing items to recommend to customers. +Let K = 200. The arms (items) are separated into 40 groups, each of size 5. Among these, 20 groups of +arms correspond to the popular items and tend to have high mean rewards. However, these arms are never +the optimal ones. The other 20 groups of arms correspond to the niche items. Most of them have low +mean rewards but a few of them (those that match the preferences of the customer) have mean rewards +that are higher than that of all the other arms. +2. iPinYou Bidding Problem. We consider here the problem of setting the bid price in auctions. Let v = 300 +be the value of the item. Each arm corresponds to a bid price b ∈ {0, ..., 299}, and the reward is either +v − b when the learner wins the auction or 0 otherwise. The reward distribution of a task is then solely +determined by the winning rates which are functions of the learner’s bid and the distribution of the highest +bid from competitors. For the latter we use the corresponding empirical distributions of 1352 ad slots +from the iPinYou bidding data set [Liao et al., 2014] (each ad slot is a single bandit task). +3. 2D Maze Problem. We consider here an online shortest path routing problem on grid graphs. We formalize +it as a reward maximization combinatorial bandit with semi-bandit feedback. As shown in Figure 4, +the supers arms are the simple paths between the source and the destination (fixed across all the tasks) +whereas the base arms are the edges of the grid graph. At each round, the learner picks a super arm and +observes the rewards of all the base arms (edges) that are contained in the super arm (path). Moreover, +the edges’ mean rewards in each task are derived from a certain 2D maze. The mean reward is −1 when +there is a wall on the associated case (marked by the black color) and −0.01 otherwise. +Training, Baselines, and Evaluation. +To train the diffusion models, for each problem we construct a +training set Dtr and a calibration set Dcal that contain the expected means of the tasks. We then conduct +experiments for the following two configurations: +13 + +1. Learn from perfect data: The priors are learned using Dtr and Dcal that contain the exact mean rewards. +Standard training procedure is applied here. +2. Learn from imperfect data: The priors are learned using ˇDtr and ˇDcal that are obtained from Dtr and Dcal +by perturbing the samples with noise of standard deviation 0.1 and then dropping each feature of a sample +with probability 0.5. To tackle this challenging situation we adopt the approach proposed in Section 4. +In terms of bandit algorithms, we compare our method, DiffTS, with UCB, Thompson sampling with +Gaussian prior using either diagonal or full covariance matrix (GTS-diag and GTS-full), and Thompson +sampling with Gaussian mixture prior [Hong et al., 2022b] with either 10 or 25 components (GMMTS-10 and +GMMTS-25).9 These priors are also learned with the same perfect / imperfect data that we use to train +diffusion models. We however skip the GMM baseline for the imperfect data setup because we are not able +to find any existing algorithm that is able to learn a good GMM on the imperfect data that we consider here. +The performance of the algorithms are then evaluated by their average regret on a standalone test +set— for a sequence of arms (at)t∈{1,...,T } pulled by an algorithm in a bandit task, the induced regret is +RegT = Tµa⋆ − �T +t=1 µat, where a⋆ ∈ arg maxa∈A µa is an optimal arm in this task. The assumed noise level +ˆσ is fixed to the same value across all the methods +Results. +The results are presented in Figure 5. For ease of readability, among the two GMM priors (10 +and 25 components), we only show the one that achieves smaller regret. We see clearly that throughout +the three problems and the two setups considered here, the proposed DiffTS algorithm always has the best +performance. The difference is particularly significant in the Popular and Niche and 2D Maze problems, in +which the regret achieved by DiffTS is around two times smaller than that achieved by the best performing +baseline method. This confirms that using diffusion prior is more advantageous in problems with complex +task distribution. +On the other hand, we also observe that the use of GMM prior in these two problems leads to performance +worse than that of GTS-full, whereas it yields performance that is as competitive as DiffTS in the iPinYou +Bidding problem. This is coherent with the visualizations we make in Appendix E, which shows that the +fitted GMM is only capable of generating good samples in the iPinYou Bidding problem. This, however, +also suggests that the use of a more complex prior is a double-edged sword, and can lead to poor performance +when the data distribution is not faithfully represented. +In Appendix C, we further present ablation studies to investigate the impacts of various components of +our algorithm. In summary, we find out both the variance calibration step and the EM-like procedure for +training with imperfect data are the most crucial to our algorithms, as dropping either of the two could lead +to severe performance degradation. We also affirm that the use of SURE-based regularization does lead to +smaller regret, but finding the optimal regularization parameter λ is a challenging problem. +Finally, while the good performance of DiffTS is itself an evidence of the effectiveness of our sampling +and training algorithms, we provide additional experiments in Appendix D to show how these methods can +actually be relevant in other contexts. +6 +Concluding Remarks +In this work, we argue that the expressivity and flexibility of diffusion models make them a promising +choice for representing complex priors in real-world online decision making problems. Through numerical +experiments, we demonstrate that using a diffusion prior in combination with a proposed Thompson sampling +algorithm can significantly reduce the achieved regret in multi-armed bandit problems. Additionally, we +propose a training procedure for diffusion models that can handle imperfect training data, addressing a +common issue in bandit scenarios, and could be applicable elsewhere too. +9For the 2D Maze problem we consider their combinatorial extensions in which the UCB index / sampled mean of a super +arm is simply the sum of the corresponding quantities of the contained base arms. +14 + +Looking ahead, our work raises a number of exciting but challenging research questions. One potential +extension is to apply our approach to meta-learning problems in contextual bandits or reinforcement learning. +This would involve modeling a distribution of functions or even Markov decision processes by diffusion models, +which remains a largely unexplored area despite a few attempts that work toward these purposes [Dutordoir +et al., 2022, Nava et al., 2022]. 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For this, we write +q(xℓ | xℓ+1, y0) = q(xℓ | xℓ+1)q(y0 | xℓ, xℓ+1) +q(y0 | xℓ+1) += q(xℓ | xℓ+1) +� +q(y0 | x0)q(x0 | xℓ, xℓ+1) dx0 +q(y0 | xℓ+1) +. +(9) +The term q(xℓ | xℓ+1) can be simply approximated with pθ,τ(xℓ | xℓ+1). As for the integral, one natural +solution is to use q(x0 | xℓ, xℓ+1) = q(x0 | xℓ) ≈ p′ +θ,τ(x0 | xℓ). Then, for example, if q(y0 | x0) = N(y0; x0, σ2I), +we can deduce +� +q(y0 | x0)p′ +θ(x0 | xℓ) dx0 = N(y0; hθ(xℓ, ℓ), σ2I + diag(τ 2 +ℓ)). +Nonetheless, as the denoiser hθ can be arbitrarily complex, this does not lead to a close form expression to +sample xℓ. Therefore, to avoid the use of involved sampling strategy in the recurrent step, we approximate +q(x0 | xℓ, xℓ+1) in a different way. We first recall that by definition of the diffusion model we may write +Xℓ = √¯αℓX0 + +√ +1 − ¯αℓ ¯Zℓ +and +Xℓ+1 = √αℓ+1Xℓ + +√ +1 − αℓZℓ+1, +where both ¯ +Zℓ and Zℓ+1 are random variable with distribution N(0, I). This leads to +Xℓ+1 = √¯αℓ+1X0 + +� +1 − ¯αℓ+1 ¯Zℓ+1 +where +¯Zℓ+1 = +� +αℓ+1(1 − ¯αℓ) +1 − ¯αℓ+1 +¯Zℓ + +� +1 − αℓ+1 +1 − ¯αℓ+1 +Zℓ+1. +Therefore, we may take ¯Zℓ+1 as a reasonable approximation of ¯Zℓ, while sampling ¯Zℓ+1 is basically the same +as sampling from p′ +θ(X0 | xℓ+1). To summarize, we write +q(x0 | xℓ, xℓ+1) = q +� +¯Zℓ = xℓ − √¯αℓx0 +√1 − ¯αℓ +��� xℓ, xℓ+1 +� +≈ q +� +¯Zℓ+1 = xℓ − √¯αℓx0 +√1 − ¯αℓ +��� xℓ, xℓ+1 +� += q +� +X0 = +1 +√¯αℓ+1 +� +xℓ+1 − +� +xℓ − √¯αℓx0 +� � +1 − ¯αℓ+1 +1 − ¯αℓ +� ��� xℓ, xℓ+1 +� +≈ p′ +θ,τ +� +X0 = +1 +√¯αℓ+1 +� +xℓ+1 − +� +xℓ − √¯αℓx0 +� � +1 − ¯αℓ+1 +1 − ¯αℓ +� ��� xℓ+1 +� += N +�� +¯αℓ(1 − ¯αℓ+1) +¯αℓ+1(1 − ¯αℓ)x0 + xℓ+1 +√¯αℓ+1 +− +� +1 − ¯αℓ+1 +¯αℓ+1(1 − ¯αℓ)xℓ ; +19 + +hθ(xℓ+1, ℓ + 1), diag(τ 2 +ℓ+1) +� += √ρℓ N +� +x0 ; +1 +√¯αℓ +(xℓ − +√ +1 − ¯αℓ¯zℓ+1), ρℓ diag(τ 2 +ℓ+1) +� +, +where ρℓ = ¯αℓ+1(1 − ¯αℓ)/(¯αℓ(1 − ¯αℓ+1)) and ¯zℓ+1 represents the noise predicted by the denoiser from xℓ+1, +that is, +¯zℓ+1 = xℓ+1 − √¯αℓ+1hθ(xℓ+1, ℓ + 1) +√1 − ¯αℓ+1 +. +In this way, we have approximated q(x0 | xℓ, xℓ+1) by a Gaussian with diagonal covariance and with mean +that depends only linearly on xℓ. In the multi-armed bandit setup that we consider here, the relation between +y0 the interaction history and x0 = µ the mean reward vector obeys (3). There exists thus C(y0) and �C(y0) +such that +� +q(y0 | x0)q(x0 | xℓ, xℓ+1) dx0 +� +�� +� +A += +� +C(y0) +t� +s=1 +N(rs; µas, σ2)q(x0 | xℓ, xℓ+1) dx0 += +� +�C(y0) +� +a∈A +N a +t >0 +N(ˆµa +t ; µa, (σa +t )2)q(x0 | xℓ, xℓ+1) dx0. +Using x0 = µ, the aforementioned approximation of q(x0 | xℓ, xℓ+1), and ignoring the multiplicative constant +that does not depend on xℓ, we get +A ∝ +� +� +a∈A +N a +t >0 +N(ˆµa +t ; xa +0, (σa +t )2)q(x0 | xℓ, xℓ+1) dx0 +≈ √ρℓ +� +� +a∈A +N a +t >0 +N(ˆµa +t ; xa +0, (σa +t )2) +� +a∈A +N +� +xa +0; +1 +√¯αℓ +(xa +ℓ − +√ +1 − ¯αℓ¯za +ℓ+1), ρℓ(τ a +ℓ+1)2 +� +dx0 += √ρℓ +� +a∈A +N a +t >0 +� +N(ˆµa +t ; xa +0, (σa +t )2)N +� +xa +0; +1 +√¯αℓ +(xa +ℓ − +√ +1 − ¯αℓ¯za +ℓ+1), ρℓ(τ a +ℓ+1)2 +� +dxa +0 += √ρℓ +� +a∈A +N a +t >0 +N +� +ˆµa +t ; +1 +√¯αℓ +(xa +ℓ − +√ +1 − ¯αℓ¯za +ℓ+1), (σa +t )2 + ρℓ(τ a +ℓ+1)2 +� +∝ +� +a∈A +N a +t >0 +N +� +xa +ℓ; √¯αℓˆµa +t + +√ +1 − ¯αℓ¯za +ℓ+1, ¯αℓ((σa +t )2 + ρℓ(τ a +ℓ+1)2). +� +Plugging the above into (9), we obtain ˜q(xℓ | xℓ+1, y0) = � +a∈A ˜q(xa +ℓ | xℓ+1, y0) where ˜q(xa +ℓ | xℓ+1, y0) = +pθ,τ(xa +ℓ | xℓ+1) if a is never pulled and otherwise it is the distribution satisfying +˜q(xa +ℓ | xℓ+1, y0) ∝ pθ,τ(xa +ℓ | xℓ+1)N +� +xa +ℓ; √¯αℓˆµa +t + +√ +1 − ¯αℓ¯za +ℓ+1, ¯αℓ((σa +t )2 + ρℓ(τ a +ℓ+1)2) +� +. +(10) +To conclude, we resort to the following lemma (see [Papandreou and Yuille, 2010] for more general results). +Lemma 1. Let µ1, µ2, σ1, σ2 ∈ R. The following two sampling algorithms are equivalent. +1. Sample x directly from the distribution whose density is proportional the product N(µ1, σ2 +1)N(µ2, σ2 +2). +2. Sample x1 from N(µ1, σ2 +1), x2 from N(µ2, σ2 +2), and compute x = σ−2 +1 x1 + σ−2 +2 x2/(σ−2 +1 ++ σ−2 +2 ). +20 + +Proof. It is well known that the product of two Gaussian PDFs is itself proportional to a Gaussian PDF. +Concretely, we have +N(µ1, σ2 +1)N(µ2, σ2 +2) ∝ N +�σ−2 +1 µ1 + σ−2 +2 µ2 +σ−2 +1 ++ σ−2 +2 +, +1 +σ−2 +1 ++ σ−2 +2 +� +. +(11) +On the other hand, the linear combination of two independent Gaussian variables is also a Gaussian variable. +For X1, X2 that follow N(µ1, σ2 +1), N(µ2, σ2 +2) and X = σ−2 +1 X1 + σ−2 +2 X2/(σ−2 +1 ++ σ−2 +2 ), we can compute +E[X] = σ−2 +1 +E[X1] + σ−2 +2 +E[X2] +σ−2 +1 ++ σ−2 +2 += σ−2 +1 µ1 + σ−2 +2 µ2 +σ−2 +1 ++ σ−2 +2 +, +Var[X] = σ−4 +1 +Var[X1] + σ−4 +2 +Var[X2] +(σ−2 +1 ++ σ−2 +2 )2 += +σ−2 +1 ++ σ−2 +2 +(σ−2 +1 ++ σ−2 +2 )2 = +1 +σ−2 +1 ++ σ−2 +2 +. +Therefore, X follows the distribution of (11) and computing the linear combination of x1 and x2 as suggested +is equivalent to sampling directly from the resulting distribution. +We obtain the algorithm presented in Section 3.2 by applying Lemma 1 to (10) with +N(µ1, σ2 +1) ← pθ,τ(xa +ℓ | xℓ+1) +N(µ2, σ2 +2) ← N +� +xa +ℓ; √¯αℓˆµa +t + +√ +1 − ¯αℓ¯za +ℓ+1, ¯αℓ((σa +t )2 + ρℓ(τ a +ℓ+1)2) +� +. +A.2 +On SURE-based Regularization +In this part we show how the loss function (7) is related to Stein’s unbiased risk estimate (SURE). We first +note that by definition of the diffusion process, we have xℓ = √¯αℓx0 + √1 − ¯αℓ ¯zℓ where ¯zℓ is a random +variable following the distribution N(0, 1). Moreover, √¯αℓhθ(xℓ, ℓ) is an estimator of √¯αℓx0 from xℓ. The +corresponding SURE thus writes +SURE(√¯αℓhθ(·, ℓ)) = ∥√¯αℓhθ(xℓ, ℓ) − xℓ∥2 − K(1 − ¯αℓ) + 2(1 − ¯αℓ) divxℓ(√¯αℓhθ(xℓ, ℓ)). +If it holds xℓ = √¯αℓy0 while y0 follows the distribution N(x0, σ2I), we get immediately 1 − ¯αℓ = ¯αℓσ2. The +above can thus be rewritten as +SURE(√¯αℓhθ(·, ℓ)) = ∥√¯αℓhθ(xℓ, ℓ) − √¯αℓy0∥2 − K¯αℓσ2 + 2¯α +3 +2 +ℓ σ2 divxℓ(hθ(xℓ, ℓ)). +Dividing the above by ¯αℓ we get an unbiased estimate of E[∥hθ(xℓ, ℓ) − x0∥2], i.e., +E[∥hθ(xℓ, ℓ) − x0∥2] = E[∥hθ(xℓ, ℓ) − y0∥2 − Kσ2 + 2√¯αℓσ2 divxℓ(hθ(xℓ, ℓ))]. +On the right hand side inside expectation we recover Eq. (7) with m = 1 and λ = 1 by replacing xℓ by ˜xℓ +and the divergence by its Monte-Carlo approximation [Ramani et al., 2008]. +B +Missing Experimental Details +In this section, we provide missing experimental details mainly concerning the construction of the problem +instances and the learning of priors. All the simulations are run on an Amazon p3.2xlarge instance equipped +with 8 NVIDIA Tesla V100 GPUs. +B.1 +Construction of Bandit Instances +We provide below more details on how the bandit instances are constructed in our problems. Besides the +three problems described in Section 5, we consider an additional Labeled Arms problem that will be used +for our ablation study. Some illustrations of the constructed instances and the vectors generated by learned +priors are provided in Appendix E. As in Popular and Niche and 2D Maze problems, in the Labeled Arms +problem we simply add Gaussian noise of standard deviation 0.1 to the mean when sampling the reward. For +these three problems we thus only explain how the means are constructed. +21 + +1. Popular and Niche (K = 200 arms). The arms are split into 40 groups of equal size. 20 of these +groups represent the ‘popular’ items while the other 20 represent the ‘niche’ items. For each bandit task, +we first construct a vector ¯µ whose coordinates’ values default to 0. However, we randomly choose 1 to +3 groups of niche items and set the value of each of these items to 1 with probability 0.7 (independently +across the selected items). Similarly, we randomly choose 15 to 17 groups of popular items and set their +values to 0.8. Then, to construct the mean reward vector µ, we perturb the values of ¯µ by independent +Gaussian noises with standard deviation of 0.1. After that, we clip the values of the popular items to +make them smaller than 0.95 and clip the entire vector to the range [0, 1]. +2. iPinYou bidding (K = 300 arms). The set of tasks is constructed with the help of the iPinYou data set +[Liao et al., 2014]. This data set contains logs of ad biddings, impressions, clicks, and final conversions, +and is separated into three different seasons. We only use the second season that contains the ads from +5 advertisers (as we are not able to find the data for the first and the third season). To form the tasks, +we further group the bids according to the associated ad slots. By keeping only those ad slots with at +least 1000 bids, we obtain a data set of 1352 ad slots. Then, the empirical distribution of the paying +price (i.e., the highest bid from competitors) of each ad slot is used to computed the success rate of +every potential bid b ∈ {0, . . . , 299} set by the learner. The reward is either 300 − b when the learner +wins the auction or 0 otherwise. Finally, we divide everything by the largest reward that the learner +can ever get in all the tasks to scale the rewards to range [0, 1]. +3. 2D Maze (K = 180 base arms). For this problem, we first use the code of the github repository +MattChanTK/gym-maze10 to generate random 2D mazes of size 19 × 19. Then, each bandit task can +be derived from a generated 2D maze by associating the maze to a weighted 10 × 10 grid graph. As +demonstrated by Figure 4, each case corresponds to either a node or an edge of the grid graph. Then, +the weight (mean reward) of an edge (base arms) is either −1 or −0.01 depending on either there is a +wall (in black color) or not (in white color) on the corresponding case. An optimal arm in this problem +would be a path that goes from the source to the destination without bumping into any walls in the +corresponding maze. +4. Labeled Arms (K = 500 arms). This problem is again inspired by applications in recommendation +systems. We are provided here a set of 50 labels L = {1, ..., 50}. Each arm is associated to a subset La +of these labels with size card(La) = 7. To sample a new bandit task B, we randomly draw a set LB ⊆ L +again with size 7. Then for each arm a, we set ¯µa = 1 − 1/4card(La ∩ LB) so that the more the two sets +intersect the higher the value. Finally, to obtain the mean rewards µ, we perturb the coordinates of ¯µ +by independent Gaussian noises of standard deviation 0.1 and scale the resulting vector to the range +[0, 1]. +Training, Calibration, and Test Sets. +Training, calibration, and test set are constructed for each of +the considered problem. Their size are fixed at 5000, 1000, 100 for the Popular and Niche, 2D Maze, and +Labeled Arms Problems, and at 1200, 100, and 52 for the iPinYou Bidding problem. +B.2 +Diffusion Models– Model Design +In all our experiments (including the ones described in Appendices C and D), we set the diffusion steps of +the diffusion models to L = 100 and adopt a linear variance schedule that varies from 1 − α1 = 10−4 to +1 − αL = 0.1. The remaining details are customized to each problem, taking into account the specificity of +the underlying data distribution. +1. Labeled Arms and Popular and Niche. These two problems have the following two important features: +(i) The expected means of the bandit instances do not exhibit any spatial correlations (see Figures 16a +and 17a). (ii) The values of the expected means are nearly binary. +10https://github.com/MattChanTK/gym-maze +22 + +The first point prevents us from using the standard U-Net architecture. Instead, we consider an architecture +adapted from Kong et al. [2020], Rasul et al. [2021], with 5 residual blocks and each block containing 6 +residual channels.11 Then, to account for the lack of spatial correlations, we add a fully connected layer at +the beginning to map the input to a vector of size 128 × 6, before reshaping these vectors into 6 channels +and feeding them to the convolutional layers. In a similar fashion, we also replace the last layer of the +architecture by a fully connected layer that maps a vector of size 128 × 6 to a vector of size K. We find +that these minimal modification already enable the model to perform well on these two problems. +As for the latter point, we follow Chen et al. [2022] and train the denoisers to predict the clean sample x0 +as it is reported in the said paper that this leads to better performance when the data are binary. +2. iPinYou Bidding. As shown in Figure 20, the pattern of this problem looks similar to that of natural +images. We therefore adopt the standard U-Net architecture, with an adaption to the 1-dimensional case +as described by [Janner et al., 2022]. The model has three feature map resolutions (from 300 to 75) and +the number of channels for each resolution is respectively 16, 32, and 64. No attention layer is used. The +denoiser is trained to predict noise as in Ho et al. [2020], Song and Ermon [2019]. +3. 2D Maze As explained in Appendix B.1 and illustrated in Figure 4, the weighted grid graphs are themselves +derived by the 2D mazes. We can accordingly establish a function that maps each 10 × 10 weighted grid +graph to an image of size 19 × 19 and vice-versa— it suffices to match the value of each associated (edge, +pixel) pair. For technical reason, we further pad the 19 × 19 images to size 20 × 20 by adding one line of +−1 at the right and one row of −1 at the bottom (see Figure 21). We then train diffusion models to learn +the distribution of the resulting images. For this, we use a 2-dimensional U-Net directly adapted from the +ones used by Ho et al. [2020]. The model has three feature map resolutions (from 20 × 20 to 5 × 5) and +the number of channels for each resolution is respectively 32, 64, and 128. A self-attention block is used at +every resolution. We again train the denoiser to predict the clean sample x0 as we have binary expected +rewards here (−0.01 or −1). +B.3 +Diffusion Models– Training +Through out our experiments, we use Adam optimizer with learning rate 5 × 10−4 and exponential decay +rates β1 = 0.9 and β2 = 0.99. The batch size and the epsilon constant in SURE-based regularization are +respectively fixed at 128 and ϵ = 10−5. When the perfect data sets Dtr and Dcal are provided, we simply +train the diffusion models for 15000 steps on the training set Dtr and apply Algorithm 2 on the calibration +set Dcal to calibrate the variances. The training procedure is more complex when only imperfect data are +available. We provide the details below. +Posterior Sampling. +As explained in Section 4 and Algorithm 5, to train from imperfect data we sample +the entire chain of diffused samples ˜x0:L from the posterior. However, while Algorithm 3 performs sampling +with predicted noise ¯zℓ+1 and as we will show in Appendix D.2, this indeed leads to improved performance +in a certain aspect, we observe that when used for training, it prevents the model from making further +progress. We believe this is because in so doing we are only reinforcing the current model with their own +predictions. Therefore, to make the method effective, in our experiments we slightly modify the posterior +sampling algorithm that is used during training. While we still construct samples x0:L following Algorithm 3, +the samples ˜x0:L used for the loss minimization phase are obtained by replacing ¯zℓ+1 (line 9) by ˜zℓ+1 sampled +from N(0, I) in the very last sampling step. That is, from xℓ+1 we sample both xℓ for further iterations of +the algorithm and ˜xℓ to be used for loss minimization. +Training Procedure Specification. +When training and validation data are incomplete and noisy, we +follow the training procedure described in Algorithm 5 with default values S = 15000 warm-up steps, J = 3 +repeats, and S′ = 3000 steps within each repeat (thus 24000 steps in total). Moreover, during the warm-up +phase we impute the missing value with constant 0.5 when constructing the diffused samples ˜xℓ. As for +11These numbers are rather arbitrary and do not seem to affect much our results. +23 + +Figure 6: The three paths (super-arms) for UCB initialization in the 2D Maze experiment. +the regularization parameter λ, we fix it at 0.1 for the Popular and Niche, 2D Maze, and Labeled Arms +problems. +Nevertheless, training from imperfect data turns out to be difficult for the iPinYou Bidding problem. We +conjecture this is both because the training set is small and because we train the denoiser to predict noise +here. Two modifications are then brought to the above procedure to address the additional difficulty. First, +as SURE-based regularization can prevent the model from learning any pattern from data when information +is scarce, we drop it for the warm-up phase and the first two repeats (i.e., the first 21000 steps). We then get +a model that has learned the noisy distribution. We then add back SURE-based regularization with λ = 0.25 +in the third repeat. After the 24000 steps, the model is good enough at reconstructing the corrupted data set, +but the unconditionally generated samples suffer from severe mode collapse. Provided that the reconstructed +samples are already of good quality, we fix the latter issue simply by applying standard training on the +reconstructed samples for another 3000 steps (thus 27000 training steps in total). +B.4 +Other Details +In this part we provide further details about the evaluation phase and the baselines. +Assumed Noise Level. +All the bandit algorithms considered in our work take as input a hyperparameter +ˆσ that should roughly be in the order of the scale of the noise. For the results presented in Section 5, we set +ˆσ = 0.1 for the Popular and Niche and 2D Maze problems and ˆσ = 0.2 for the iPinYou Bidding problem. +The former is exactly the ground truth standard deviation of the underlying noise distribution. For the +iPinYou Bidding problem the noise is however not Gaussian, and ˆσ = 0.2 is approximately the third quartile +of the empirical distribution of the expected rewards’ standard deviations (computed across tasks and arms). +In Appendix D.1, we present additional results for algorithms run with different assumed noise levels ˆσ. +UCB. +The most standard implementation of the UCB algorithm sets the upper confidence bound to +U a +t = ˆµa +t + ˆσ +� +2 log t +N a +t +. +(12) +Instead, in our experiments we use U a +t = ˆµa +t +ˆσ/ +� +N a +t . Eq. (12) is more conservative than our implementation, +and we thus do not expect it to yield smaller regret within the time horizon of our experiments. +UCB Initialization. +In contrary to Thompson sampling-based methods, UCB typically requires an +initialization phase. For vanilla multi-armed bandits (Popular and Niche, iPinYou Bidding, and Labeled +Arms) this simply consists in pulling each arm once. For combinatorial bandits we need to pull a set of super +arms that covers all the base arms. In the 2D Maze experiment we choose the three paths shown in Figure 6. +24 + +Gaussian Prior with Imperfect Data. +To fit a Gaussian on incomplete and noisy data, we proceed as +follows: First, we compute the mean of arm a from those samples that have observation for a. Next, in a +similar fashion, the covariance between any two arms are only computed with samples that have observations +for both arms. Let the resulting matrix be ˆΣ. Since the covariance matrix of the sum of two independently +distributed random vectors (in our case X0 and noise) is the sum of the covariance matrices of the two +random vectors, we further compute ˆΣ′ = ˆΣ − σ2I as an estimate of the covariance matrix of X0. Finally, as +ˆΣ′ is not necessarily positive semi-definite and can even have negative diagonal entries, for TS with diagonal +covariance matrix we threshold the estimated variances to be at least 0 and for TS with full covariance matrix +we threshold the eigenvalues of the estimated covariance matrix ˆΣ′ to be at least 10−4.12 +Arm Selection in 2D Maze Problem. +All the algorithms we use in the 2D Maze problem first com- +pute/sample some values for each base arm (edge) and then select the super arm (path) that maximizes the +sum of its base arms’ values (for DiffTS we first map the sampled 20 × 20 image back to a weighted graph +and the remaining is the same). Concretely, we implement this via Dijkstra’s shortest path algorithm applied +to the weighted graphs with weights defined as the opposite of the computed/sampled values. However, these +weights are not guaranteed to be non-negative, and we thus clip all the negative values to 0 before computing +the shortest path. +C +Ablation Study +In this appendix, we perform ablation studies on the Popular and Niche and Labeled Arms problems to +explore the impacts of various design choices of our algorithms. +C.1 +Predicted versus Sampled Noise in Posterior Sampling +In the DiffTS scheme that we develop (Algorithms 3 and 4), we propose to use the predicted noise ¯zℓ+1 in the +construction of the diffused observation ˜yℓ. Alternatively, we can replace it by the sampled noise vector ˜zℓ+1 +(the resulting algorithm then becomes very similar to the one proposed in Song et al. [2021]). In Figure 9, we +investigate how this decision affects the performance of DiffTS with diffusion priors trained on perfect data +set Dtr. It turns out that for the two problems considered here, there is not clear winner between the two +options. However, it seems that using only sampled noise produces noisier samples, which leads to significant +increase in regret in the Labeled Arms problem. We further confirm this intuition in Appendix D.2, where +we show on a toy problem that the use of predicted noise often leads to samples that are more consistent +with the learned prior. However, this does not always lead to performance improvement in bandit problems +as the learned prior is never perfect. +C.2 +Importance of Variance Calibration +Throughout our work, we have highlighted multiple times the importance of equipping the diffusion model +with a suitable variance estimate. We demonstrate this in Figure 8. We consider diffusion priors trained +on the perfect data set Dtr along with three different reverse variance schedules: (i) calibrated, i.e., Eq. (2); +(ii) non-calibrated, i.e., Eq. (1); (iii) partially calibrated– precisely, only the variance of X0 | x1 is calibrated. +We see clearly that a non-calibrated reverse variance schedule leads catastrophic regret performance. This is +because the sampling process relies too much on the learned model; in particular, the variance of pθ(X0 | x1) +is fixed at zero. Instead, calibrating X0 | x1 itself already leads to significant decrease in regret, making it +as competitive as (and sometimes even better than) the fully calibrated alternative. This suggests that the +trade-off between the learned model and the observations mainly occurs at the last reverse step, whereas +enlarging the variance of the remaining reverse steps has little to no effect. [Yet, it is also clear from the +12Our implementation requires the prior covariance matrix to be positive definite. +25 + +0 +1000 +2000 +3000 +4000 +5000 +# Iterations +0 +50 +100 +150 +200 +Regret +GTS-full +DiffTS predicted noise +DiffTS sampled noise +Labeled Arms ˆσ = 0.1 +0 +1000 +2000 +3000 +4000 +5000 +# Iterations +0 +20 +40 +60 +80 +100 +120 +140 +160 +Regret +GTS-full +DiffTS predicted noise +DiffTS sampled noise +Labeled Arms ˆσ = 0.05 +0 +1000 +2000 +3000 +4000 +5000 +# Iterations +0 +20 +40 +60 +80 +100 +120 +140 +Regret +GTS-full +DiffTS predicted noise +DiffTS sampled noise +Popular & Niche ˆσ = 0.1 +0 +1000 +2000 +3000 +4000 +5000 +# Iterations +0 +20 +40 +60 +80 +100 +120 +140 +Regret +GTS-full +DiffTS predicted noise +DiffTS sampled noise +Popular & Niche ˆσ = 0.05 +Figure 7: Regret comparison for DiffTS with predicted or independently sampled noise in the construction of diffused +observation ˜yℓ. +0 +1000 +2000 +3000 +4000 +5000 +# Iterations +0 +100 +200 +300 +400 +500 +600 +700 +Regret +GTS-full +DiffTS calibrated +DiffTS non-calibrated +DiffTS p(x0|x1) calibrated +Labeled Arms ˆσ = 0.1 +0 +1000 +2000 +3000 +4000 +5000 +# Iterations +0 +100 +200 +300 +400 +500 +600 +700 +Regret +GTS-full +DiffTS calibrated +DiffTS non-calibrated +DiffTS p(x0|x1) calibrated +Labeled Arms ˆσ = 0.05 +0 +1000 +2000 +3000 +4000 +5000 +# Iterations +0 +20 +40 +60 +80 +100 +120 +140 +Regret +GTS-full +DiffTS calibrated +DiffTS non-calibrated +DiffTS p(x0|x1) calibrated +Popular & Niche ˆσ = 0.1 +0 +1000 +2000 +3000 +4000 +5000 +# Iterations +0 +20 +40 +60 +80 +100 +120 +140 +Regret +GTS-full +DiffTS calibrated +DiffTS non-calibrated +DiffTS p(x0|x1) calibrated +Popular & Niche ˆσ = 0.05 +Figure 8: Regret comparison for DiffTS with three different types of reverse variance schedules. +0 +1000 +2000 +3000 +4000 +5000 +# Iterations +0 +50 +100 +150 +200 +Regret +GTS-full +DiffTS perfect data +DiffTS λ = 0 +DiffTS λ = 0.1 +DiffTS λ = 0.5 +Labeled Arms ˆσ = 0.1 +0 +1000 +2000 +3000 +4000 +5000 +# Iterations +0 +20 +40 +60 +80 +100 +120 +140 +Regret +GTS-full +DiffTS perfect data +DiffTS λ = 0 +DiffTS λ = 0.1 +DiffTS λ = 0.5 +Labeled Arms ˆσ = 0.05 +0 +1000 +2000 +3000 +4000 +5000 +# Iterations +0 +20 +40 +60 +80 +100 +120 +140 +160 +Regret +GTS-full +DiffTS perfect data +DiffTS λ = 0 +DiffTS λ = 0.1 +DiffTS λ = 0.5 +Popular & Niche ˆσ = 0.1 +0 +1000 +2000 +3000 +4000 +5000 +# Iterations +0 +20 +40 +60 +80 +100 +120 +140 +Regret +GTS-full +DiffTS perfect data +DiffTS λ = 0 +DiffTS λ = 0.1 +DiffTS λ = 0.5 +Popular & Niche ˆσ = 0.05 +Figure 9: Regret comparison for DiffTS trained on noisy data with different regularization weight λ. +0 +1000 +2000 +3000 +4000 +5000 +# Iterations +0 +50 +100 +150 +200 +Regret +GTS-full +DiffTS perfect data +DiffTS without EM +DiffTS EM +Labeled Arms ˆσ = 0.1 +0 +1000 +2000 +3000 +4000 +5000 +# Iterations +0 +20 +40 +60 +80 +100 +120 +140 +Regret +GTS-full +DiffTS perfect data +DiffTS without EM +DiffTS EM +Labeled Arms ˆσ = 0.05 +0 +1000 +2000 +3000 +4000 +5000 +# Iterations +0 +20 +40 +60 +80 +100 +120 +140 +160 +Regret +GTS-full +DiffTS perfect data +DiffTS without EM +DiffTS EM +Popular & Niche ˆσ = 0.1 +0 +1000 +2000 +3000 +4000 +5000 +# Iterations +0 +20 +40 +60 +80 +100 +120 +140 +Regret +GTS-full +DiffTS perfect data +DiffTS without EM +DiffTS EM +Popular & Niche ˆσ = 0.05 +Figure 10: Regret comparison for DiffTS trained on incomplete data with or without EM. +0 +1000 +2000 +3000 +4000 +5000 +# Iterations +0 +50 +100 +150 +200 +Regret +GTS-full +DiffTS perfect data +DiffTS without EM +DiffTS EM +DiffTS EM without SURE +Labeled Arms ˆσ = 0.1 +0 +1000 +2000 +3000 +4000 +5000 +# Iterations +0 +20 +40 +60 +80 +100 +120 +140 +160 +Regret +GTS-full +DiffTS perfect data +DiffTS without EM +DiffTS EM +DiffTS EM without SURE +Labeled Arms ˆσ = 0.05 +0 +1000 +2000 +3000 +4000 +5000 +# Iterations +0 +25 +50 +75 +100 +125 +150 +175 +Regret +GTS-full +DiffTS perfect data +DiffTS without EM +DiffTS EM +DiffTS EM without SURE +Popular & Niche ˆσ = 0.1 +0 +1000 +2000 +3000 +4000 +5000 +# Iterations +0 +20 +40 +60 +80 +100 +120 +140 +Regret +GTS-full +DiffTS perfect data +DiffTS without EM +DiffTS EM +DiffTS EM without SURE +Popular & Niche ˆσ = 0.05 +Figure 11: Regret comparison for DiffTS trained on noisy and incomplete data with or without EM / SURE-based +regularization. +26 + +experiment on the Popular and Niche problem with presumed noise standard deviation 0.5 that calibrating +the variance of all the reverse steps may still be beneficial in some situation.] +C.3 +Ablation Study for Training from Imperfect Data +Our algorithm for training from imperfect data (Algorithm 5) makes two important modifications to the +original training scheme: the Expectation Maximization-like procedure (abbreviated as EM hereinafter) and +the use of SURE-based regularization. Below we discuss their effects for three types of data: noisy data, +incomplete data, and noisy and incomplete data. We fix all the hyper-parameters to the ones used in the +main experiment unless otherwise specified. In particular, we set the noise standard deviation to 0.1 for noisy +data and the missing rate to 0.5 for incomplete data. +For comparison, we also plot the regrets for the full covariance Gaussian prior baseline. The means and +the covariance of the prior are fitted with the three types of imperfect data that are used to train and calibrate +the diffusion models, following the procedure detailed in Appendix B.4. +Training from Noisy Data. +To cope with noisy data, we add SURE-based regularization with weight +λ to our training objective (7). In this part, we focus on how the choice of λ affects the regret when the +data are noisy. For the sake of simplicity, we only complete the warm-up phase of the algorithm, that is, +the models are only trained for 15000 steps with loss function L and xℓ sampled from Xℓ | X0 = y0. In our +experiments we note this is generally good enough for noisy data without missing entries. +The results are shown in Figure 9. As we can see, the value of λ has a great influence on the regret +achieved with the learned prior. However, finding the most appropriate λ for each problem is a challenging +task. Using a larger value of λ helps greatly for the Labeled Arms problem when it is given the ground-truth +standard deviation σ = 0.1, but is otherwise harmful for the Popular and Niche problem. We believe that +finding a way to determine the adequate value of λ will be an important step to make our method more +practically relevant. +Training from Incomplete Data. +The EM step is mainly designed to tackle missing data. In Figure 10 +we show how the induced regrets differ when the models are trained with and without it and when the +observations are missing at random but not noisy. To make a fair comparison, we also train the model for a +total of 24000 (instead of 15000) steps when EM is not employed. As we can see, in all the setups the use of +EM results in lower regret. +Training from Incomplete and Noisy Data. +To conclude this section we investigate the effects of EM +and SURE-based regularization when the data are both noisy and incomplete, as in our main experiment. +We either drop totally the regularization term, i.e., set λ = 0, or skip the EM step (but again we train the +models for 24000 steps with the configuration of the warm-up phase in this case). We plot the resulting +regrets in Figure 11. For the models without EM, the variance calibration algorithm proposed in Section 4.2 +(Algorithm 6) does not work well so we calibrate it with a perfect calibration set Dcal.13 However, even +with this the absence of EM consistently leads to the worst performance. On the other hand, dropping the +regularization term only causes clear performance degradation for the Labeled Arms problem. This is in line +with our results in Figure 9. +D +Additional Experiments +In this appendix, we first supplement our numerical section Section 5 with results obtained under different +assumed noise levels. After that, we present additional experiments for the posterior sampling and the +13Indeed, by design Algorithm 6 only gives good result when the posterior sampling step provides a reasonable approximation +of x0. How to calibrate the variance of a poorly performed model from imperfect data is yet another difficult question to be +addressed. +27 + +0 +1000 +2000 +3000 +4000 +5000 +# Iterations +0 +25 +50 +75 +100 +125 +150 +175 +Regret +DiffTS (ours) +UCB +GTS-diag +GTS-full +GMMTS-10 +GMMTS-25 +Perfect data ˆσ = 0.05 +0 +1000 +2000 +3000 +4000 +5000 +# Iterations +0 +50 +100 +150 +200 +Regret +DiffTS (ours) +UCB +GTS-diag +GTS-full +GMMTS-10 +GMMTS-25 +Perfect data ˆσ = 0.1 +0 +1000 +2000 +3000 +4000 +5000 +# Iterations +0 +25 +50 +75 +100 +125 +150 +175 +Regret +DiffTS (Ours) +UCB +GTS-diag +GTS-full +Imperfect data ˆσ = 0.05 +0 +1000 +2000 +3000 +4000 +5000 +# Iterations +0 +50 +100 +150 +200 +Regret +DiffTS (Ours) +UCB +GTS-diag +GTS-full +Imperfect data ˆσ = 0.1 +(a) Popular and Niche +0 +1000 +2000 +3000 +4000 +5000 +# Iterations +0 +20 +40 +60 +80 +100 +120 +140 +160 +Regret +DiffTS (ours) +UCB +GTS-diag +GTS-full +GMMTS-10 +GMMTS-25 +Perfect data ˆσ = 0.1 +0 +1000 +2000 +3000 +4000 +5000 +# Iterations +0 +50 +100 +150 +200 +Regret +DiffTS (ours) +UCB +GTS-diag +GTS-full +GMMTS-10 +GMMTS-25 +Perfect data ˆσ = 0.2 +0 +1000 +2000 +3000 +4000 +5000 +# Iterations +0 +50 +100 +150 +200 +250 +Regret +DiffTS (ours) +UCB +GTS-diag +GTS-full +GMMTS-10 +GMMTS-25 +Perfect data ˆσ = 0.3 +0 +1000 +2000 +3000 +4000 +5000 +# Iterations +0 +20 +40 +60 +80 +100 +120 +140 +160 +Regret +DiffTS (ours) +UCB +GTS-diag +GTS-full +Imperfect data ˆσ = 0.1 +0 +1000 +2000 +3000 +4000 +5000 +# Iterations +0 +50 +100 +150 +200 +Regret +DiffTS (ours) +UCB +GTS-diag +GTS-full +Imperfect data ˆσ = 0.2 +0 +1000 +2000 +3000 +4000 +5000 +# Iterations +0 +50 +100 +150 +200 +250 +Regret +DiffTS +UCB +GTS-diag +GTS-full +Imerfect data ˆσ = 0.3 +(b) iPinYou Bidding +0 +1000 +2000 +3000 +4000 +5000 +# Iterations +0 +200 +400 +600 +800 +1000 +1200 +1400 +Regret +DiffTS (ours) +UCB +GTS-diag +GTS-full +GMMTS-10 +GMMTS-25 +Perfect data ˆσ = 0.05 +0 +1000 +2000 +3000 +4000 +5000 +# Iterations +0 +200 +400 +600 +800 +1000 +1200 +Regret +DiffTS (ours) +UCB +GTS-diag +GTS-full +GMMTS-10 +GMMTS-25 +Perfect data ˆσ = 0.1 +0 +1000 +2000 +3000 +4000 +5000 +# Iterations +0 +200 +400 +600 +800 +1000 +1200 +Regret +DiffTS (Ours) +UCB +GTS-diag +GTS-full +Imperfect data ˆσ = 0.05 +0 +1000 +2000 +3000 +4000 +5000 +# Iterations +0 +200 +400 +600 +800 +1000 +1200 +Regret +DiffTS (Ours) +UCB +GTS-diag +GTS-full +Imperfect data ˆσ = 0.1 +(c) 2D Maze +0 +1000 +2000 +3000 +4000 +5000 +# Iterations +0 +50 +100 +150 +200 +250 +Regret +DiffTS (ours) +UCB +GTS-diag +GTS-full +GMMTS-10 +GMMTS-25 +Perfect data ˆσ = 0.05 +0 +1000 +2000 +3000 +4000 +5000 +# Iterations +0 +50 +100 +150 +200 +250 +300 +350 +Regret +DiffTS (ours) +UCB +GTS-diag +GTS-full +GMMTS-10 +GMMTS-25 +Perfect data ˆσ = 0.1 +0 +1000 +2000 +3000 +4000 +5000 +# Iterations +0 +50 +100 +150 +200 +250 +Regret +DiffTS (Ours) +UCB +GTS-diag +GTS-full +Imperfect data ˆσ = 0.05 +0 +1000 +2000 +3000 +4000 +5000 +# Iterations +0 +50 +100 +150 +200 +250 +300 +350 +Regret +DiffTS (Ours) +UCB +GTS-diag +GTS-full +Imperfect data ˆσ = 0.1 +(d) Labeled Arms +Figure 12: Regret performances on four different problems with priors fitted/trained on either exact expected +rewards (perfect data) or partially observed noisy rewards (imperfect data) and with different assumed noise levels ˆσ. +The results are averaged over tasks of a test set and shaded areas represent standard errors. +28 + +training algorithms. +D.1 +Experimental Results with Different Assumed Noise Levels +To further validate the benefit of diffusion priors, we conduct experiments for the four problems introduced +in Appendix B.1 under different assumed noise levels. The results are shown in Figure 12. We see that +DiffTS achieves the smallest regret in 15 out of the 18 plots, confirming again the advantage of using diffusion +priors. Moreover, although DiffTS performs worse than either GMMTS or GTS-full in iPinYou Bidding and +Labled Arms for a certain assumed noise level, the smallest regret is still achieved by DiffTS when taking all +the noise levels that we have experimented with into account. +Finally, it is clear from Figure 12 that the choice of the assumed noise level ˆσ also has a great influence +on the induced regret. The problem of choosing an appropriate ˆσ is however beyond the scope of our work. +D.2 +Comparison of Posterior Sampling Strategies on a Toy Problem +In this part, we demonstrate on a toy problem that using predicted noise ¯zℓ+1 to construct the diffused +observation ˜yℓ leads to more consistent examples compared to using independently sampled noise vectors. +Data Set and Diffusion Model Training. +We consider a simple data distribution over R200. The 200 +features are grouped into 20 groups. For each sample, we randomly select up to 6 groups and set the values of +the corresponding features to 1. The remaining features take the value 0. Some samples from this distribution +are illustrated in Figure 13a. As for the diffusion model, the model architecture, hyper-parameters, and +training procedure are taken to be the same as those for the Popular and Niche problem (Appendix B). In +Figure 13b we see that the data distribution is perfectly learned. +Posterior Sampling. +We proceed to investigate the performance of our posterior sampling algorithm on +this example. For this, we form a test set of 100 samples drawn from the same distribution and drop each +single feature with probability 0.5 as shown in Figure 13c. We then conduct posterior sampling with the +learned model using Algorithm 3. To define the diffused observation ˜yℓ, we either follow (4) or replace the +predicted noise ¯zℓ+1 by the sampled noise ˜zℓ+1 in the formula. The corresponding results are shown in +Figures 13d and 13e. As we can see, using predicted noise clearly leads to samples that are more consistent +with both the observations and the learned prior. +To provide a quantitative measure, in the constructed samples we define a group to be ‘relevant’ if the +values of all its features are greater than 0.8. We then compute the recall and precision by comparing the +ground-truth selected groups and the ones identified as relevant. When predicted noise is used, the average +recall and precision are both at 100%. On the other hand, when independently sampled noise is used, the +average recall falls to around 85% (this value varies due to the randomness of the sampling procedure but +never exceeds 90%) while the average precision remains at around 98%. +D.3 +Training from Imperfect Image Data +To illustrate the potential of the training procedure introduced in Section 4.1, we further conduct experiments +on the MNIST and Fashion-MNIST [Xiao et al., 2017] data sets. Both data sets are composed of gray-scale +images of size 28 × 28. MNIST contains hand-written digits whereas Fashion-MNIST contain fashion items +taken from Zalando shopping catalog. Some images of the two data sets are shown in Figures 14a and 15a. +Data Corruption and Experimental Setup. +For our experiments, we scale the images to range [0, 1] +and corrupt the resulting data with missing rate 0.5 (i.e., each pixel is dropped with 50%) and noise of +standard deviation 0.1. As we only use training images, this results in 60000 corrupted images for each of +the two data sets. We further separate 1000 images from the 60000 to form the calibration sets. We then +29 + +(a) 30 samples from the training set. +(b) 30 feature vectors generated by the learned diffusion model. +(c) 30 samples from the test set. Red squares indicate missing values. +(d) Feature vectors reconstructed with learned diffusion model and Algorithm 3 using predicted noise vectors ¯zℓ. The +inputs are the ones shown in 13c. +(e) Feature vectors reconstructed with learned diffusion model and Algorithm 3 using independently sampled noise +vectors ˜zℓ. The inputs are the ones shown in 13c. +Figure 13: Feature vectors of the toy problems presented in Appendix D.2. Rows and columns correspond respectively +to features and samples. For visualization purpose, the features are ordered in a way that those of the same group are +put together. The darker the color the higher the value, with white and black representing respectively 0 and 1. +30 + +0 +les +10 +idw +20 +S +0 +25 +50 +75 +100 +125 +150 +175 +features0 +e +10 +idw +20 +S +0 +25 +50 +75 +100 +125 +150 +175 +featuressample +10 +20 +25 +50 +75 +100 +125 +150 +1/ +features0 +e +10 +idw +20 +S +0 +25 +50 +75 +100 +125 +150 +175 +features0 +les +- +10 +- +20 +S +0 +25 +50 +75 +100 +125 +150 +175 +features(a) Original images +(b) Corrupted images +(c) Modelorig generated +(d) Modelorig reconstructed +(e) Modelcor14 generated +(f) Modelcor14 reconstructed +(g) Modelcor16 generated +(h) Modelcor16 reconstructed +Figure 14: Various images related to the MNIST data set. The three models Modelorig, Modelcor14, and Modelcor16 +are respectively trained on the original data set, on the corrupted data set for 14000 steps, and on the corrupted data +set for 16000 steps (Modelcor16 is trained on top of Modelcor14 for another 2000 steps; see the text for more details). +‘Generated’ means unconditional sampling while ‘reconstructed’ means posterior sampling with Algorithm 3 applied +to the corrupted images shown in (b). +train the diffusion models from these corrupted images following Algorithm 5, with S = 5000 warm-up steps, +J = 3 repeats of the EM procedure, and S′ = 3000 inner steps for each repeat (the total number of training +steps is thus 14000). The learning rate and the batch size are respectively fixed at 10−4 and 128. +For the regularization term, we take λ = 0.2 for MNIST and λ = 0.1 for Fashion-MNIST. The constant ε +is set to 10−5 as before. As in Ho et al. [2020], Song et al. [2020], we note that the use of exponential moving +average (EMA) can lead to better performance. Therefore, we use the EMA model for the posterior sampling +step. The EMA rate is 0.995 with an update every 10 training steps. For comparison, we also train diffusion +models on the original data sets with the aforementioned learning rate and batch size for 10000 steps. Finally, +to examine the influence of the regularization weight λ on the generated images, we consider a third model +for MNIST trained on top of the 14000-step model with corrupted data. For this model, we perform an +additional posterior sampling step and then train for another 2000 steps with λ = 1. The remaining details, +including the model architecture, are the same as those for the 2D Maze experiment. +Results. +In Figures 14 and 15, we show images from the original data set, from the corrupted data set, and +produced by the trained models either by unconditional sampling or data reconstruction with Algorithm 3. +Overall, our models manage to generate images that resemble the ones from the original data set without +overly sacrificing the diversity. +Nonetheless, looking at the samples for Fashion-MNIST we clearly see that a lot of details are lost in the +images generated by or reconstructed with diffusion models. In the case of training from perfect data, this can +clearly be improved with various modifications to the model including change in model architecture, number +of diffusion steps, and/or sampling algorithms [Karras et al., 2022]. This would become more challenging in +the case of training from imperfect data as the image details can be heavily deteriorated by noise or missing +pixels. +On the other hand, the effect of the regularization parameter λ can be clearly seen in the MNIST +31 + +3232 +8 +2 +83 +3 +C3DO +2 +3 +Y3 +3(a) Original images +(b) Corrupted images +(c) Modelorig generated +(d) Modelorig reconstructed +(e) Modelcor generated +(f) Modelcor reconstructed +Figure 15: Various images related to the Fashion-MNIST data set. The two models Modelorig and Modelcor are +respectively trained on the original data set and the corrupted data set. ‘Generated’ means unconditional sampling +while ‘reconstructed’ means posterior sampling with Algorithm 3 applied to the corrupted images shown in (b). +experiment from Figure 14. Larger λ enables the model to produce digits that are more ‘connected’ but could +cause other artifacts. As in any data generation task, the definition of a good model, and accordingly the +appropriate choice of λ, varies according to the context. +To summarize, we believe that the proposed training procedure has a great potential to be applied in +various areas, including training from noisy and/or incomplete image data, as demonstrated in Figures 14 +and 15. However, there is still some way to go in making the algorithm being capable of producing high-equality +samples for complex data distribution. +E +Expected Reward Visualization +In Figures 16 to 21 we provide various visualizations of the bandit mean reward vectors either of the training +sets or generated by the learned priors. +32 + +(a) 40 samples from the perfect training set Dtr. +(b) 40 samples from the perfect training set Dtr, reordered to put the arms of the same group together. The +popular arms are on the right side of the figure. +(c) 40 mean reward vectors generated the diffusion model trained on perfect data, reordered to put the arms of +the same group together. The popular arms are on the right side of the figure. +(d) 50 mean reward vectors generated by the 25-component GMM fitted on perfect data, reordered to put the +arms of the same group together. The popular arms are on the right side of the figure. +Figure 16: Visualization of the mean reward vectors of the Popular and Niche problem. Rows and columns +correspond to tasks and arms. The darker the color the higher the value, with white and black representing respectively +0 and 1. Diffusion models manage to learn the underlying patterns that become recognizable by humans only when +the arms are grouped in a specific way. +33 + +0 +Ks +task +bandit +30 +0 +25 +50 +75 +100 +125 +150 +175 +arms10 + task +bandit +20 +30 +0 +25 +50 +75 +100 +125 +150 +175 +arms0 +10 + task +bandit +20 +30 +0 +25 +50 +75 +100 +125 +150 +175 +arms10 +bandit task +20 +30 +0 +25 +50 +75 +100 +125 +150 +175 +arms(a) 100 samples from the perfect training set Dtr. +(b) 60 samples from the perfect training set Dtr, grouped by labels and showing only 5 labels. Note that each +arm has multiple labels and thus appears in multiple groups. +(c) 60 mean reward vectors generated by the diffusion model trained on perfect data, grouped by labels and +showing only 5 labels. Note that each arm has multiple labels and thus appears in multiple groups. +(d) 60 mean reward vectors generated by the 25-component GMM fitted on perfect data, grouped by labels and +showing only 5 labels. Note that each arm has multiple labels and thus appears in multiple groups. +Figure 17: Visualization of the mean reward vectors of the Labeled Arms problem. Rows and columns correspond +to tasks and arms. The darker the color the higher the value, with white and black representing respectively 0 and 1. +While human eyes can barely recognize any pattern in the constructed vectors, diffusion models manage to learn the +underlying patterns that become recognizable by humans only when the arms are grouped in a specific way. +34 + +tasks +20 +40 +andit +09 +b +80 +0 +100 +200 +300 +400 +armsasks +ta +20 +andit +40 +b +0 +50 +100 +150 +200 +250 +300 +armssks +20 +andit +40 +b +0 +50 +100 +150 +200 +250 +300 +armssks +ta +20 +andit +40 +b +0 +50 +100 +150 +200 +250 +300 +arms(a) 40 samples from the imperfect training set ˇDtr. Red squares indicate missing values. +(b) 40 mean reward vectors generated by the diffusion model trained on imperfect data. +Figure 18: Mean reward vectors of the Popular and Niche problem. Rows and columns correspond to tasks and +arms. For ease of visualization, the arms are reordered so that arms of the same group are put together and popular +arms are on the right of the figures. The darker the color the higher the value, with white and black representing +respectively 0 and 1. +(a) 60 samples from the imperfect training set ˇDtr. Red squares indicate missing values. +(b) 60 mean reward vectors generated by the diffusion model trained on imperfect data. +Figure 19: Mean reward vectors of the Labeled Arms problem. Rows and columns correspond to tasks and arms. +For ease of visualization, the arms are grouped by labels and only arms that are associated to 5 labels are shown. The +darker the color the higher the value, with white and black representing respectively 0 and 1. +35 + +10 +bandit task +30 +0 +25 +50 +100 +125 +150 +175 +arms0 + task +10 +bandit +20 +30 +0 +25 +50 +75 +100 +125 +150 +175 +arms tasks +20 +bandit +40 +0 +50 +100 +150 +200 +250 +300 +armsasks +tas +20 +andit +40 +b +0 +50 +100 +150 +200 +250 +300 +arms(a) 50 samples from the perfect training set Dtr. +(b) 50 mean reward vectors generated by the diffusion model trained on perfect data. +(c) 50 mean reward vectors generated by the 25-component GMM fitted on perfect data. +(d) 50 samples from the imperfect training set ˇDtr. Red squares indicate missing values. +(e) 50 mean reward vectors generated by the diffusion model trained on imperfect data. +Figure 20: Mean reward vectors of the iPinYou Bidding problem. Rows and columns correspond respectively to +tasks and arms. For visualization purpose, we order the tasks by the position of their optimal arm. The darker the +color the higher the value, with white and black representing respectively 0 and 1. +36 + +0 +tasks +20 +bandit +40 +0 +50 +100 +150 +200 +250 +arms0 +: tasks +20 +bandit +40 +0 +50 +100 +150 +200 +250 +arms0 +tasks +20 +bandit +40 +0 +50 +100 +150 +200 +250 +armstasks +2 +bandit +50 +100 +150 +200 +250 +arms0 +tasks +20 +bandit +40 +0 +50 +100 +150 +200 +250 +arms(a) Sample from the perfect training set Dtr. +(b) Sample generated by the diffusion model trained on +perfect data. +(c) Sample generated by the 25-component GMM fitted +on prefect data. +(d) Sample from the imperfect training set ˇDtr. +Red +squares and edges indicate missing values. +(e) Sample generated by the diffusion model trained on +imperfect data. +Figure 21: The weighted grid graphs and the corresponding 2D maze representations of the 2D Maze problem. For +visualization, the weights (mean rewards) are first clipped to [−1, 0]. Then, for the grid graphs darker the color higher +the mean reward (i.e., closer to 0) while for the maze representations it is the opposite. Also note that for the maze +representations only a part of the pixels correspond the the edges of the grid graphs, while the remaining pixels are +filled with default colors (black or white). The red paths indicate the optimal (super-)arms. +37 + +9259.25 +1110555 \ No newline at end of file diff --git a/a9E4T4oBgHgl3EQfoQ0b/content/tmp_files/load_file.txt b/a9E4T4oBgHgl3EQfoQ0b/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9d705863f930b8e9d50b4617029492257fbc21e2 --- /dev/null +++ b/a9E4T4oBgHgl3EQfoQ0b/content/tmp_files/load_file.txt @@ -0,0 +1,1956 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf,len=1955 +page_content='Thompson Sampling with Diffusion Generative Prior Yu-Guan Hsieh ∗ Université Grenoble Alpes yu-guan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='hsieh@univ-grenoble-alpes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='fr Shiva Kasiviswanathan Amazon kasivisw@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='com Branislav Kveton AWS AI Labs bkveton@amazon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='com Patrick Blöbaum Amazon bloebp@amazon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='com Abstract In this work, we initiate the idea of using denoising diffusion models to learn priors for online decision making problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Our special focus is on the meta-learning for bandit framework, with the goal of learning a strategy that performs well across bandit tasks of a same class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' To this end, we train a diffusion model that learns the underlying task distribution and combine Thompson sampling with the learned prior to deal with new tasks at test time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Our posterior sampling algorithm is designed to carefully balance between the learned prior and the noisy observations that come from the learner’s interaction with the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' To capture realistic bandit scenarios, we also propose a novel diffusion model training procedure that trains even from incomplete and/or noisy data, which could be of independent interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Finally, our extensive experimental evaluations clearly demonstrate the potential of the proposed approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' ∗Work done during internship at Amazon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='05182v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='LG] 12 Jan 2023 Contents 1 Introduction 3 2 Preliminaries and Problem Description 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='1 Denoising Diffusion Probabilistic Model .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' 29 E Expected Reward Visualization 32 2 1 Introduction Uncertainty quantification is an integral part of online decision making and forms the basis of various online algorithms that trade-off exploration against exploitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Among these methods, Bayesian approaches allow us to quantify the uncertainty using probability distributions, with the help of the powerful tools of Bayesian inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Nonetheless, their performance is known to be sensitive to the choice of prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' For concreteness, let us consider the problem of stochastic multi-armed bandits (MABs) [Bubeck and Cesa-Bianchi, 2012, Lattimore and Szepesvári, 2020], in which a learner repeatedly pulls one of the K arms from a given set A = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=', K} and receives rewards that depend on the learner’s choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' More precisely, when arm a is pulled at round t, the learner receives reward rt ∈ R drawn from an arm-dependent distribution Pa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' The goal of the learner is either to i) accumulate the highest possible reward over time (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' regret-minimization) or to ii) find the arm with the highest expected reward within a prescribed number of rounds (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' best-arm identification).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' For both purposes, we need to have a reasonable estimate of the arms’ mean rewards µa = Era∼Pa[ra].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' In general, this would require us to pull each arm a certain number of times, which becomes inefficient when K is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' While the no-free-lunch principle prevents us from improving upon this bottleneck in general situations, it is worth noticing that the bandit instances (referred as tasks hereinafter) that we encounter in most practical problems are far from arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' To name a few examples, in recommendation systems, each task corresponds to a user with certain underlying preferences that affect how much they like each item;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' in online shortest path routing, we operate in real-world networks that feature specific characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' In this regard, introducing such inductive bias to the learning algorithm would be beneficial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' In Bayesian models, this can be expressed through the choice of the prior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Moreover, as suggested by the mete-learning paradigm, the prior itself can also be learned from data, which often leads to superior performance [Rothfuss et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=', 2021, Hospedales et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=', 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' This has led to the idea of meta-learning a prior for bandits [Peleg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=', 2022, Cella et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=', 2020, Basu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=', 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' On the other hand, we have recently witnessed the success of deep generative modeling in producing high-quality synthetic data across various modalities [Saharia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=', 2022, Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=', 2021, Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=', 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' The impressive results shows that these models come out as a powerful tool for modeling complex distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' While different models have their own strength and weakness, diffusion models [Sohl-Dickstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=', 2015, Ho et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=', 2020] turn out to be particularly appealing for our use case as its iterative sampling scheme makes it much more flexible to be applied on a downstream task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' In this regard, this paper attempts to answer the following question: Can diffusion models provide better priors to address the exploration-exploitation trade-off in bandits?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Our Contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' In this work, we initiate the idea of using diffusion models to meta-learn a prior for bandit problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Working towards this direction, we make the following contributions: We propose a new Thompson sampling scheme that incorporates a prior represented by a diffusion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' The designed algorithm strikes a delicate balance between the learned prior and bandit observations, bearing in mind the importance of having an accurate uncertainty estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' In particular, the deployment of the diffusion model begins with a variance calibration step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Then, in each round of the interaction we summarize the interaction history by a masked vector of dimension K, and perform posterior sampling with a modified iterative sampling process that makes use of this vector in each step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Standard diffusion model training assumes access to noise-free samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' This is however nearly impossible in most bandit applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' To overcome this limitation, we propose a novel diffusion model training procedure which only utilizes incomplete and/or noisy observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Our method alternates between sampling from the posterior distribution and minimizing a modified loss function that is suited to imperfect data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' We believe that this training procedure could be of interest beyond bandit setup for example in deep generative modeling scenarios where noise-free training data are not accessible or expensive to get.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' 3 We perform extensive experimental evaluations on various synthetic and real datasets to demonstrate the benefit of the considered approach against several baseline methods including Thomspon sampling with Gaussian priors [Thompson, 1933], Thompson sampling with Gaussian mixture model (GMM) priors [Hong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=', 2022b], and UCB [Auer, 2002].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' The results confirm that the use of diffusion prior always leads to improved performance and the improvement is particularly significant when the underlying problem has a complex structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Related Work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Prior to our work, the use of diffusion models in decision making has been explored by Janner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' [2022], Ajay et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' [2022], who used conditional diffusion models to synthesize trajectories in offline decision making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Their approaches demonstrated good performance on various benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' In contrast, our focus is on online decision making, where exploration is crucial for the success of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Additionally, we use diffusion models to learn a task prior, rather than a distribution specific to a single task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' More generally, diffusion models have been used as priors in various areas, primarily for the goal of inverse problem solving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Our posterior sampling algorithm shares some similarity with those presented in previous studies by Sohl-Dickstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' [2015], Song et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' [2021], Chung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' [2022a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Essentially, these algorithms combine each unconditional sampling step with a step that ensures coherence with the observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Alternatively, close form expression for the conditional score function and the conditional reverse step can be derived if we assume the observed noise is carved from the noise of the diffusion process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' This approach was taken by Kawar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' [2021, 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Another solution is to approximate the posterior with a Gaussian distribution, as proposed by Graikos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' [2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' In this case, samples are reconstructed by minimizing a weighted sum of the denoising loss and a constraint loss, rather than using an iterative sampling scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Regarding the algorithmic framework, we build upon the well-known Thompson sampling idea introduced by Thompson [1933] nearly a century ago.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' It has reemerged as one of the most popular algorithms for bandit problems in the last decade due to its simplicity and generality [Chapelle and Li, 2012, Russo and Van Roy, 2014, Russo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=', 2018].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Nonetheless, it is only until more recently that a series of work [Lu and Van Roy, 2019, Simchowitz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=', 2021] provides a through investigation into the influence of the algorithm’s prior, and confirms the benefit of learning a meta-prior in bandits via both empirical and theoretical evidence [Cella et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=', 2020, Basu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=', 2021, Kveton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=', 2021, Peleg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=', 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' The main difference between our work and the above is the use of a more complex prior, which also goes beyond the previously studied mixture prior [Hong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=', 2022b] and multi-layered Gaussian prior [Hong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=', 2022a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' On a slightly different note, a large corpus of work have investigated other ways to encode prior knowledge, including the use of arm hierarchy [Sen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=', 2021], graphs [Valko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=', 2014], or more commonly a latent parameter shared by the arms [Lattimore and Munos, 2014, Maillard and Mannor, 2014, Hong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=', 2020, Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=', 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' All the variables are multi-dimensional unless otherwise specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' For a vector x, xa represents the a-th coordinate of the vector, and x2 represents its coordinate-wise square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' A sequence of vectors (xl)l∈{l1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=',l2} is written as xl1:l2 for conciseness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' [n] denotes the sequence of integers {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=', n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' To distinguish random variables from their instantiation, we represent the former with capital letters and the latter with the corresponding lowercase letters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Conditioning on X = x is then abbreviated as · | x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' A Gaussian distribution centered at µ ∈ Rd with covariance Σ ∈ Rd×d is written as N(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' µ, Σ) or simply N(µ, Σ) if the random variable in question is clear from the context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' 2 Preliminaries and Problem Description In this section, we briefly review denoising diffusion models and introduce our meta-learning for bandits framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='1 Denoising Diffusion Probabilistic Model First introduced by Sohl-Dickstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' [2015] and recently popularized by Ho et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' [2020] and Song and Ermon [2019], denoising diffusion models (or the closely related score-based models) have demonstrated 4 state-of-the-art performance in various data generation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' A large number of variants of these models have been proposed since then.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' In this paper, we primarily follow the notation and formulation of Ho et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' [2020], with minor modifications to suit our purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Intuitively speaking, diffusion models learn to approximate a distribution Q0 over Rd by training a series of denoisers with samples drawn from this distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Writing q for the probability density function (assume everything is Lebesgue measurable for simplicity) and X0 for the associated random variable, we define the forward diffusion process with respect to a sequence of scale factors (αℓ) ∈ (0, 1)L by q(x1:L | x0) = L−1 � ℓ=0 q(xℓ+1 | xℓ), q(Xℓ+1 | xℓ) = N(Xℓ+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' √αℓ+1xℓ, (1 − αℓ+1)Id).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' The first equality suggests that the forward process forms a Markov chain that starts at x0 ∈ Rd, while the second equality implies that the transition kernel is Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Further denoting the product of the scale factors by ¯αℓ = �ℓ i=1 αi, we then have q(Xℓ | x0) = N(Xℓ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' √¯αℓx0, (1 − ¯αℓ)Id).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' The sequence (αℓ) ∈ (0, 1)L is chosen to be decreasing and such that ¯αL ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' We thus expect q(Xℓ) ≈ N(0, Id).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' A denoising diffusion model learns to reverse the diffusion process by optimizing a certain parameter θ that defines a distribution Pθ over random variables X′ 0:L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' The hope is that the marginal distribution Pθ(X′ 0) would be a good approximation of Q0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' In practice, this is achieved by setting pθ(Xℓ) = N(0, Id), enforcing the learned reverse process to be Markovian, and modeling pθ(Xℓ | xℓ+1) as a Gaussian parameterized by1 pθ(Xℓ | xℓ+1) = q(Xℓ | xℓ+1, X0 = hθ(xℓ+1, ℓ + 1)) � �� � ˆx0 = N � Xℓ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' √¯αℓ(1 − αℓ+1) 1 − ¯αℓ+1 ˆx0 + √αℓ+1(1 − ¯αℓ) 1 − ¯αℓ+1 xℓ+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' 1 − ¯αℓ 1 − ¯αℓ+1 (1 − αℓ+1)Id � (1) In the above hθ is the learned denoiser and hθ(xℓ+1, ℓ + 1) is the predicted clean sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='2 Meta-Learning of Bandit Tasks Our work focuses on meta-learning problems in which the tasks are bandit instances drawn from an underlying distribution that we denote by T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' As in standard meta-learning, the goal is to learn an inductive bias from the meta training set that would improve the overall performance of an algorithm on new tasks drawn from the same distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' In the context of this paper, the inductive bias is encoded in the form of a prior distribution that would be used by the Thompson sampling algorithm when the learner interacts with new bandit instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' For the sake of simplicity, we restrict our attention to the multi-armed bandit scenario presented in Section 1, with the additional assumption that the noise in the rewards are Gaussian with known variance σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='3 The only unknown information is thus the vector of the mean rewards µ = (µa)a∈A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' For this specific situation, Thompson sampling takes as input a prior distribution over RK, samples a guess ˜µt of the mean reward vector from the posterior distribution at each round, and pulls arm at ∈ arg maxa∈A ˜µa t in that round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' The posterior distribution itself is determined by both the prior and the interaction history, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=', the sequence of the action-reward pairs (as, rs)s∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=',t−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' As for the meta-training phase, we consider two situations that are distinguished by whether the learner has access to perfect data or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' In the former case, the meta-training set is composed of the exact means Dtr = {µB}B of training tasks B drawn from the distribution T , whereas in the latter case the training set is 1With a slight abuse of notation, we drop the prime from X′ 0:L in the remaining of the work, but one should keep in mind that the distributions of X0:L induced by the forward process and of X′ 0:L modeled by the diffusion model are distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' 2To obtain hθ we typically train a neural network with a U-Net architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' In [Ho et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=', 2020], this network is trained to output the predicted noise ¯zℓ = (xℓ − √¯αℓhθ(xℓ, ℓ))/√1 − ¯αℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' 3We make this assumption as we are using diffusion prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' As far as we are aware, all the existing diffusion model posterior sampling algorithms for the case of Gaussian noise either rely on this assumption or circumvent it by adding some adjustable hyperparameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' How to extend these algorithms to cope with unknown noise variance properly is an interesting open question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' 5 Model Training Variance Calibration Bandit Deployment perfect / imperfect observations of from different tasks expected rewards diffusion model calibrated variances diffusion prior Task Distribution Task action feedback Figure 1: Overview of the meta-learning for bandits with diffusion prior framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' composed of incomplete and/or noisy observations of these vectors (see Section 4 for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' We use the term imperfect data to informally refer to the scenario where the data is incomplete and/or noisy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' The entire algorithm flow is summarized in Figure 1 and Algorithm 1, where both the model training and the variance calibration blocks together define the diffusion prior that is used by Thompson sampling in the deployment phase, as we will immediately see in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='Algorithm 1 Meta-learning for Bandits with Diffusion Models ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='1: Meta-Training Phase a): ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='Diffusion Model Training ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='2: Input: Training set containing reward observations from different tasks ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='3: Train a diffusion model hθ to model the distribution of the mean rewards (in case of imperfect data use ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='Algorithm 5) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='4: Meta-Training Phase b): ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='Variance Calibration ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='5: Input: Diffusion model hθ and calibration set containing reward observations from different tasks ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='6: Use Algorithm 2 to estimate the mean squared reconstruction errors τ 1:L of the model hθ from different ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='diffusion steps to calibrate the variance of each reverse step (in case of imperfect data use Algorithm 6) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='7: Meta-Deployment Phase ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='8: Input: Diffusion model hθ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' reconstruction error τ 1:L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' and assumed noise level ˆσ 9: For any new task,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' run Thompson sampling with diffusion prior (Algorithm 4) with provided parameters 3 Using Trained Diffusion Models in Thompson Sampling In this section,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' we describe how a learned diffusion model can be incorporated as a prior for Thompson sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' For sake of presenting the core ideas, we focus here on the case where clean datasets Dtr and Dcal are used for training and calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' The case where only imperfect datasets are available is addressed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' With a clean dataset Dtr, diffusion model can be trained using well-known techniques, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=', [Ho et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=', 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Then, as outlined in Algorithm 1, given a trained model, the two remaining steps are: a) variance calibration, and b) Thompson sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='1 Variance Calibration While Ho et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' [2020] fixed the variance of pθ(Xℓ | xℓ+1) to that of q(Xℓ | xℓ+1, x0) as expressed by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' (1), it was recently shown by Bao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' [2021] that this choice was sub-optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' This defect turns out to be critical when we use diffusion model as prior in online decision problems, as it falls short in quantifying the right level of uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' To remedy this problem, we follow closely the approach of Bao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' [2022] and calibrate the variances of the reverse process with a separate calibration set Dcal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Precisely, we write 6 pθ,τ(Xℓ | xℓ+1) = � q(Xℓ | xℓ+1, x0)p′ θ,τ(x0 | xℓ+1) dx0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' (2) In the above, p′ θ,τ(X0 | xℓ+1) is a Gaussian distribution centered at the denoiser output ˆx0 = hθ(xℓ+1, ℓ + 1) and τ = τ 1:L is the optimized variance parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' This is different from (1) where instead of p′ θ,τ(x0 | xℓ+1)dx0 we only have a Dirac concentrated at x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' The covariance of p′ θ,τ(X0 | xℓ+1) is taken as the diagonal matrix diag(τ 2 ℓ+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' As for the variance parameter τ ℓ+1, it represents the (coordinate-wise) root mean squared reconstruction error and is computed on the calibration set Dcal by constructing Dcal,ℓ of pairs (x0, xℓ) with x0 ∈ Dcal and xℓ sampled from Xℓ | x0, and setting τ a ℓ = � � x0,xℓ∈Dcal,ℓ ∥xa 0 − ha θ(xℓ, ℓ)∥2/ card(Dcal,ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' The above procedure is summarized in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Intuitively, the calibration step automatically adjusts how much we rely on the learned model in the upcoming tasks by taking the reconstruction error as a proxy for the model’s quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' We opt for a simple model here in which the covariance matrix is the same at all points, whereas Bao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' [2022] fit a neural network to predict the mean squared residual at every xℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Once the reconstruction errors are computed, the covariance of pθ,τ can be derived from (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Algorithm 2 Diffusion Model Variance Calibration 1: Input: Diffusion model hθ, calibration set Dcal = {xi,0}i, noise standard deviation σ 2: Output: Reconstructions errors τ 1:L 3: for ℓ = 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' L do 4: Construct Dcal,ℓ = {xi,0, xi,ℓ}i by sampling xi,ℓ from Xℓ | xi,0 5: for a = 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' K do 6: Set τ a ℓ = �� x0,xℓ∈Dcal,ℓ∥xa 0 − ha θ(xℓ, ℓ)∥2/ card(Dcal,ℓ) 7: end for 8: end for 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='2 Thompson Sampling with Diffusion Prior We next proceed to discuss how to perform Thompson sampling with a diffusion model learned prior pθ,τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' For this, we need to sample from the posterior when the prior is specified as such.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Concretely, for a given evidence y0 of x0 with known q(y0 | x0), we are interested in sampling from X0 | y0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' While an exact solution does not exist in general, we may look at this problem as sampling from the prior mixed with evidence y0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' In this regard, our algorithm gradually guides the sample towards the evidence during the sampling process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' This is achieved by conditioning the reverse Markovian process on Y 0 = y0 and seeks an approximation for each conditional reverse step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' In the case of multi-armed bandits, y0 = (as, rs)s∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=',t} is the interaction history and x0 = µ ∈ RK is the mean reward vector of the task, and it holds that q(y0 | x0) ∝ t� s=1 q(rs | µ, as) = t� s=1 N(rs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' µas, σ2) [as a function of x0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' (3) By the proportionality we hide all the randomness in the learner’s actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' This is legitimate because the learner’s actions only depend on the mean reward vector via their interaction history with the environment, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=', q(as | a1, r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' , as−1, rs−1, µ) = q(as | a1, r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' , as−1, rs−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' The initialization and the recursive steps of our conditional sampling scheme tailored to this situation is then provided below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Detailed derivation behind the algorithm is provided in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Sampling from XL | y0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' For this part, we simply ignore y0 and sample from N(0, Id) as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' 7 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Unconditional generation Conditional generation Result Predicted noise Diffused observation Observation Unknown latent Figure 2: Illustration of the proposed posterior sampling with diffusion prior algorithm (Algorithm 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Warm-up Posterior Sampling Loss Minimization minimize minimize sample sample with current model EM Figure 3: Overview of the proposed training procedure to deal with incomplete and/or noisy data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Sampling from Xℓ | xℓ+1, y0 We first create an unconditional latent variable x′ ℓ by sampling from the unconditional reverse process pθ,τ(Xℓ | xℓ+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' We then perform coordinate-wise operation by distinguishing between the following two situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Arm a has never been pulled in the first t rounds: In this case we just set xa ℓ to be x′a ℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Arm a has been pulled in the first t rounds: We denote by ˆµa t = �t s=1 rs 1{as = a}/N a t as the empirical mean and σa t = σ/ � N a t as the adjusted standard deviation, where N a t is the number of times that arm a has been pulled up to time t (included).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' We also write ζa ℓ,1 for the standard deviation of pθ,τ(Xa ℓ | xℓ+1) and define the diffused observation ˜ya ℓ = √¯αℓˆµa t + √ 1 − ¯αℓ¯za ℓ+1 + ζa ℓ,2˜za ℓ+1 (4) that contains a predicted noise component ¯zℓ+1 satisfying xℓ+1 = √¯αℓ+1hθ(xℓ+1, ℓ+1)+√1 − ¯αℓ+1¯zℓ+1 and an independent noise component with ˜za ℓ+1 sampled from N(0, 1) and further multiplied by ζa ℓ,2 = � ¯αℓ � (σa t )2 + ¯αℓ+1(1 − ¯αℓ) ¯αℓ(1 − ¯αℓ+1)(τ a ℓ+1)2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' (5) The output of the conditional reverse step is then a weighted sum of the diffused observation and the unconditional latent variable45 xa ℓ = (ζa ℓ,1)−2x′a ℓ + (ζa ℓ,2)−2˜ya ℓ (ζa ℓ,1)−2 + (ζa ℓ,2)−2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' As we see above, while the case of single observation with missing entries is clearly a special case of bandit observations, in terms of algorithmic scheme, it becomes equivalent when we summarize the interaction history of bandits with mean ˆµa t and standard deviation σt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Therefore, we present the posterior sampling algorithm for the former situation in Algorithm 3, and depict the induced Thompson sampling algorithm in Algorithm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' It is also important to note that while our algorithms shares similarity with existing posterior sampling methods [Song et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=', 2021, Chung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=', 2022b], we supplement our diffused observation ˜ya ℓ with a predicted noise component that improves the coherence between the observation and the generated sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' 4 Training Diffusion Models from Imperfect Data Standard training procedure of diffusion models require access to a dataset of clean samples Dtr = {xi,0}i∈[n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Nonetheless, in most bandit applications, it is nearly impossible to obtain such dataset as the exact mean 4We set xa ℓ = ˜ya ℓ if ζa ℓ,2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' 5The weighted average is also equivalent to sampling xa ℓ from a certain Gaussian distribution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' see Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='1 for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' 8 Algorithm 3 Posterior Sampling with Diffusion Prior 1: Input: Observation y0 ∈ RK, noise standard deviation σ ∈ RK, binary mask m ∈ {0, 1}K, diffusion model hθ and associated reconstruction errors τ 1:L 2: Output: Posterior sample x0 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' x0:L) approximately sampled from X0 | y0 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' X0:L | y0) 3: Sample xL ∼ N(0, Id) 4: for ℓ ∈ L − 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' 0 do 5: Predict clean sample ˆx0 ← hθ(xℓ+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' ℓ + 1) and associated noise ¯zℓ+1 6: Sample unconditional latent variable xℓ from pθ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='τ(Xℓ | xℓ+1) 7: for a ∈ A such that ma = 1 do 8: Sample ˜za ℓ+1 ∼ N(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' 1) and compute ζa ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='2 following (5) 9: Compute diffused observation ˜ya ℓ ← √¯αℓya 0 + √1 − ¯αℓ¯za ℓ+1 + ζa ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='2˜za ℓ+1 10: Set xa ℓ ← (ζa ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='1)−2xa ℓ +(ζa ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='2)−2˜ya ℓ (ζa ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='1)−2+(ζa ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='2)−2 ▷ ζa ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='1 is the standard deviation of pθ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='τ(Xa ℓ | xℓ+1) 11: end for 12: end for Algorithm 4 Thompson Sampling with Diffusion Prior (DiffTS) 1: Input: Diffusion model hθ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' reconstruction errors τ 1:L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' assumed noise level ˆσ ∈ R 2: for t = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' do 3: Sample ˜x0 using Algorithm 3 with y0 ← ˆµt−1, σ ← σt−1, m defined by ma = 1{N a t−1 > 0} 4: Pull arm at ∈ arg maxa∈A ˜xa 0 5: Update number of pulls N a t , scaled standard deviation σa t , and empirical reward ˆµa t for a ∈ A 6: end for reward vector µ of each single task is never directly observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Instead, one can collect imperfect observations of these vectors, either through previous bandit interactions or forced exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Taking this into account, in this section, we build towards a systematic procedure to train (and calibrate) diffusion models from imperfect (incomplete and/or noisy) data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' It is worth noticing that the application scope of our methodology goes beyond the bandit setup and covers any situation where imperfect data are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' As an example, we apply our approach to train from imperfect images (corrupted MNIST and Fashion-MNIST [Xiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=', 2017] datasets) and obtain promising results (details are provided in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' For ease of exposition, we first focus on the case of uniform noise variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Extension to deal with non-uniform noise variance is later presented in Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' When all the observed noises have the same variance σ2 ∈ R, the samples of the imperfect dataset ˇDtr = {yi,0}i∈[n] can be written as yi,0 = mi ⊙(xi,0 +zi) where mi ∈ {0, 1}K is binary mask, zi is a noise vector sampled from N(0, σ2Id), and ⊙ denotes element-wise multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='6 In the considered bandit problem, such dataset can be obtained by randomly pulling a subset of arms once for each arm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' We also assume that the associated masks {mi}i∈[n] and the noise standard deviation σ are known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' We can thus rewrite the dataset as ˇDtr = {yi,0, mi}i∈[n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='1 Training with Imperfect Data In presence of perfect data, diffusion model training optimizes the denoising objective Eℓ∼Uniform({1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=',L}),x0∼Q0,xℓ∼Xℓ | x0[∥x0 − hθ(xℓ, ℓ)∥2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' (6) Nonetheless, neither x0 nor xℓ are available when we only have access to an imperfect dataset ˇDtr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' To tackle these challenges, we propose an expectation-maximization (EM)-type procedure where in the place of the expectation step we perform sampling of latent variables and in the place of the maximization step we 6As we will see Remark 1, the masking of an entry can also be viewed as an observation with infinite variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' 9 minimize a tailored loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' An indicative algorithmic scheme is summarized in Algorithm 5 (without mini-batching, dataset shuffling, and the use of specific optimization algorithm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Posterior Sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Due to the absence of a clean observation of x0, it is impossible to sample xℓ via the forward diffusion process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Nonetheless, we can perform posterior sampling with the current model as done in several variants of stochastic EM [Fort and Moulines, 2003].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' In fact, as explained in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='1, a diffusion model can be regarded as a probability model over the random variables X0:L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' A typical expectation step in EM for a given parameter θ′ requires us to compute the expected log likelihood function Q(θ) = n � i=1 EXi,0:L | yi,0,mi,θ′ log pθ(Xi,0:L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Nonetheless, this is intractable in general due to the use of neural network in the definition of pθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' To circumvent this issue, we can instead sample ˜xi,0:L from the posterior with density pθ′(· | yi,0, mi) and use stochastic gradient ascent to maximize the log likelihood function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Loss Minimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Having obtained the posterior samples, we have the option to either maximize the log-likelihood of �Dtr or minimize the denoising loss � ˜x0:L∈ � Dtr �L ℓ=1∥˜x0 − hθ(˜xℓ, ℓ)∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' However, both of these approaches heavily rely on the samples generated in the posterior sampling step, which can bias the model towards generating low-quality samples during early stages of training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' To address this issue, we propose to consider a loss function that utilizes the actual observation y0 and not the reconstructed sample ˜x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Fix a small value ε and a regularization parameter λ, the new loss function for a sample pair (y0, ˜xℓ) at diffusion step ℓ with associated mask m is defined as L(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' y0, ˜xℓ, m, ℓ) = ∥m ⊙ y0 − m ⊙ hθ(˜xℓ, ℓ)∥2 + 2λ√¯αℓσ2 Eb∼N (0,I) b⊤ �hθ(˜xℓ + εb, ℓ) − hθ(˜xℓ, ℓ) ε � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' (7) The above loss function is composed of two components that address respectively the incompleteness and the noise in the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' First, it handles incomplete (missing) data by only considering the observed entries as determined by the element-wise product with the mask in the first term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Next, to account for noise, we include a regularization term that penalizes the denoiser from varying too much when the input changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Overall, our denoising loss find its roots in a series of work [Metzler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=', 2018, Zhussip et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=', 2019] that investigates the training of denoiser in the absence of ground-truth clean data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' In particular, the expectation here is an approximation of the divergence div˜xℓ(hθ(˜xℓ, ℓ)) that appears in Stein’s unbiased risk estimate (SURE) [Stein, 1981, Eldar, 2008], an unbiased estimator of the mean squared error whose computation only requires the use of noisy samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='7 From a practical viewpoint, the regularization term provides a trade-off between the bias and the variance of the learned model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' When λ is set to 0, the model learns to generate noisy samples, which corresponds to a flatter prior that encourages exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' When λ gets larger, the model tries to denoise from the observed noisy samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' This can however deviate the model from the correct prior and accordingly jeopardize the online learning procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Overall Procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' The complete algorithm for training from imperfect data is presented in Algorithm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' It alternates between the posterior sampling and the loss minimization steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' While any posterior sam- pling algorithm can be used for the former, in our experiments we simply rely on the one presented in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='2 (note that we actually sample the entire chain ˜x0:L in the procedure of sampling ˜x0) to ac- quire posterior samples �Dtr = {˜xi,0:L}i∈S⊆[n] for (a subset of) the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Then, in the loss minimization step we sample from the dataset �D ′ tr = {˜xi,0:L, yi,0, mi}i∈S and use stochastic gradient descent to mini- mize � (˜x0:L,y0,m)∈�D ′ tr �L ℓ=1 L(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' y0, ˜xℓ, m, ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Moreover, we begin with a warm-up phrase where we sample 7When λ = 1, xℓ = ˜xℓ = √¯αℓy0, m = 1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=', all the entries are observed), and the expectation is replaced by the divergence, we recover SURE up to additive constant −Kσ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' See Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='2 for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' 10 Algorithm 5 Diffusion Model Training from Imperfect (Incomplete and/or Noisy) Data 1: Input: Training set ˇDtr = {yi,0, mi}i, calibration set ˇDcal, noise standard deviation σ, number of warm-up, outer, and inner training steps S, J, and S′ 2: Output: Diffusion model hθ 3: Warm-up 4: for s = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' , S do 5: Sample y0, m from ˇDtr 6: Sample ℓ from the uniform distribution over {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=', L} 7: Sample yℓ from Xℓ | X0 = y0 8: Take gradient step to minimize L(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' y0, yℓ, m, ℓ) (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' (7)) 9: end for 10: Main Training Procedure 11: for j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' , J do 12: Posterior Sampling 13: Compute reconstructions errors τ 1:L with Algorithm 6 using ˇDcal 14: Construct �D ′ tr = {˜xi,0:L, yi,0, mi}i with Algorithm 3 15: Loss Minimization 16: for s = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' , S′ do 17: Sample ˜x0:L, y0, m from ˇDtr 18: Sample ℓ from the uniform distribution over {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=', L} 19: Take gradient step to minimize L(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' y0, ˜xℓ, m, ℓ) (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' (7)) 20: end for 21: end for yℓ = √¯αℓy0 + √1 − ¯αℓ˜zℓ with ˜zℓ sampled from N(0, Id) as in standard diffusion model training but replace the mean squared error by the loss function L introduced in (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='8 This initial phase produces better training samples than posterior sampling with randomly initialized model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Remark 1 (Bandit observations / observations with varying variances).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' When the observations come from bandit interactions and each arm can be pulled more than once, we can first summarize the interaction history by the empirical mean and the vector of adjusted standard deviation as suggested in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Therefore, it remains to address the case where the noise vector zi is sampled from N(0, diag(σi2)) for some vector σi ∈ RK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' As the design of our posterior sampling algorithm already takes this into account, the posterior sampling steps of the algorithm remains unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' The only difference would thus lie in the definition of the modified loss (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Intuitively, we would like to give more weights to samples that are less uncertain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' This can be achieved by weighting the loss by the inverse of the variances, that is, we set L′(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' y0, ˜xℓ, m, σ, ℓ) = K � a=1 ma|ya 0 − ha θ(˜xℓ, ℓ)| (σa)2 + 2λ√¯αℓ Eb∼N (0,I) b⊤ �hθ(˜xℓ + εb, ℓ) − hθ(˜xℓ, ℓ) ε � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' (8) To make sure the above loss is always well defined, we may further replace (σa)2 by (σa)2 + δ for some small δ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' It is worth noticing that one way to interpret the absence of observation ma = 0 is to set the corresponding variance to infinite, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=', σa = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' In this case we see there is even no need of m anymore as the coordinates with σa = +∞ would already be given 0 weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Finally, to understand why we choose to weight with the inverse of the variance, we consider a scalar x, and a set of noisy observations y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' , yn respectively drawn from N(x, σ2 1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' , N(x, σ2 n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Then, the maximum likelihood estimate of x is �n i=1 σ2 i yi/(�n i=1 σ2 i ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' 8In our experiments, we impute the missing values of y0 by a non-zero constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' 11 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='2 Variance Calibration with Imperfect Data As mentioned in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='1, a reliable variance estimate of the reverse process is essential for building a good diffusion prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' This holds true not only for the online learning process at test phase, but also for the posterior sampling step of our training procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' The algorithm introduced in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='1 calibrates the variance through perfect data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' In this part, we extend it to operate with imperfect data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Let ˇDcal be a set of imperfect data constructed in the same way as ˇDtr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' We write ˇD a cal = {(y0, m) ∈ ˇDcal : ma = 1} as the subset of ˇDcal for which a noisy observation of the feature at position a is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Our algorithm (outlined in Algorithm 6) is inspired by the following two observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' First, if the entries are missing completely at random, observed ya 0 of ˇD a cal and sampled xa 0 + za with x0 ∼ Q0 and z ∼ N(0, σ2I) have the same distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Moreover, for any triple (x0, y0, xℓ) with y0 = x0 + z, xℓ = √¯αℓx0 + √1 − ¯αℓ ¯zℓ and x0, z, and ¯zℓ sampled independently from Q0, N(0, σ2I), and N(0, I), it holds that E[∥ya 0 − ha θ(xℓ, ℓ)∥2] = E[∥xa 0 − ha θ(xℓ, ℓ)∥2] + σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' We can thus estimate E[∥xa 0 − ha θ(xℓ, ℓ)∥2] if we manage to pair each ya 0 ∈ ˇD a cal with a such xℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' We again resort to Algorithm 3 for the construction of xℓ (referred to as ˜xℓ in Algorithm 6 and hereinafter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Unlike the training procedure, here we first construct ˜x0 and sample ˜xℓ from Xℓ | ˜x0 to decrease the mutual information between ˜xℓ and y0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Nonetheless, the use of our posterior sampling algorithm itself requires a prior with calibrated variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' To resolve the chicken-and-egg dilemma, we add a warm-up step where we precompute the reconstruction errors with Algorithm 2 by treating ˇDcal as the perfect dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' In our experiments, we observe this step yields estimates of the right order of magnitude but not good enough to be used with Thompson sampling, while the second step brings the relative error to as small as 5% compare to the estimate obtained with perfect validation data using Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Algorithm 6 Diffusion Model Variance Calibration from Imperfect (incomplete and/or noisy) Data 1: Input: Diffusion model hθ, calibration set ˇDcal = {yi,0, mi}i, noise standard deviation σ 2: Output: Reconstructions errors τ 1:L 3: Data Set Preprocessing 4: Precompute reconstructions errors τ 1:L with Algorithm 2 and Dcal ← ˇDcal (masks ignored) 5: Construct �Dcal = {˜xi,0, yi,0, mi}i with Algorithm 3 6: Variance Calibration 7: for ℓ = 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' L do 8: Construct �Dcal,ℓ = {˜xi,ℓ, yi,0, mi}i by sampling ˜xi,ℓ from Xℓ | ˜xi,0 9: for a = 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' K do 10: Let �Da cal,ℓ = {˜xℓ, y0 : (˜xℓ, y0, m) ∈ �Dcal,ℓ, ma = 1} 11: Set τ a ℓ = � (� ˜xℓ,y0∈ � Da cal,ℓ∥xa 0 − ha θ(xℓ, ℓ)∥2/ card( �Da cal,ℓ)) − σ2 12: end for 13: end for 5 Numerical Experiments Figure 4: An example task of the 2D Maze prob- lem presented be- low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' The red path indicates the opti- mal (super-)arm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' In this section, we illustrate the benefit of using diffusion prior through numerical experiments on both real and synthetic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Missing experimental details, ablation studies, and additional experiments are presented in Appendices B to D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Problem Construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' To demonstrate the wide applica- bility of our technique,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' we consider here three bandit problems ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='respectively inspired by the applications in recommendation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='3000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='4000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='5000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='# Iterations ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='150 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='Regret ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='DiffTS (ours) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='UCB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='GTS-diag ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='GTS-full ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='GMMTS-10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='3000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='4000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='5000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='# Iterations ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='150 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='Regret ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='DiffTS (ours) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='UCB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='GTS-diag ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='GTS-full ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='GMMTS-25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='3000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='4000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='5000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='# Iterations ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='600 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='800 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='1200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='Regret ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='DiffTS (ours) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='UCB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='GTS-diag ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='GTS-full ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='GMMTS-25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='3000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='4000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='5000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='# Iterations ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='150 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='Regret ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='DiffTS (Ours) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='UCB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='GTS-diag ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='GTS-full ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='(a) Popular and Niche ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='3000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='4000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='5000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='# Iterations ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='150 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='Regret ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='DiffTS (ours) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='UCB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='GTS-diag ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='GTS-full ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='(b) iPinYou Bidding ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='3000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='4000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='5000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='# Iterations ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='600 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='800 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='1200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='Regret ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='DiffTS (Ours) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='UCB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='GTS-diag ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='GTS-full ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='(c) 2D Maze ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='Figure 5: Regret performances on three different problems with priors fitted/trained on either exact expected rewards ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='(top) or partially observed noisy rewards (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' The results are averaged over tasks of a test set and shaded areas represent standard errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' system, online pricing, and online shortest path routing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Detailed description of the problems and some visualization that help understand the problem structures are provided in Appendices B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='1 and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' The first and the third problems listed below rely on synthetic data, where we only specify the construction of the means and the rewards are obtained by perturbing the means with Gaussian noise of standard deviation σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='1, and for the second problem we use the iPinYou dataset [Liao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=', 2014].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Popular and Niche Problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' We consider here the problem of choosing items to recommend to customers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Let K = 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' The arms (items) are separated into 40 groups, each of size 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Among these, 20 groups of arms correspond to the popular items and tend to have high mean rewards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' However, these arms are never the optimal ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' The other 20 groups of arms correspond to the niche items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Most of them have low mean rewards but a few of them (those that match the preferences of the customer) have mean rewards that are higher than that of all the other arms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' iPinYou Bidding Problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' We consider here the problem of setting the bid price in auctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Let v = 300 be the value of the item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Each arm corresponds to a bid price b ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=', 299}, and the reward is either v − b when the learner wins the auction or 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' The reward distribution of a task is then solely determined by the winning rates which are functions of the learner’s bid and the distribution of the highest bid from competitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' For the latter we use the corresponding empirical distributions of 1352 ad slots from the iPinYou bidding data set [Liao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=', 2014] (each ad slot is a single bandit task).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' 2D Maze Problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' We consider here an online shortest path routing problem on grid graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' We formalize it as a reward maximization combinatorial bandit with semi-bandit feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' As shown in Figure 4, the supers arms are the simple paths between the source and the destination (fixed across all the tasks) whereas the base arms are the edges of the grid graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' At each round, the learner picks a super arm and observes the rewards of all the base arms (edges) that are contained in the super arm (path).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Moreover, the edges’ mean rewards in each task are derived from a certain 2D maze.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' The mean reward is −1 when there is a wall on the associated case (marked by the black color) and −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='01 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Training, Baselines, and Evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' To train the diffusion models, for each problem we construct a training set Dtr and a calibration set Dcal that contain the expected means of the tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' We then conduct experiments for the following two configurations: 13 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Learn from perfect data: The priors are learned using Dtr and Dcal that contain the exact mean rewards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Standard training procedure is applied here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Learn from imperfect data: The priors are learned using ˇDtr and ˇDcal that are obtained from Dtr and Dcal by perturbing the samples with noise of standard deviation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='1 and then dropping each feature of a sample with probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' To tackle this challenging situation we adopt the approach proposed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' In terms of bandit algorithms, we compare our method, DiffTS, with UCB, Thompson sampling with Gaussian prior using either diagonal or full covariance matrix (GTS-diag and GTS-full), and Thompson sampling with Gaussian mixture prior [Hong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=', 2022b] with either 10 or 25 components (GMMTS-10 and GMMTS-25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='9 These priors are also learned with the same perfect / imperfect data that we use to train diffusion models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' We however skip the GMM baseline for the imperfect data setup because we are not able to find any existing algorithm that is able to learn a good GMM on the imperfect data that we consider here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' The performance of the algorithms are then evaluated by their average regret on a standalone test set— for a sequence of arms (at)t∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=',T } pulled by an algorithm in a bandit task, the induced regret is RegT = Tµa⋆ − �T t=1 µat, where a⋆ ∈ arg maxa∈A µa is an optimal arm in this task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' The assumed noise level ˆσ is fixed to the same value across all the methods Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' The results are presented in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' For ease of readability, among the two GMM priors (10 and 25 components), we only show the one that achieves smaller regret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' We see clearly that throughout the three problems and the two setups considered here, the proposed DiffTS algorithm always has the best performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' The difference is particularly significant in the Popular and Niche and 2D Maze problems, in which the regret achieved by DiffTS is around two times smaller than that achieved by the best performing baseline method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' This confirms that using diffusion prior is more advantageous in problems with complex task distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' On the other hand, we also observe that the use of GMM prior in these two problems leads to performance worse than that of GTS-full, whereas it yields performance that is as competitive as DiffTS in the iPinYou Bidding problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' This is coherent with the visualizations we make in Appendix E, which shows that the fitted GMM is only capable of generating good samples in the iPinYou Bidding problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' This, however, also suggests that the use of a more complex prior is a double-edged sword, and can lead to poor performance when the data distribution is not faithfully represented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' In Appendix C, we further present ablation studies to investigate the impacts of various components of our algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' In summary, we find out both the variance calibration step and the EM-like procedure for training with imperfect data are the most crucial to our algorithms, as dropping either of the two could lead to severe performance degradation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' We also affirm that the use of SURE-based regularization does lead to smaller regret, but finding the optimal regularization parameter λ is a challenging problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Finally, while the good performance of DiffTS is itself an evidence of the effectiveness of our sampling and training algorithms, we provide additional experiments in Appendix D to show how these methods can actually be relevant in other contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' 6 Concluding Remarks In this work, we argue that the expressivity and flexibility of diffusion models make them a promising choice for representing complex priors in real-world online decision making problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Through numerical experiments, we demonstrate that using a diffusion prior in combination with a proposed Thompson sampling algorithm can significantly reduce the achieved regret in multi-armed bandit problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Additionally, we propose a training procedure for diffusion models that can handle imperfect training data, addressing a common issue in bandit scenarios, and could be applicable elsewhere too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' 9For the 2D Maze problem we consider their combinatorial extensions in which the UCB index / sampled mean of a super arm is simply the sum of the corresponding quantities of the contained base arms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' 14 Looking ahead, our work raises a number of exciting but challenging research questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' One potential extension is to apply our approach to meta-learning problems in contextual bandits or reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' This would involve modeling a distribution of functions or even Markov decision processes by diffusion models, which remains a largely unexplored area despite a few attempts that work toward these purposes [Dutordoir et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=', 2022, Nava et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=', 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Another factor we did not address in our work is the uncertainty of the learned model itself (in contrast to the uncertainty modeled by the model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' When the diffusion model is trained on a limited amount of data, its uncertainty is high, and using it as a fixed prior may lead to poor results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Finally, the posterior sampling algorithm for the diffusion model is a key bottleneck in terms of scaling our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' There has been significant work on accelerating unconditional sampling of diffusion models [Salimans and Ho, 2021, Dockhorn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=', 2022, Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=', 2022], but it is still an open question how to incorporate these techniques into posterior sampling schemes.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' 18 Appendix A Mathematics of Algorithm Design In this appendix we provide mathematical derivations that inspire the design of several components of our algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='1 Recurrent Step in Posterior Sampling from Diffusion Prior We provide below the derivation of the recurrent step of our posterior sampling algorithm (Algorithm 3) that samples from XL | xℓ+1, y0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' For this, we write q(xℓ | xℓ+1, y0) = q(xℓ | xℓ+1)q(y0 | xℓ, xℓ+1) q(y0 | xℓ+1) = q(xℓ | xℓ+1) � q(y0 | x0)q(x0 | xℓ, xℓ+1) dx0 q(y0 | xℓ+1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' (9) The term q(xℓ | xℓ+1) can be simply approximated with pθ,τ(xℓ | xℓ+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' As for the integral, one natural solution is to use q(x0 | xℓ, xℓ+1) = q(x0 | xℓ) ≈ p′ θ,τ(x0 | xℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Then, for example, if q(y0 | x0) = N(y0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' x0, σ2I), we can deduce � q(y0 | x0)p′ θ(x0 | xℓ) dx0 = N(y0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' hθ(xℓ, ℓ), σ2I + diag(τ 2 ℓ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Nonetheless, as the denoiser hθ can be arbitrarily complex, this does not lead to a close form expression to sample xℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Therefore, to avoid the use of involved sampling strategy in the recurrent step, we approximate q(x0 | xℓ, xℓ+1) in a different way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' We first recall that by definition of the diffusion model we may write Xℓ = √¯αℓX0 + √ 1 − ¯αℓ ¯Zℓ and Xℓ+1 = √αℓ+1Xℓ + √ 1 − αℓZℓ+1, where both ¯ Zℓ and Zℓ+1 are random variable with distribution N(0, I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' This leads to Xℓ+1 = √¯αℓ+1X0 + � 1 − ¯αℓ+1 ¯Zℓ+1 where ¯Zℓ+1 = � αℓ+1(1 − ¯αℓ) 1 − ¯αℓ+1 ¯Zℓ + � 1 − αℓ+1 1 − ¯αℓ+1 Zℓ+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Therefore, we may take ¯Zℓ+1 as a reasonable approximation of ¯Zℓ, while sampling ¯Zℓ+1 is basically the same as sampling from p′ θ(X0 | xℓ+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' To summarize, we write q(x0 | xℓ, xℓ+1) = q � ¯Zℓ = xℓ − √¯αℓx0 √1 − ¯αℓ ��� xℓ, xℓ+1 � ≈ q � ¯Zℓ+1 = xℓ − √¯αℓx0 √1 − ¯αℓ ��� xℓ, xℓ+1 � = q � X0 = 1 √¯αℓ+1 � xℓ+1 − � xℓ − √¯αℓx0 � � 1 − ¯αℓ+1 1 − ¯αℓ � ��� xℓ, xℓ+1 � ≈ p′ θ,τ � X0 = 1 √¯αℓ+1 � xℓ+1 − � xℓ − √¯αℓx0 � � 1 − ¯αℓ+1 1 − ¯αℓ � ��� xℓ+1 � = N �� ¯αℓ(1 − ¯αℓ+1) ¯αℓ+1(1 − ¯αℓ)x0 + xℓ+1 √¯αℓ+1 − � 1 − ¯αℓ+1 ¯αℓ+1(1 − ¯αℓ)xℓ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' 19 hθ(xℓ+1, ℓ + 1), diag(τ 2 ℓ+1) � = √ρℓ N � x0 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' 1 √¯αℓ (xℓ − √ 1 − ¯αℓ¯zℓ+1), ρℓ diag(τ 2 ℓ+1) � , where ρℓ = ¯αℓ+1(1 − ¯αℓ)/(¯αℓ(1 − ¯αℓ+1)) and ¯zℓ+1 represents the noise predicted by the denoiser from xℓ+1, that is, ¯zℓ+1 = xℓ+1 − √¯αℓ+1hθ(xℓ+1, ℓ + 1) √1 − ¯αℓ+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' In this way, we have approximated q(x0 | xℓ, xℓ+1) by a Gaussian with diagonal covariance and with mean that depends only linearly on xℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' In the multi-armed bandit setup that we consider here, the relation between y0 the interaction history and x0 = µ the mean reward vector obeys (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' There exists thus C(y0) and �C(y0) such that � q(y0 | x0)q(x0 | xℓ, xℓ+1) dx0 � �� � A = � C(y0) t� s=1 N(rs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' µas, σ2)q(x0 | xℓ, xℓ+1) dx0 = � �C(y0) � a∈A N a t >0 N(ˆµa t ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' µa, (σa t )2)q(x0 | xℓ, xℓ+1) dx0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Using x0 = µ, the aforementioned approximation of q(x0 | xℓ, xℓ+1), and ignoring the multiplicative constant that does not depend on xℓ, we get A ∝ � � a∈A N a t >0 N(ˆµa t ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' xa 0, (σa t )2)q(x0 | xℓ, xℓ+1) dx0 ≈ √ρℓ � � a∈A N a t >0 N(ˆµa t ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' xa 0, (σa t )2) � a∈A N � xa 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' 1 √¯αℓ (xa ℓ − √ 1 − ¯αℓ¯za ℓ+1), ρℓ(τ a ℓ+1)2 � dx0 = √ρℓ � a∈A N a t >0 � N(ˆµa t ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' xa 0, (σa t )2)N � xa 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' 1 √¯αℓ (xa ℓ − √ 1 − ¯αℓ¯za ℓ+1), ρℓ(τ a ℓ+1)2 � dxa 0 = √ρℓ � a∈A N a t >0 N � ˆµa t ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' 1 √¯αℓ (xa ℓ − √ 1 − ¯αℓ¯za ℓ+1), (σa t )2 + ρℓ(τ a ℓ+1)2 � ∝ � a∈A N a t >0 N � xa ℓ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' √¯αℓˆµa t + √ 1 − ¯αℓ¯za ℓ+1, ¯αℓ((σa t )2 + ρℓ(τ a ℓ+1)2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' � Plugging the above into (9), we obtain ˜q(xℓ | xℓ+1, y0) = � a∈A ˜q(xa ℓ | xℓ+1, y0) where ˜q(xa ℓ | xℓ+1, y0) = pθ,τ(xa ℓ | xℓ+1) if a is never pulled and otherwise it is the distribution satisfying ˜q(xa ℓ | xℓ+1, y0) ∝ pθ,τ(xa ℓ | xℓ+1)N � xa ℓ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' √¯αℓˆµa t + √ 1 − ¯αℓ¯za ℓ+1, ¯αℓ((σa t )2 + ρℓ(τ a ℓ+1)2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' (10) To conclude, we resort to the following lemma (see [Papandreou and Yuille, 2010] for more general results).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Let µ1, µ2, σ1, σ2 ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' The following two sampling algorithms are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Sample x directly from the distribution whose density is proportional the product N(µ1, σ2 1)N(µ2, σ2 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Sample x1 from N(µ1, σ2 1), x2 from N(µ2, σ2 2), and compute x = σ−2 1 x1 + σ−2 2 x2/(σ−2 1 + σ−2 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' 20 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' It is well known that the product of two Gaussian PDFs is itself proportional to a Gaussian PDF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Concretely, we have N(µ1, σ2 1)N(µ2, σ2 2) ∝ N �σ−2 1 µ1 + σ−2 2 µ2 σ−2 1 + σ−2 2 , 1 σ−2 1 + σ−2 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' (11) On the other hand, the linear combination of two independent Gaussian variables is also a Gaussian variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' For X1, X2 that follow N(µ1, σ2 1), N(µ2, σ2 2) and X = σ−2 1 X1 + σ−2 2 X2/(σ−2 1 + σ−2 2 ), we can compute E[X] = σ−2 1 E[X1] + σ−2 2 E[X2] σ−2 1 + σ−2 2 = σ−2 1 µ1 + σ−2 2 µ2 σ−2 1 + σ−2 2 , Var[X] = σ−4 1 Var[X1] + σ−4 2 Var[X2] (σ−2 1 + σ−2 2 )2 = σ−2 1 + σ−2 2 (σ−2 1 + σ−2 2 )2 = 1 σ−2 1 + σ−2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Therefore, X follows the distribution of (11) and computing the linear combination of x1 and x2 as suggested is equivalent to sampling directly from the resulting distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' We obtain the algorithm presented in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='2 by applying Lemma 1 to (10) with N(µ1, σ2 1) ← pθ,τ(xa ℓ | xℓ+1) N(µ2, σ2 2) ← N � xa ℓ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' √¯αℓˆµa t + √ 1 − ¯αℓ¯za ℓ+1, ¯αℓ((σa t )2 + ρℓ(τ a ℓ+1)2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='2 On SURE-based Regularization In this part we show how the loss function (7) is related to Stein’s unbiased risk estimate (SURE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' We first note that by definition of the diffusion process, we have xℓ = √¯αℓx0 + √1 − ¯αℓ ¯zℓ where ¯zℓ is a random variable following the distribution N(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Moreover, √¯αℓhθ(xℓ, ℓ) is an estimator of √¯αℓx0 from xℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' The corresponding SURE thus writes SURE(√¯αℓhθ(·, ℓ)) = ∥√¯αℓhθ(xℓ, ℓ) − xℓ∥2 − K(1 − ¯αℓ) + 2(1 − ¯αℓ) divxℓ(√¯αℓhθ(xℓ, ℓ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' If it holds xℓ = √¯αℓy0 while y0 follows the distribution N(x0, σ2I), we get immediately 1 − ¯αℓ = ¯αℓσ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' The above can thus be rewritten as SURE(√¯αℓhθ(·, ℓ)) = ∥√¯αℓhθ(xℓ, ℓ) − √¯αℓy0∥2 − K¯αℓσ2 + 2¯α 3 2 ℓ σ2 divxℓ(hθ(xℓ, ℓ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Dividing the above by ¯αℓ we get an unbiased estimate of E[∥hθ(xℓ, ℓ) − x0∥2], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=', E[∥hθ(xℓ, ℓ) − x0∥2] = E[∥hθ(xℓ, ℓ) − y0∥2 − Kσ2 + 2√¯αℓσ2 divxℓ(hθ(xℓ, ℓ))].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' On the right hand side inside expectation we recover Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' (7) with m = 1 and λ = 1 by replacing xℓ by ˜xℓ and the divergence by its Monte-Carlo approximation [Ramani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=', 2008].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' B Missing Experimental Details In this section, we provide missing experimental details mainly concerning the construction of the problem instances and the learning of priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' All the simulations are run on an Amazon p3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='2xlarge instance equipped with 8 NVIDIA Tesla V100 GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='1 Construction of Bandit Instances We provide below more details on how the bandit instances are constructed in our problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Besides the three problems described in Section 5, we consider an additional Labeled Arms problem that will be used for our ablation study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Some illustrations of the constructed instances and the vectors generated by learned priors are provided in Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' As in Popular and Niche and 2D Maze problems, in the Labeled Arms problem we simply add Gaussian noise of standard deviation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='1 to the mean when sampling the reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' For these three problems we thus only explain how the means are constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' 21 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Popular and Niche (K = 200 arms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' The arms are split into 40 groups of equal size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' 20 of these groups represent the ‘popular’ items while the other 20 represent the ‘niche’ items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' For each bandit task, we first construct a vector ¯µ whose coordinates’ values default to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' However, we randomly choose 1 to 3 groups of niche items and set the value of each of these items to 1 with probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='7 (independently across the selected items).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Similarly, we randomly choose 15 to 17 groups of popular items and set their values to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Then, to construct the mean reward vector µ, we perturb the values of ¯µ by independent Gaussian noises with standard deviation of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' After that, we clip the values of the popular items to make them smaller than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='95 and clip the entire vector to the range [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' iPinYou bidding (K = 300 arms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' The set of tasks is constructed with the help of the iPinYou data set [Liao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=', 2014].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' This data set contains logs of ad biddings, impressions, clicks, and final conversions, and is separated into three different seasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' We only use the second season that contains the ads from 5 advertisers (as we are not able to find the data for the first and the third season).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' To form the tasks, we further group the bids according to the associated ad slots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' By keeping only those ad slots with at least 1000 bids, we obtain a data set of 1352 ad slots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Then, the empirical distribution of the paying price (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=', the highest bid from competitors) of each ad slot is used to computed the success rate of every potential bid b ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' , 299} set by the learner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' The reward is either 300 − b when the learner wins the auction or 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Finally, we divide everything by the largest reward that the learner can ever get in all the tasks to scale the rewards to range [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' 2D Maze (K = 180 base arms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' For this problem, we first use the code of the github repository MattChanTK/gym-maze10 to generate random 2D mazes of size 19 × 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Then, each bandit task can be derived from a generated 2D maze by associating the maze to a weighted 10 × 10 grid graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' As demonstrated by Figure 4, each case corresponds to either a node or an edge of the grid graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Then, the weight (mean reward) of an edge (base arms) is either −1 or −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='01 depending on either there is a wall (in black color) or not (in white color) on the corresponding case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' An optimal arm in this problem would be a path that goes from the source to the destination without bumping into any walls in the corresponding maze.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Labeled Arms (K = 500 arms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' This problem is again inspired by applications in recommendation systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' We are provided here a set of 50 labels L = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=', 50}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Each arm is associated to a subset La of these labels with size card(La) = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' To sample a new bandit task B, we randomly draw a set LB ⊆ L again with size 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Then for each arm a, we set ¯µa = 1 − 1/4card(La ∩ LB) so that the more the two sets intersect the higher the value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Finally, to obtain the mean rewards µ, we perturb the coordinates of ¯µ by independent Gaussian noises of standard deviation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='1 and scale the resulting vector to the range [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Training, Calibration, and Test Sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Training, calibration, and test set are constructed for each of the considered problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Their size are fixed at 5000, 1000, 100 for the Popular and Niche, 2D Maze, and Labeled Arms Problems, and at 1200, 100, and 52 for the iPinYou Bidding problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='2 Diffusion Models– Model Design In all our experiments (including the ones described in Appendices C and D), we set the diffusion steps of the diffusion models to L = 100 and adopt a linear variance schedule that varies from 1 − α1 = 10−4 to 1 − αL = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' The remaining details are customized to each problem, taking into account the specificity of the underlying data distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Labeled Arms and Popular and Niche.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' These two problems have the following two important features: (i) The expected means of the bandit instances do not exhibit any spatial correlations (see Figures 16a and 17a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' (ii) The values of the expected means are nearly binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' 10https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='com/MattChanTK/gym-maze 22 The first point prevents us from using the standard U-Net architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Instead, we consider an architecture adapted from Kong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' [2020], Rasul et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' [2021], with 5 residual blocks and each block containing 6 residual channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='11 Then, to account for the lack of spatial correlations, we add a fully connected layer at the beginning to map the input to a vector of size 128 × 6, before reshaping these vectors into 6 channels and feeding them to the convolutional layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' In a similar fashion, we also replace the last layer of the architecture by a fully connected layer that maps a vector of size 128 × 6 to a vector of size K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' We find that these minimal modification already enable the model to perform well on these two problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' As for the latter point, we follow Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' [2022] and train the denoisers to predict the clean sample x0 as it is reported in the said paper that this leads to better performance when the data are binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' iPinYou Bidding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' As shown in Figure 20, the pattern of this problem looks similar to that of natural images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' We therefore adopt the standard U-Net architecture, with an adaption to the 1-dimensional case as described by [Janner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=', 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' The model has three feature map resolutions (from 300 to 75) and the number of channels for each resolution is respectively 16, 32, and 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' No attention layer is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' The denoiser is trained to predict noise as in Ho et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' [2020], Song and Ermon [2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' 2D Maze As explained in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='1 and illustrated in Figure 4, the weighted grid graphs are themselves derived by the 2D mazes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' We can accordingly establish a function that maps each 10 × 10 weighted grid graph to an image of size 19 × 19 and vice-versa— it suffices to match the value of each associated (edge, pixel) pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' For technical reason, we further pad the 19 × 19 images to size 20 × 20 by adding one line of −1 at the right and one row of −1 at the bottom (see Figure 21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' We then train diffusion models to learn the distribution of the resulting images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' For this, we use a 2-dimensional U-Net directly adapted from the ones used by Ho et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' [2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' The model has three feature map resolutions (from 20 × 20 to 5 × 5) and the number of channels for each resolution is respectively 32, 64, and 128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' A self-attention block is used at every resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' We again train the denoiser to predict the clean sample x0 as we have binary expected rewards here (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='01 or −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='3 Diffusion Models– Training Through out our experiments, we use Adam optimizer with learning rate 5 × 10−4 and exponential decay rates β1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='9 and β2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' The batch size and the epsilon constant in SURE-based regularization are respectively fixed at 128 and ϵ = 10−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' When the perfect data sets Dtr and Dcal are provided, we simply train the diffusion models for 15000 steps on the training set Dtr and apply Algorithm 2 on the calibration set Dcal to calibrate the variances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' The training procedure is more complex when only imperfect data are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' We provide the details below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Posterior Sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' As explained in Section 4 and Algorithm 5, to train from imperfect data we sample the entire chain of diffused samples ˜x0:L from the posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' However, while Algorithm 3 performs sampling with predicted noise ¯zℓ+1 and as we will show in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='2, this indeed leads to improved performance in a certain aspect, we observe that when used for training, it prevents the model from making further progress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' We believe this is because in so doing we are only reinforcing the current model with their own predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Therefore, to make the method effective, in our experiments we slightly modify the posterior sampling algorithm that is used during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' While we still construct samples x0:L following Algorithm 3, the samples ˜x0:L used for the loss minimization phase are obtained by replacing ¯zℓ+1 (line 9) by ˜zℓ+1 sampled from N(0, I) in the very last sampling step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' That is, from xℓ+1 we sample both xℓ for further iterations of the algorithm and ˜xℓ to be used for loss minimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Training Procedure Specification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' When training and validation data are incomplete and noisy, we follow the training procedure described in Algorithm 5 with default values S = 15000 warm-up steps, J = 3 repeats, and S′ = 3000 steps within each repeat (thus 24000 steps in total).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Moreover, during the warm-up phase we impute the missing value with constant 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='5 when constructing the diffused samples ˜xℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' As for 11These numbers are rather arbitrary and do not seem to affect much our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' 23 Figure 6: The three paths (super-arms) for UCB initialization in the 2D Maze experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' the regularization parameter λ, we fix it at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='1 for the Popular and Niche, 2D Maze, and Labeled Arms problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Nevertheless, training from imperfect data turns out to be difficult for the iPinYou Bidding problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' We conjecture this is both because the training set is small and because we train the denoiser to predict noise here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Two modifications are then brought to the above procedure to address the additional difficulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' First, as SURE-based regularization can prevent the model from learning any pattern from data when information is scarce, we drop it for the warm-up phase and the first two repeats (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=', the first 21000 steps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' We then get a model that has learned the noisy distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' We then add back SURE-based regularization with λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='25 in the third repeat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' After the 24000 steps, the model is good enough at reconstructing the corrupted data set, but the unconditionally generated samples suffer from severe mode collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Provided that the reconstructed samples are already of good quality, we fix the latter issue simply by applying standard training on the reconstructed samples for another 3000 steps (thus 27000 training steps in total).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='4 Other Details In this part we provide further details about the evaluation phase and the baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Assumed Noise Level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' All the bandit algorithms considered in our work take as input a hyperparameter ˆσ that should roughly be in the order of the scale of the noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' For the results presented in Section 5, we set ˆσ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='1 for the Popular and Niche and 2D Maze problems and ˆσ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='2 for the iPinYou Bidding problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' The former is exactly the ground truth standard deviation of the underlying noise distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' For the iPinYou Bidding problem the noise is however not Gaussian, and ˆσ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='2 is approximately the third quartile of the empirical distribution of the expected rewards’ standard deviations (computed across tasks and arms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' In Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='1, we present additional results for algorithms run with different assumed noise levels ˆσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' UCB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' The most standard implementation of the UCB algorithm sets the upper confidence bound to U a t = ˆµa t + ˆσ � 2 log t N a t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' (12) Instead, in our experiments we use U a t = ˆµa t +ˆσ/ � N a t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' (12) is more conservative than our implementation, and we thus do not expect it to yield smaller regret within the time horizon of our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' UCB Initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' In contrary to Thompson sampling-based methods, UCB typically requires an initialization phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' For vanilla multi-armed bandits (Popular and Niche, iPinYou Bidding, and Labeled Arms) this simply consists in pulling each arm once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' For combinatorial bandits we need to pull a set of super arms that covers all the base arms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' In the 2D Maze experiment we choose the three paths shown in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' 24 Gaussian Prior with Imperfect Data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' To fit a Gaussian on incomplete and noisy data, we proceed as follows: First, we compute the mean of arm a from those samples that have observation for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Next, in a similar fashion, the covariance between any two arms are only computed with samples that have observations for both arms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Let the resulting matrix be ˆΣ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Since the covariance matrix of the sum of two independently distributed random vectors (in our case X0 and noise) is the sum of the covariance matrices of the two random vectors, we further compute ˆΣ′ = ˆΣ − σ2I as an estimate of the covariance matrix of X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Finally, as ˆΣ′ is not necessarily positive semi-definite and can even have negative diagonal entries, for TS with diagonal covariance matrix we threshold the estimated variances to be at least 0 and for TS with full covariance matrix we threshold the eigenvalues of the estimated covariance matrix ˆΣ′ to be at least 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='12 Arm Selection in 2D Maze Problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' All the algorithms we use in the 2D Maze problem first com- pute/sample some values for each base arm (edge) and then select the super arm (path) that maximizes the sum of its base arms’ values (for DiffTS we first map the sampled 20 × 20 image back to a weighted graph and the remaining is the same).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Concretely, we implement this via Dijkstra’s shortest path algorithm applied to the weighted graphs with weights defined as the opposite of the computed/sampled values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' However, these weights are not guaranteed to be non-negative, and we thus clip all the negative values to 0 before computing the shortest path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' C Ablation Study In this appendix, we perform ablation studies on the Popular and Niche and Labeled Arms problems to explore the impacts of various design choices of our algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='1 Predicted versus Sampled Noise in Posterior Sampling In the DiffTS scheme that we develop (Algorithms 3 and 4), we propose to use the predicted noise ¯zℓ+1 in the construction of the diffused observation ˜yℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Alternatively, we can replace it by the sampled noise vector ˜zℓ+1 (the resulting algorithm then becomes very similar to the one proposed in Song et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' [2021]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' In Figure 9, we investigate how this decision affects the performance of DiffTS with diffusion priors trained on perfect data set Dtr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' It turns out that for the two problems considered here, there is not clear winner between the two options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' However, it seems that using only sampled noise produces noisier samples, which leads to significant increase in regret in the Labeled Arms problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' We further confirm this intuition in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='2, where we show on a toy problem that the use of predicted noise often leads to samples that are more consistent with the learned prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' However, this does not always lead to performance improvement in bandit problems as the learned prior is never perfect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='2 Importance of Variance Calibration Throughout our work, we have highlighted multiple times the importance of equipping the diffusion model with a suitable variance estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' We demonstrate this in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' We consider diffusion priors trained on the perfect data set Dtr along with three different reverse variance schedules: (i) calibrated, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=', Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' (2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' (ii) non-calibrated, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=', Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' (1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' (iii) partially calibrated– precisely, only the variance of X0 | x1 is calibrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' We see clearly that a non-calibrated reverse variance schedule leads catastrophic regret performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' This is because the sampling process relies too much on the learned model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' in particular, the variance of pθ(X0 | x1) is fixed at zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Instead, calibrating X0 | x1 itself already leads to significant decrease in regret, making it as competitive as (and sometimes even better than) the fully calibrated alternative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' This suggests that the trade-off between the learned model and the observations mainly occurs at the last reverse step, whereas enlarging the variance of the remaining reverse steps has little to no effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' [Yet, it is also clear from the 12Our implementation requires the prior covariance matrix to be positive definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' 25 0 1000 2000 3000 4000 5000 # Iterations 0 50 100 150 200 Regret GTS-full DiffTS predicted noise DiffTS sampled noise Labeled Arms ˆσ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='1 0 1000 2000 3000 4000 5000 # Iterations 0 20 40 60 80 100 120 140 160 Regret GTS-full DiffTS predicted noise DiffTS sampled noise Labeled Arms ˆσ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='05 0 1000 2000 3000 4000 5000 # Iterations 0 20 40 60 80 100 120 140 Regret GTS-full DiffTS predicted noise DiffTS sampled noise Popular & Niche ˆσ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='1 0 1000 2000 3000 4000 5000 # Iterations 0 20 40 60 80 100 120 140 Regret GTS-full DiffTS predicted noise DiffTS sampled noise Popular & Niche ˆσ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='05 Figure 7: Regret comparison for DiffTS with predicted or independently sampled noise in the construction of diffused observation ˜yℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' 0 1000 2000 3000 4000 5000 # Iterations 0 100 200 300 400 500 600 700 Regret GTS-full DiffTS calibrated DiffTS non-calibrated DiffTS p(x0|x1) calibrated Labeled Arms ˆσ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='1 0 1000 2000 3000 4000 5000 # Iterations 0 100 200 300 400 500 600 700 Regret GTS-full DiffTS calibrated DiffTS non-calibrated DiffTS p(x0|x1) calibrated Labeled Arms ˆσ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='05 0 1000 2000 3000 4000 5000 # Iterations 0 20 40 60 80 100 120 140 Regret GTS-full DiffTS calibrated DiffTS non-calibrated DiffTS p(x0|x1) calibrated Popular & Niche ˆσ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='1 0 1000 2000 3000 4000 5000 # Iterations 0 20 40 60 80 100 120 140 Regret GTS-full DiffTS calibrated DiffTS non-calibrated DiffTS p(x0|x1) calibrated Popular & Niche ˆσ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='05 Figure 8: Regret comparison for DiffTS with three different types of reverse variance schedules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' 0 1000 2000 3000 4000 5000 # Iterations 0 50 100 150 200 Regret GTS-full DiffTS perfect data DiffTS λ = 0 DiffTS λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='1 DiffTS λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='5 Labeled Arms ˆσ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='1 0 1000 2000 3000 4000 5000 # Iterations 0 20 40 60 80 100 120 140 Regret GTS-full DiffTS perfect data DiffTS λ = 0 DiffTS λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='1 DiffTS λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='5 Labeled Arms ˆσ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='05 0 1000 2000 3000 4000 5000 # Iterations 0 20 40 60 80 100 120 140 160 Regret GTS-full DiffTS perfect data DiffTS λ = 0 DiffTS λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='1 DiffTS λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='5 Popular & Niche ˆσ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='1 0 1000 2000 3000 4000 5000 # Iterations 0 20 40 60 80 100 120 140 Regret GTS-full DiffTS perfect data DiffTS λ = 0 DiffTS λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='1 DiffTS λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='5 Popular & Niche ˆσ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='05 Figure 9: Regret comparison for DiffTS trained on noisy data with different regularization weight λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' 0 1000 2000 3000 4000 5000 # Iterations 0 50 100 150 200 Regret GTS-full DiffTS perfect data DiffTS without EM DiffTS EM Labeled Arms ˆσ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='1 0 1000 2000 3000 4000 5000 # Iterations 0 20 40 60 80 100 120 140 Regret GTS-full DiffTS perfect data DiffTS without EM DiffTS EM Labeled Arms ˆσ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='05 0 1000 2000 3000 4000 5000 # Iterations 0 20 40 60 80 100 120 140 160 Regret GTS-full DiffTS perfect data DiffTS without EM DiffTS EM Popular & Niche ˆσ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='1 0 1000 2000 3000 4000 5000 # Iterations 0 20 40 60 80 100 120 140 Regret GTS-full DiffTS perfect data DiffTS without EM DiffTS EM Popular & Niche ˆσ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='05 Figure 10: Regret comparison for DiffTS trained on incomplete data with or without EM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' 0 1000 2000 3000 4000 5000 # Iterations 0 50 100 150 200 Regret GTS-full DiffTS perfect data DiffTS without EM DiffTS EM DiffTS EM without SURE Labeled Arms ˆσ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='1 0 1000 2000 3000 4000 5000 # Iterations 0 20 40 60 80 100 120 140 160 Regret GTS-full DiffTS perfect data DiffTS without EM DiffTS EM DiffTS EM without SURE Labeled Arms ˆσ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='05 0 1000 2000 3000 4000 5000 # Iterations 0 25 50 75 100 125 150 175 Regret GTS-full DiffTS perfect data DiffTS without EM DiffTS EM DiffTS EM without SURE Popular & Niche ˆσ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='1 0 1000 2000 3000 4000 5000 # Iterations 0 20 40 60 80 100 120 140 Regret GTS-full DiffTS perfect data DiffTS without EM DiffTS EM DiffTS EM without SURE Popular & Niche ˆσ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='05 Figure 11: Regret comparison for DiffTS trained on noisy and incomplete data with or without EM / SURE-based regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' 26 experiment on the Popular and Niche problem with presumed noise standard deviation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='5 that calibrating the variance of all the reverse steps may still be beneficial in some situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='3 Ablation Study for Training from Imperfect Data Our algorithm for training from imperfect data (Algorithm 5) makes two important modifications to the original training scheme: the Expectation Maximization-like procedure (abbreviated as EM hereinafter) and the use of SURE-based regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Below we discuss their effects for three types of data: noisy data, incomplete data, and noisy and incomplete data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' We fix all the hyper-parameters to the ones used in the main experiment unless otherwise specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' In particular, we set the noise standard deviation to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='1 for noisy data and the missing rate to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='5 for incomplete data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' For comparison, we also plot the regrets for the full covariance Gaussian prior baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' The means and the covariance of the prior are fitted with the three types of imperfect data that are used to train and calibrate the diffusion models, following the procedure detailed in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Training from Noisy Data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' To cope with noisy data, we add SURE-based regularization with weight λ to our training objective (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' In this part, we focus on how the choice of λ affects the regret when the data are noisy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' For the sake of simplicity, we only complete the warm-up phase of the algorithm, that is, the models are only trained for 15000 steps with loss function L and xℓ sampled from Xℓ | X0 = y0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' In our experiments we note this is generally good enough for noisy data without missing entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' The results are shown in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' As we can see, the value of λ has a great influence on the regret achieved with the learned prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' However, finding the most appropriate λ for each problem is a challenging task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Using a larger value of λ helps greatly for the Labeled Arms problem when it is given the ground-truth standard deviation σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='1, but is otherwise harmful for the Popular and Niche problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' We believe that finding a way to determine the adequate value of λ will be an important step to make our method more practically relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Training from Incomplete Data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' The EM step is mainly designed to tackle missing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' In Figure 10 we show how the induced regrets differ when the models are trained with and without it and when the observations are missing at random but not noisy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' To make a fair comparison, we also train the model for a total of 24000 (instead of 15000) steps when EM is not employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' As we can see, in all the setups the use of EM results in lower regret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Training from Incomplete and Noisy Data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' To conclude this section we investigate the effects of EM and SURE-based regularization when the data are both noisy and incomplete, as in our main experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' We either drop totally the regularization term, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=', set λ = 0, or skip the EM step (but again we train the models for 24000 steps with the configuration of the warm-up phase in this case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' We plot the resulting regrets in Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' For the models without EM, the variance calibration algorithm proposed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='2 (Algorithm 6) does not work well so we calibrate it with a perfect calibration set Dcal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='13 However, even with this the absence of EM consistently leads to the worst performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' On the other hand, dropping the regularization term only causes clear performance degradation for the Labeled Arms problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' This is in line with our results in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' D Additional Experiments In this appendix, we first supplement our numerical section Section 5 with results obtained under different assumed noise levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' After that, we present additional experiments for the posterior sampling and the 13Indeed, by design Algorithm 6 only gives good result when the posterior sampling step provides a reasonable approximation of x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' How to calibrate the variance of a poorly performed model from imperfect data is yet another difficult question to be addressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' 27 0 1000 2000 3000 4000 5000 # Iterations 0 25 50 75 100 125 150 175 Regret DiffTS (ours) UCB GTS-diag GTS-full GMMTS-10 GMMTS-25 Perfect data ˆσ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='05 0 1000 2000 3000 4000 5000 # Iterations 0 50 100 150 200 Regret DiffTS (ours) UCB GTS-diag GTS-full GMMTS-10 GMMTS-25 Perfect data ˆσ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='1 0 1000 2000 3000 4000 5000 # Iterations 0 25 50 75 100 125 150 175 Regret DiffTS (Ours) UCB GTS-diag GTS-full Imperfect data ˆσ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='05 0 1000 2000 3000 4000 5000 # Iterations 0 50 100 150 200 Regret DiffTS (Ours) UCB GTS-diag GTS-full Imperfect data ˆσ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='1 (a) Popular and Niche 0 1000 2000 3000 4000 5000 # Iterations 0 20 40 60 80 100 120 140 160 Regret DiffTS (ours) UCB GTS-diag GTS-full GMMTS-10 GMMTS-25 Perfect data ˆσ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='1 0 1000 2000 3000 4000 5000 # Iterations 0 50 100 150 200 Regret DiffTS (ours) UCB GTS-diag GTS-full GMMTS-10 GMMTS-25 Perfect data ˆσ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='2 0 1000 2000 3000 4000 5000 # Iterations 0 50 100 150 200 250 Regret DiffTS (ours) UCB GTS-diag GTS-full GMMTS-10 GMMTS-25 Perfect data ˆσ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='3 0 1000 2000 3000 4000 5000 # Iterations 0 20 40 60 80 100 120 140 160 Regret DiffTS (ours) UCB GTS-diag GTS-full Imperfect data ˆσ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='1 0 1000 2000 3000 4000 5000 # Iterations 0 50 100 150 200 Regret DiffTS (ours) UCB GTS-diag GTS-full Imperfect data ˆσ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='2 0 1000 2000 3000 4000 5000 # Iterations 0 50 100 150 200 250 Regret DiffTS UCB GTS-diag GTS-full Imerfect data ˆσ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='3 (b) iPinYou Bidding 0 1000 2000 3000 4000 5000 # Iterations 0 200 400 600 800 1000 1200 1400 Regret DiffTS (ours) UCB GTS-diag GTS-full GMMTS-10 GMMTS-25 Perfect data ˆσ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='05 0 1000 2000 3000 4000 5000 # Iterations 0 200 400 600 800 1000 1200 Regret DiffTS (ours) UCB GTS-diag GTS-full GMMTS-10 GMMTS-25 Perfect data ˆσ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='1 0 1000 2000 3000 4000 5000 # Iterations 0 200 400 600 800 1000 1200 Regret DiffTS (Ours) UCB GTS-diag GTS-full Imperfect data ˆσ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='05 0 1000 2000 3000 4000 5000 # Iterations 0 200 400 600 800 1000 1200 Regret DiffTS (Ours) UCB GTS-diag GTS-full Imperfect data ˆσ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='1 (c) 2D Maze 0 1000 2000 3000 4000 5000 # Iterations 0 50 100 150 200 250 Regret DiffTS (ours) UCB GTS-diag GTS-full GMMTS-10 GMMTS-25 Perfect data ˆσ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='05 0 1000 2000 3000 4000 5000 # Iterations 0 50 100 150 200 250 300 350 Regret DiffTS (ours) UCB GTS-diag GTS-full GMMTS-10 GMMTS-25 Perfect data ˆσ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='1 0 1000 2000 3000 4000 5000 # Iterations 0 50 100 150 200 250 Regret DiffTS (Ours) UCB GTS-diag GTS-full Imperfect data ˆσ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='05 0 1000 2000 3000 4000 5000 # Iterations 0 50 100 150 200 250 300 350 Regret DiffTS (Ours) UCB GTS-diag GTS-full Imperfect data ˆσ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='1 (d) Labeled Arms Figure 12: Regret performances on four different problems with priors fitted/trained on either exact expected rewards (perfect data) or partially observed noisy rewards (imperfect data) and with different assumed noise levels ˆσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' The results are averaged over tasks of a test set and shaded areas represent standard errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' 28 training algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='1 Experimental Results with Different Assumed Noise Levels To further validate the benefit of diffusion priors, we conduct experiments for the four problems introduced in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='1 under different assumed noise levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' The results are shown in Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' We see that DiffTS achieves the smallest regret in 15 out of the 18 plots, confirming again the advantage of using diffusion priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Moreover, although DiffTS performs worse than either GMMTS or GTS-full in iPinYou Bidding and Labled Arms for a certain assumed noise level, the smallest regret is still achieved by DiffTS when taking all the noise levels that we have experimented with into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Finally, it is clear from Figure 12 that the choice of the assumed noise level ˆσ also has a great influence on the induced regret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' The problem of choosing an appropriate ˆσ is however beyond the scope of our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='2 Comparison of Posterior Sampling Strategies on a Toy Problem In this part, we demonstrate on a toy problem that using predicted noise ¯zℓ+1 to construct the diffused observation ˜yℓ leads to more consistent examples compared to using independently sampled noise vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Data Set and Diffusion Model Training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' We consider a simple data distribution over R200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' The 200 features are grouped into 20 groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' For each sample, we randomly select up to 6 groups and set the values of the corresponding features to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' The remaining features take the value 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Some samples from this distribution are illustrated in Figure 13a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' As for the diffusion model, the model architecture, hyper-parameters, and training procedure are taken to be the same as those for the Popular and Niche problem (Appendix B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' In Figure 13b we see that the data distribution is perfectly learned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Posterior Sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' We proceed to investigate the performance of our posterior sampling algorithm on this example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' For this, we form a test set of 100 samples drawn from the same distribution and drop each single feature with probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='5 as shown in Figure 13c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' We then conduct posterior sampling with the learned model using Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' To define the diffused observation ˜yℓ, we either follow (4) or replace the predicted noise ¯zℓ+1 by the sampled noise ˜zℓ+1 in the formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' The corresponding results are shown in Figures 13d and 13e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' As we can see, using predicted noise clearly leads to samples that are more consistent with both the observations and the learned prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' To provide a quantitative measure, in the constructed samples we define a group to be ‘relevant’ if the values of all its features are greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' We then compute the recall and precision by comparing the ground-truth selected groups and the ones identified as relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' When predicted noise is used, the average recall and precision are both at 100%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' On the other hand, when independently sampled noise is used, the average recall falls to around 85% (this value varies due to the randomness of the sampling procedure but never exceeds 90%) while the average precision remains at around 98%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='3 Training from Imperfect Image Data To illustrate the potential of the training procedure introduced in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='1, we further conduct experiments on the MNIST and Fashion-MNIST [Xiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=', 2017] data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Both data sets are composed of gray-scale images of size 28 × 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' MNIST contains hand-written digits whereas Fashion-MNIST contain fashion items taken from Zalando shopping catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Some images of the two data sets are shown in Figures 14a and 15a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Data Corruption and Experimental Setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' For our experiments, we scale the images to range [0, 1] and corrupt the resulting data with missing rate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='5 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=', each pixel is dropped with 50%) and noise of standard deviation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' As we only use training images, this results in 60000 corrupted images for each of the two data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' We further separate 1000 images from the 60000 to form the calibration sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' We then 29 (a) 30 samples from the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' (b) 30 feature vectors generated by the learned diffusion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' (c) 30 samples from the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Red squares indicate missing values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' (d) Feature vectors reconstructed with learned diffusion model and Algorithm 3 using predicted noise vectors ¯zℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' The inputs are the ones shown in 13c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' (e) Feature vectors reconstructed with learned diffusion model and Algorithm 3 using independently sampled noise vectors ˜zℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' The inputs are the ones shown in 13c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Figure 13: Feature vectors of the toy problems presented in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Rows and columns correspond respectively to features and samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' For visualization purpose, the features are ordered in a way that those of the same group are put together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' The darker the color the higher the value, with white and black representing respectively 0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='175 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='features(a) Original images ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='(b) Corrupted images ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='(c) Modelorig generated ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='(d) Modelorig reconstructed ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='(e) Modelcor14 generated ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='(f) Modelcor14 reconstructed ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='(g) Modelcor16 generated ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='(h) Modelcor16 reconstructed ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='Figure 14: Various images related to the MNIST data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' The three models Modelorig, Modelcor14, and Modelcor16 are respectively trained on the original data set, on the corrupted data set for 14000 steps, and on the corrupted data set for 16000 steps (Modelcor16 is trained on top of Modelcor14 for another 2000 steps;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' see the text for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' ‘Generated’ means unconditional sampling while ‘reconstructed’ means posterior sampling with Algorithm 3 applied to the corrupted images shown in (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' train the diffusion models from these corrupted images following Algorithm 5, with S = 5000 warm-up steps, J = 3 repeats of the EM procedure, and S′ = 3000 inner steps for each repeat (the total number of training steps is thus 14000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' The learning rate and the batch size are respectively fixed at 10−4 and 128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' For the regularization term, we take λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='2 for MNIST and λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='1 for Fashion-MNIST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' The constant ε is set to 10−5 as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' As in Ho et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' [2020], Song et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' [2020], we note that the use of exponential moving average (EMA) can lead to better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Therefore, we use the EMA model for the posterior sampling step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' The EMA rate is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='995 with an update every 10 training steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' For comparison, we also train diffusion models on the original data sets with the aforementioned learning rate and batch size for 10000 steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Finally, to examine the influence of the regularization weight λ on the generated images, we consider a third model for MNIST trained on top of the 14000-step model with corrupted data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' For this model, we perform an additional posterior sampling step and then train for another 2000 steps with λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' The remaining details, including the model architecture, are the same as those for the 2D Maze experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' In Figures 14 and 15, we show images from the original data set, from the corrupted data set, and produced by the trained models either by unconditional sampling or data reconstruction with Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Overall, our models manage to generate images that resemble the ones from the original data set without overly sacrificing the diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Nonetheless, looking at the samples for Fashion-MNIST we clearly see that a lot of details are lost in the images generated by or reconstructed with diffusion models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' In the case of training from perfect data, this can clearly be improved with various modifications to the model including change in model architecture, number of diffusion steps, and/or sampling algorithms [Karras et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=', 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' This would become more challenging in the case of training from imperfect data as the image details can be heavily deteriorated by noise or missing pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' On the other hand, the effect of the regularization parameter λ can be clearly seen in the MNIST 31 3232 8 2 83 3 C3DO 2 3 Y3 3(a) Original images (b) Corrupted images (c) Modelorig generated (d) Modelorig reconstructed (e) Modelcor generated (f) Modelcor reconstructed Figure 15: Various images related to the Fashion-MNIST data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' The two models Modelorig and Modelcor are respectively trained on the original data set and the corrupted data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' ‘Generated’ means unconditional sampling while ‘reconstructed’ means posterior sampling with Algorithm 3 applied to the corrupted images shown in (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' experiment from Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Larger λ enables the model to produce digits that are more ‘connected’ but could cause other artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' As in any data generation task, the definition of a good model, and accordingly the appropriate choice of λ, varies according to the context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' To summarize, we believe that the proposed training procedure has a great potential to be applied in various areas, including training from noisy and/or incomplete image data, as demonstrated in Figures 14 and 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' However, there is still some way to go in making the algorithm being capable of producing high-equality samples for complex data distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' E Expected Reward Visualization In Figures 16 to 21 we provide various visualizations of the bandit mean reward vectors either of the training sets or generated by the learned priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' 32 (a) 40 samples from the perfect training set Dtr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' (b) 40 samples from the perfect training set Dtr, reordered to put the arms of the same group together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' The popular arms are on the right side of the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' (c) 40 mean reward vectors generated the diffusion model trained on perfect data, reordered to put the arms of the same group together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' The popular arms are on the right side of the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' (d) 50 mean reward vectors generated by the 25-component GMM fitted on perfect data, reordered to put the arms of the same group together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' The popular arms are on the right side of the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Figure 16: Visualization of the mean reward vectors of the Popular and Niche problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Rows and columns correspond to tasks and arms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' The darker the color the higher the value, with white and black representing respectively 0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Diffusion models manage to learn the underlying patterns that become recognizable by humans only when the arms are grouped in a specific way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' 33 0 Ks task bandit 30 0 25 50 75 100 125 150 175 arms10 task bandit 20 30 0 25 50 75 100 125 150 175 arms0 10 task bandit 20 30 0 25 50 75 100 125 150 175 arms10 bandit task 20 30 0 25 50 75 100 125 150 175 arms(a) 100 samples from the perfect training set Dtr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' (b) 60 samples from the perfect training set Dtr, grouped by labels and showing only 5 labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Note that each arm has multiple labels and thus appears in multiple groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' (c) 60 mean reward vectors generated by the diffusion model trained on perfect data, grouped by labels and showing only 5 labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Note that each arm has multiple labels and thus appears in multiple groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' (d) 60 mean reward vectors generated by the 25-component GMM fitted on perfect data, grouped by labels and showing only 5 labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Note that each arm has multiple labels and thus appears in multiple groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Figure 17: Visualization of the mean reward vectors of the Labeled Arms problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Rows and columns correspond to tasks and arms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' The darker the color the higher the value, with white and black representing respectively 0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' While human eyes can barely recognize any pattern in the constructed vectors, diffusion models manage to learn the underlying patterns that become recognizable by humans only when the arms are grouped in a specific way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' 34 tasks 20 40 andit 09 b 80 0 100 200 300 400 armsasks ta 20 andit 40 b 0 50 100 150 200 250 300 armssks 20 andit 40 b 0 50 100 150 200 250 300 armssks ta 20 andit 40 b 0 50 100 150 200 250 300 arms(a) 40 samples from the imperfect training set ˇDtr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Red squares indicate missing values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' (b) 40 mean reward vectors generated by the diffusion model trained on imperfect data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Figure 18: Mean reward vectors of the Popular and Niche problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Rows and columns correspond to tasks and arms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' For ease of visualization, the arms are reordered so that arms of the same group are put together and popular arms are on the right of the figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' The darker the color the higher the value, with white and black representing respectively 0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' (a) 60 samples from the imperfect training set ˇDtr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Red squares indicate missing values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' (b) 60 mean reward vectors generated by the diffusion model trained on imperfect data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Figure 19: Mean reward vectors of the Labeled Arms problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Rows and columns correspond to tasks and arms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' For ease of visualization, the arms are grouped by labels and only arms that are associated to 5 labels are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' The darker the color the higher the value, with white and black representing respectively 0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' 35 10 bandit task 30 0 25 50 100 125 150 175 arms0 task 10 bandit 20 30 0 25 50 75 100 125 150 175 arms tasks 20 bandit 40 0 50 100 150 200 250 300 armsasks tas 20 andit 40 b 0 50 100 150 200 250 300 arms(a) 50 samples from the perfect training set Dtr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' (b) 50 mean reward vectors generated by the diffusion model trained on perfect data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' (c) 50 mean reward vectors generated by the 25-component GMM fitted on perfect data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' (d) 50 samples from the imperfect training set ˇDtr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Red squares indicate missing values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' (e) 50 mean reward vectors generated by the diffusion model trained on imperfect data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Figure 20: Mean reward vectors of the iPinYou Bidding problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Rows and columns correspond respectively to tasks and arms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' For visualization purpose, we order the tasks by the position of their optimal arm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' The darker the color the higher the value, with white and black representing respectively 0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' 36 0 tasks 20 bandit 40 0 50 100 150 200 250 arms0 : tasks 20 bandit 40 0 50 100 150 200 250 arms0 tasks 20 bandit 40 0 50 100 150 200 250 armstasks 2 bandit 50 100 150 200 250 arms0 tasks 20 bandit 40 0 50 100 150 200 250 arms(a) Sample from the perfect training set Dtr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' (b) Sample generated by the diffusion model trained on perfect data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' (c) Sample generated by the 25-component GMM fitted on prefect data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' (d) Sample from the imperfect training set ˇDtr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Red squares and edges indicate missing values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' (e) Sample generated by the diffusion model trained on imperfect data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Figure 21: The weighted grid graphs and the corresponding 2D maze representations of the 2D Maze problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' For visualization, the weights (mean rewards) are first clipped to [−1, 0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Then, for the grid graphs darker the color higher the mean reward (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=', closer to 0) while for the maze representations it is the opposite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' Also note that for the maze representations only a part of the pixels correspond the the edges of the grid graphs, while the remaining pixels are filled with default colors (black or white).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' The red paths indicate the optimal (super-)arms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content=' 37 9259.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} +page_content='25 1110555' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf'} diff --git a/b9FPT4oBgHgl3EQfBTTg/content/tmp_files/2301.12985v1.pdf.txt b/b9FPT4oBgHgl3EQfBTTg/content/tmp_files/2301.12985v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..9571571c95846e1c5255e52df9d1a1b75766a790 --- /dev/null +++ b/b9FPT4oBgHgl3EQfBTTg/content/tmp_files/2301.12985v1.pdf.txt @@ -0,0 +1,1564 @@ +Integrating Earth Observation Data into Causal +Inference: Challenges and Opportunities +Connor T. Jerzak +Department of Government +University of Texas at Austin +Email: connor.jerzak@austin.utexas.edu +Website: ConnorJerzak.com +Fredrik Johansson +Data Science and AI Division +Chalmers University of Technology +Email: fredrik.johansson@chalmers.se +Website: fredjo.com +Adel Daoud +Institute for Analytical Sociology +Linköping University +Email: adel.daoud@liu.se +Website: AdelDaoud.se +AI and Global Development Lab: global-lab.ai +Abstract +Observational studies require adjustment for confounding factors that are corre- +lated with both the treatment and outcome. In the setting where the observed +variables are tabular quantities such as average income in a neighborhood, tools +have been developed for addressing such confounding. However, in many parts +of the developing world, features about local communities may be scarce. In this +context, satellite imagery can play an important role, serving as a proxy for the +confounding variables otherwise unobserved. In this paper, we study confounder +adjustment in this non-tabular setting, where patterns or objects found in satellite +images contribute to the confounder bias. Using the evaluation of anti-poverty aid +programs in Africa as our running example, we formalize the challenge of perform- +ing causal adjustment with such unstructured data—what conditions are sufficient +to identify causal effects, how to perform estimation, and how to quantify the ways +in which certain aspects of the unstructured image object are most predictive of +the treatment decision. Via simulation, we also explore the sensitivity of satellite +image-based observational inference to image resolution and to misspecification of +the image-associated confounder. Finally, we apply these tools in estimating the +effect of anti-poverty interventions in African communities from satellite imagery. +Keywords: Earth observation; Causal inference; Neighborhood dynamics +Word count: 12,207 +Note: This work largely subsumes +Jerzak, Connor T., Fredrik Johansson, and Adel Daoud. “Estimating Causal Effects +Under Image Confounding Bias with an Application to Poverty in Africa.” arXiv +preprint arXiv:2206.06410 (2022). +Preprint. +arXiv:2301.12985v1 [stat.ML] 30 Jan 2023 + +1 +Introduction +The causal revolution in the social sciences has entered a new phase where scholars are increasingly +combining traditional tabular data with novel data sources (Daoud and Dubhashi, 2023; Morgan +and Winship, 2015; Imai, 2022; Pearl, 2009). And, as a response to increasing data digitization and +availability, a growing literature has emerged more recently that seeks to provide valid estimation of +treatment effects in the presence of high-dimensional confounders, where the number of variables is +large relative to the number of observations. (Li and Pearl, 2021; Mozer et al., 2020; Chernozhukov +et al., 2018; Yoon et al., 2018; Wager and Athey, 2018; Shalit et al., 2017; Hill, 2011; Schneeweiss +et al., 2009; Belloni et al., 2014; Alexander et al., 2021). Yet, merely adding more covariates +describing each unit’s context does not itself address concerns about non-random missingness within +that confounder data, about the difficulty in obtaining information, about historical interventions, and +about the lack or unreliability of data in the most economically disadvantaged places (Jerven and +Johnston, 2015; Dinku, 2019)—precisely those in need of effective interventions. +In this article, we argue that satellite data and remote sensing information provide an important +resource for expanding the reach of causal inquiry in otherwise data-scarce environments. Satellite +data is available for every corner of the globe. This breadth of information is available on a historical +basis stretching back to the 1970s. Moreover, unlike a sizable proportion of datasets which are a +single snapshot of a social system at a particular point in time, many earth observation satellites return +to every place on earth every two weeks or even more frequently—providing 26 or more temporal +slices per year. The temporally-resolved information contained in satellite data has been shown +to be associated with variables of social science importance often characterized as neighborhood +features—features such as the development of transportation networks (Nagne and Gawali, 2013), +the degree of urbanness (Schneider et al., 2009), health and material conditions (Daoud et al., 2021), +living standards (Jean et al., 2016; Yeh et al., 2020), and a host of other neighborhood features +(Sowmya and Trinder, 2000). For these reasons, data from space-borne instruments may provide +critical information for causal inference analyses, especially given the rapid proliferation of earth +observation satellites from the hundreds into the thousands Tatem et al. (2008) and sub-100 cm +resolution now widely available on the latest generation of satellites (Hallas, 2019). +Despite the potential offered by satellite images in causal inference, there is a lack of methodological +guidance for causal estimation when confounding is induced by patterns or objects observed in +an image (Castro et al., 2020). To help fill this need, we examine observational causal inference +in the presence of confounding captured as latent image patterns. A story-based intuition for we +have in mind is the following: an actor examines a neighborhood, looking for certain aspects of +that neighborhood (such as the presence of poverty) to guide the choice of intervention, T. Those +neighborhood aspects, U, are observed by the decision maker but unobserved by outside analysts. +Some of that neighborhood-level information is also embedded in the satellite data representation +(M) of that same neighborhood. Thus, observed image patterns indicate the existence of real-world +objects that provide information about the confounding factors associated with both treatment and +outcome. We may not directly observe the true latent variables about which the treatment decision +was made, but we can readily observe, from earth observation resources, and adjust for inferred image +patterns that correlate with the treatment, even if these patterns are difficult to adjust for directly +(Voigt et al., 2016). +Our focus in this paper on causal inference with earth observation data complements a social-science +research trend, especially in sociology and political science, where scholars increasingly leverage +visual data. For example, such visual data are used in qualitatively analyzing photos (Pauwels, 2010; +O’Hara and Higgins, 2019), estimating image similarities (Zhang and Peng, 2022), and approximating +the number of demonstrating people from news photos (Cruz and González-Villa, 2021). Recently, +the use of visual data also includes video data for analyzing social processes, such as police violence +(Nassauer and Legewie, 2021). For larger visual data and in a quantitative design, scholars have to +train algorithms “learning to see” objects of interest (Torres and Cantú, 2022). Nonetheless, although +these contributions are critical for research designs relying on image data, there is a need for deeper +grounding in the causal inference literature (Daoud and Dubhashi, 2023; Morgan and Winship, 2015; +Imai, 2022; Pearl, 2009), creating a knowledge gap about how to leverage images for causal inference. +Our article contributes to filling that gap. +In what follows, we first in Section 3 describe several causal structures relevant to satellite image- +based causal inference and discuss their implications for identification and estimation. We focus +on under what conditions the image alone is sufficient to adjust for the confounding introduced +2 + +by U. This holds, for example, when U may be derived deterministically from M. An important +special case of image pattern confounding occurs when decision-makers make choices based on the +(translation-invariant) existence of a pattern in the image, which motivates adjustment techniques +based on convolutional models developed in the machine learning community (Goodfellow et al., +2016). We also analyze the complementary case where the image confounder is itself the cause of the +image. +We next study in Section 4 finite-sample estimation of average treatment effects in the fully identified +case by conducting a simulation in which confounders are derived from the observed image. In +this setting, we investigate the impact of model misspecification on estimates, as well as the role of +image resolution—a key aspect of image data with no perfect analog in tabular, network, or text data. +Finally, in Section 5, we demonstrate the use of the proposed estimation framework in an application +in which we evaluate the effectiveness of international aid programs on neighborhood-level poverty +by estimating treatment propensity using geo-referenced satellite data to proxy for confounding +factors. +2 +Related Work and Contribution +Observational causal inference methods primarily evolved in the context of tabular covariates (e.g., +see Imbens and Rubin (2016)), where separate, human understandable features are used as covariates +in adjusting for confounding via regression (Best and Wolf, 2013), weighting Jung et al. (2020), or +doubly robust (Funk et al., 2011) methods. While tabular covariates are readily interpretable and can +be tailored by researchers at the time of data collection, they face several limitations. For example, +tabular covariates such as gender, income, and ethnicity are typically collected at the time of data +collection by researchers. Thus, they cannot capture historical confounders (such as income pre-data +collection) without additional effort. Moreover, they are subject to missingness or mismeasurement +due to unit-level behavior: a unit may decide–or not—to answer a survey question or may mask their +true opinions, both of which induce bias into the resulting causal analysis. +The causal text analysis literature has recently developed interesting insight into how text can +supplement tabular covariates in observational causal inference, emphasizing how text information +can provide insight into why treatments were assigned (Grimmer and Stewart, 2013; Egami et al., +2018; Keith et al., 2020), even if there is a risk posed by the open-ended quality of text that can +lead to bias in observational analyses Daoud et al. (2022). While text does pose an opportunity for +investigators, it is limited by availability pre-intervention: much text is gathered via web resources, +so historical interventions can be difficult to model via text, not to mention the fact that some +interventions might not have relevant text associated with them. While text is a promising new data +source for causal inference, it is not available universally for all spatially defined interventions. +Social network data has also been increasingly emphasized in the causal inference setting. The +typical use of this network data is not to adjust for confounding bias but instead to model how +units respond to the treatment status of social connections, primarily to estimate spillover effects +VanderWeele and An (2013). Network data must be gathered for every unit about every other unit: +this kind of data source scales non-linearly—with a polynomial growth in the number of possible +connections as the number of units. As a consequence, network data has generally been difficult to +obtain outside a few contexts such as trade relations, alliances, or online social networks, where social +links are assumed to be online website links (e.g., Lewis et al. (2012)). Because of the computational +complexity, the temporal resolution of network data used in social science research is generally +refined at best on a yearly basis, as in international relations network analyses (see, for example, Lebo +and Box-Steffensmeier (2008)). +In this context, satellite image data potentially addresses a number of the limitations of other data +streams for observational causal inference (see Table 1 for a summary). Satellite image data— +obtained from space-born instruments funded by NASA, the European Space Agency, and others—is +high-dimensional and readily interpretable as a raw data source in that human observers can generally +identify key features of the data, such as the presence of a city, forest, river, soil moisture content, +and so forth. Satellite data are also available for all geo-referenced interventions—not just those +about which a human observer decided to write a document. Moreover, while satellite data does have +missingness due to cloud cover, experimental units cannot affect the availability of satellite images +about their neighborhood, an inability that reduces the risk of bias due to systematic data missingness +3 + +(Kenward and Carpenter, 2007). Finally, remote sensing data also has a temporal resolution of two +weeks or better for many leading satellite image providers—meaning that, in principle, one can +examine the evolution of a neighborhood with 26 or more data slices per year. +Table 1: Comparing satellite imagery data with other data types in the context of observational causal +inference. ∗ The European Space Agency’s Sentinel earth observation satellites revisit locations at +least once every five days; Landsat revisits locations around once every 14 days. +Tabular +Text +Network +Earth +Observation +Origin & Character +Common origin +Researcher +surveys/gov’t +data +Web +resources +Online social +network +databases +Space-born +instruments +High-dimensional +Not usually +Yes +Yes +Yes +Unstructured in raw +form +No +Yes +Yes +Yes +Readily interpretable +as a raw data source +Yes +Yes +No +Yes +Scaling with # of units +Linear +Linear +Polynomial +Linear +Availability +Available for all +geo-referenced +interventions +No +No +No +Yes +Readily available +pre-intervention +No +No +No +Yes +Typical temporal +resolution +Singular/1 +year +Singular +Singular +5-14 days∗ +Subject to missingness +due to unit behavior +Yes +Yes +Yes +No +The machine learning community has begun exploring the opportunities of images in the causal +image context (Castro et al., 2020; Reddy and Balasubramanian, 2019; Du et al., 2021). Related work +has been done on estimating counterfactual outcomes under spatially defined counterfactual treatment +strategies (Papadogeorgou et al., 2020), on accounting for spatial interdependence in causal effect +estimation (Reich et al., 2021), and on balancing observed covariate representations using adversarial +networks and image data (Kallus, 2020). Other works address images as treatments (Kaddour et al., +2021), counterfactual inference and interpretability (Pawlowski et al., 2020; Singla et al., 2020), and +image-based treatment effect heterogeneity (Jerzak et al., 2023). There are also several important +works on how algorithms may discover causal structures from images in the causal discovery tradition +(Chalupka et al., 2016b,a, 2015; Schölkopf et al., 2021; Yi et al., 2020; Ding et al., 2021). +What we can add to this literature is twofold. First, we provide a systematic analysis on how to +understand the latent confounding structure proxied for or induced by images in the causal inference +setting, where latent objects in the image may affect treatment and outcome (Castro et al., 2020). +Developing techniques for causal inference under image pattern confounding could open up new +avenues for observational studies (Pawlowski et al., 2020; Singla et al., 2020; Kaddour et al., 2021). +(See Section A.1 for a more expansive connection developed with the proxy literature.) +Second, this analysis also specifically provides guidance for social scientists interested in using +earth observation resources to perform observational causal inference in neighborhood or developing +contexts. As mentioned, earth observation data provides a large amount of raw information about +4 + +potential confounding variables of importance in social science research: we here analyze the promise +and pitfalls of this emerging analytic pipeline. +To conclude this section, we also note that this work on integrating satellite images into causal +pipelines in data-poor environments also has policy and practical resonance. Policymakers may +consult maps to evaluate where to allocate aid to villages that otherwise may remain poor (Holmgren +and Thuresson, 1998; Bedi et al., 2007). For example, since 2000, policymakers frequently rely on +raw satellite images to evaluate damage due to natural disasters or war (Voigt et al., 2016; Burke +et al., 2021; Kino et al., 2021). Based on these images, they decide where to intervene in helping the +poor (Borie et al., 2019). Thus, the use of satellite images in causal inference may help those in the +policy world as well as evaluate anti-poverty programs, especially since areas experiencing poverty +are also often affected by weak state capacity (Besley and Persson, 2010), meaning that the poorest +places are often those about which we know least in the absence of remote sensing information. +3 +Characterizing Challenges and Opportunities of Satellite Image-based +Observational Inference +A major reason why satellite images can serve as a useful tool for observational causal inference +is that the satellite images can proxy for confounders that explain why some places but not others +received a treatment of some kind, such as an anti-poverty aid program. +With this motivation in mind, we study the identification and estimation of the average treatment +effect (ATE) of T on a real-valued outcome of interest, Y , based on observational data (suppressing +the unit subscripts of the various random variables except when important for expository purposes). +With Y (1) being the potential outcome (Rubin, 2005) under intervention with T = 1 and Y (0) for +T = 0, +ATE = E[Y (1) − Y (0)] . +The ATE can represent, for example, the difference in average wealth in villages after anti-poverty +interventions and after no intervention, respectively. In that setting, historical interventions can be +thought of as being determined by a decision-maker using information about neighborhoods that are +proxied by a satellite image representation, M. +Whether we think that the image patterns cause the confounder or the confounder dynamics cause the +image pattern is of little consequence in practice: in the latter, the image is proxy; in the former, the +image is a driver; the same type of analysis will be done in either case (Pearl, 2013b) as described +in the proxy literature (Kuroki and Pearl, 2014; Louizos et al., 2017). To understand the estimation +dynamics of the ATE from observational images and adjust for induced confounding bias, we analyze +the data-generating process next. +3.1 +Baseline Model of Confounding Bias +To understand similarities and differences of observational causal inference in the tabular case and +in the satellite image context, first consider the causal graph in Figure 1a. Note that we reintroduce +unit-level subscripts in the Figure to emphasize some distinctions: s denotes a scene (e.g., village), +and w and h denote the width and height indices of a spatially resolved context like an image. +This figure depicts classical confounding: a treatment of interest (Tswh), such as an anti-poverty +intervention, is associated with factors (such as the presence of mineral extraction sites) that affect +both the treatment and the outcome (Yswh). Observed confounding variables are grouped in Xswh; +unobserved confounders in Uswh. In the general, s need not be spatially defined when working with +non-satellite-derived images, like if the images were to be X-ray snapshots. In that case, the scene +index would refer to a patient or body part. However, in the social science setting with satellite data, +s will usually refer to some neighborhood or the neighborhood context around an observational unit. +3.2 +Pixel-based Image Confounding +We now turn to the causal model in Figure 1a, which depicts the kind of confounding dealt with in +much of observational research (Rosenbaum and Rubin, 1983). We extend this model to describe +image-based confounding. +5 + +Uswh +Tswh +Yswh +Xswh +(a) Baseline causal diagram. +Uswh +Msw′h′ +Tswh +Yswh +Xswh +w′, h′ ∈ Πs(wh) +(b) Diagram illustrating image-based confounding. +Figure 1: Diagram representing variables associated with a scene s. In our running example, Uswh +represents unobserved confounders, Xswh observed confounders, Tswh treatment and Yswh the +outcome, all at location w, h in scene s. In the right-hand diagram, latent confounders Uswg are +determined by a neighborhood Πs(wh) of the location h, w in the image M representing scene s. +The arrow between Uswh and Msw′h′ is bi-directional to indicate that we are agnostic about the +direction of causality (whether the image “causes” the confounder or the confounder “causes” the +image). +First, we define this model at the local level, where treatments are implemented at specific locations +w, h (in, for example, the precision agriculture context where treatments are applied to small sections +of land; Liaghat et al. (2010)). +To describe causal dependencies at the neighborhood level, we introduce the following notation. +Let Πs(wh) ∈ N2 denote the set of location indices involved in generating the confounder value +from the neighborhood around swh. For example, if the confounder at swh were generated using +neighborhood information in a z × z region around swh, +Πs(wh) = +� +w − ⌊z/2⌋, ..., w + ⌊z/2⌋ +� +× +� +h − ⌊z/2⌋, ..., h + ⌊z/2⌋ +� +. +For example, if z = 2, Πs(wh) = {w − 1, w, w + 1} × {h − 1, h, h + 1}, where the × symbol here +denotes the Cartesian product capturing all ordered pairs of the left and right set. In other words, +Πs(wh) simply characterizes the set of width and height indices centered around location wh in +scene s. +With this notation, we can illustrate the confounding structure induced by image-based confounding +at the neighborhood level in Figure 1b. Figure 1b is a formulation of spatial interdependence, as the +image-information for indices in Πs(wh) affects the confounder Uswh. Conditional on the value of +the confounder, the treatment decision for each unit is made. This confounding dynamic would be +invoked if a decision maker who scans a scene looking for similarity of the neighborhood around +swh to some mental image, defined, for example, by an image filter l, and makes a decision on this +basis. +We illustrate this process on satellite image data from Landsat, a U.S. Geological Survey and NASA +satellite (see Section 5.1 for details). We perform the illustration using a particular parametric +model for the neighborhood-induced confounding. The parametric model used for illustration is +convolutional: we let a single convolutional filter in the form of a diagonal matrix represent an image +pattern used by a decision maker to determine the treatment probability, as depicted in Figure 2. After +applying fl to the raw image shown in the right panel of Figure 2, we obtain the resulting image- +derived confounder values. “Applying the filter to the image” here means calculating a similarity +score at every place in the original image with the filter (the “image pattern”). This score generates +a new, latent image representing the similarity structure which, here, underlies the confounding +dynamic. The similarity score is calculated by summing up the result of multiplying the filter with +the relevant image pixels (similar to how the leading term of the covariance between A and B would +be calculated by taking the average of the product of A and B). This simple example shows how +the presence of objects or patterns in images (as represented by the diagonal line here) can generate +confounder values in the context of satellite-based observational inference. +6 + +Raw Image +After Convolution +Figure 2: Left. The kernel filter pattern used to generate Uswh in the right-hand image. Center, right. +Illustration of image-based confounding using Landsat data for Nigeria. The center panel depicts +the raw reflectance; the right panel depicts the transformed values after convolution with the filter, +values which enter the model for treatment/outcome to generate confounding. +3.3 +Scene-based Image Confounding +While the pixel-based confounding structure may be relevant if small sections of land (e.g., houses) +receive treatment, in many social science applications, the image is defined at one resolution, but +the treatment, outcome, and confounder potentially at another. In the aid context, the treatment +and outcome data are often defined at the neighborhood or village level, but the satellite data itself +contains additional height and width dimensions (in a word, we can “peek inside” the village). To +take another example, a policymaker may examine an entire village, looking for the maximum or +average similarity to some target pattern: the village is the unit to which treatment is allocated, and +the confounder is defined at that level but is created using more granular information. +Satellite image-based observational inference can accommodate situations such as these, where the +confounder, treatment, and outcome are defined at different scales. In particular, we can add the +following to our causal model. Let Πs ∈ N2 denote the height and width indices (locations) used +when aggregating up information to the final scene-based unit of analysis, s. The right panel of +Figure 2 then illustrates a scene-based confounding structure generated by Πs, where Πs intuitively +represents the image indices associated with the whole neighborhood context that was used in +deciding upon treatment. +To further explain the role of Πs, consider how, if investigators are interested in a village, s, the width +of the satellite image collected around s fixes the maximum size of Πs that can be accommodated +in the estimation procedure. If the image is, say, 1024 by 1024 pixels, then the cap on Πs is +{1, ..., 1024} × {1, ..., 1024}, which specifies a square of 14.59 km in width using the Landsat +images used throughout this paper. In short, the maximum neighborhood around the village that can +be used to recover the latent confounder is fixed by the width of the satellite image input. +In general, this scene-based causal model is significant because, while some studies are able to +obtain resolved (e.g., household-level) outcome data, this data may, in other cases, be costly or +even impossible to obtain due to privacy reasons. In these situations, treatment and outcome data +can only be measured at the scene level. We next turn to the question of identification in this +image-confounding context. +3.4 +Identification in Satellite Image-based Observational Inference +We can confirm that under image-based confounding as formalized in Figures 1b and 3a, treatment +effects may be identified by adjusting for the image M. Here, we suppress dependence on the indexes +s, w, and h since the same arguments will apply if T, U, or Y are defined at the swh (pixel) or s +(scene) levels. +To begin, we assume that the confounder U is a deterministic function of M and return to the case +where U has multiple causes later. This is justified, for example, in applications where confounding +is based on the existence of an object—either if the policymaker scanned M for the object prompting +the policymaker to allocate a treatment in that area of interest, or if M is more generally associated +with important confounder patterns/objects without error. +7 + +Uswh +Us +Msw′h′ +Ts +Ys +Xs +w, h ∈ Πs +w′, h′ ∈Πs(wh) +(a) Image-based confounding at the scene level. +Uswh +Us +Rs +Msw′h′ +Ts +Ys +Xs +w, h ∈ Πs +w′, h′ ∈Πs(wh) +Objects not in image +(b) Some confounding not observed in the image. +Figure 3 +As U is determined fully by M, ruling out other potential noise sources, there exists a deterministic +function f such that U = f(M). The aforementioned case of U being the (pooled) convolution of a +2D filter with the image M satisfies this assumption. +Proposition 1. Suppose the confounder U is deterministic given the image M, such that U = f(M), +(with f unknown), and that the causal structure obeys either of Figures 1b & 3a. Then, p(Y (t)) and +therefore ATE of T on Y is identifiable from p(M, X, T, Y ). +Proof. For simplicity of exposition, we give the proof for the case without additional confounding +variables X. The proof generalizes readily to non-empty X, by marginalization and conditioning. +The claim follows from U being a deterministic function of M. By the backdoor criterion applied to +the graphs in Figures 1b & 3a, X, U is an adjustment set for the effect of T on Y , which implies the +exchangeability of potential outcomes: Y (t) ⊥ T | X (see, e.g., Hernán MA (2020)). In the case of +empty X, +p(Y (t)) = +� +u +p(Y |T = t, U = u)p(U = u). +(1) +Since U is a deterministic, but not necessarily invertible function of M, U = f(M), we have that +p(Y | T = t, U = u) = p(Y | T = t, M ∈ f −1(u)) +and +p(U = u) += +� +m∈f −1(u) +p(M = m) +(2) +where f −1 is the inverse map of f, so that +p(Y (t)) = +� +u +� +m∈f −1(u) +p(Y |T = t, M = m)p(M = m) = +� +m +p(Y |T = t, M = m)p(M = m) . +Hence, M is also an adjustment set for T on Y . From a similar proof, we see that X, M is an +adjustment set in the case of non-empty X. From here, standard arguments (Rosenbaum and Rubin, +1983) show that +E[Y (t)] = E +� +Y +p(T = t) +p(T = t | M = m) +���� T = t +� +(3) +which justifies the use of inverse-propensity weighting with respect to M. +The argument of Proposition 1 rests on the assumption that the image contains all information +about the latent confounder. When treatment decisions are made based on object detection, this +assumption would be satisfied if the image contains all objects that are relevant to the outcome and +treatment decision. This is violated if, for example, unlabeled objects, depicted as Rs in Figure 3b, +8 + +are themselves the driver of the treatment decision but are not possible to reconstruct from the image +data. If the image data imperfectly depicts those objects, full identification is no longer possible, as +there is a possibility of residual confounding. Specifically, the inverse map f −1 in Equation 2 is no +longer uniquely or well defined as a set of images; Rs must be adjusted for as well. In this imperfect +case, the image becomes a driver of the confounding, and thus, has similar properties to proxies +(Pearl, 2013b; Peña, 2020). +We have shown how, under assumptions, the image is itself an adjustment set for estimating the effect +of programs on outcomes in the context of image-based confounding. However, non-parametric +inference is difficult in the image context because no two images are the same. Thus, the probability +of seeing identical treatment/control images is zero, violating overlap assumptions necessary for +model-free inference (D’Amour, 2019). Machine learning models for the image may seek to estimate +U, forming latent representations for the image. In this lower-dimensional space, there is more +likely to be empirical overlap between treatment/control, justifying the use of modeling approaches +like the ones discussed in Section 4 and Section 5. Thus, while adjusting directly for U would +fulfill the overlap assumptions optimally, this is infeasible; when adjusting for M instead, a critical +argument for our approach to work is that the propensities depend only on aspects of M that capture +U, aspects assumed to be compressible to a lower dimensional representation. As such, the situation +is more benign than if propensities depended freely on all patterns in M. Nevertheless, results from +parametric models can be sensitive to the details of model specification—something that can partially +be addressed by robustness checks varying key parameters but cannot be conclusively settled. +3.5 +Interpretation in Satellite Image-based Observational Inference +Having discussed identification, we now turn to important questions around estimation and interpreta- +tion in satellite image-based observational inference. +With tabular data, interpreting results is generally straightforward for two reasons. First, features in a +tabular dataset are human interpretable: we have measurements on pre-treatment variables such as +age and ethnicity in a tabular dataset, and those quantities can be readily communicated via language. +Second, linear models predominate in much of observational inference—both in weighting methods +(which often involve a logistic regression step) or in methods modeling the outcome (where OLS is +often applied). Linear models have a particularly simple structure where the relationship between the +inputs and the predictions can be conveyed simply via regression coefficients. For example, in the +OLS context (with covariate vector, X, and with no interaction/polynomial terms), we can readily +communicate how one thing relates to another: +∂ �E [Y | X] +∂ Xd += βd. +With linearity, gradients are the same for all values of Xd, leading to the interpretation of OLS +coefficients: a one unit change in the predictor Xd is associated with a βd unit change in the outcome, +holding all else equal. Interpretation (in the sense of understanding how covariate inputs relate to +model outputs) is arguably straightforward—at least if we sidestep the subtle issue of what it truly +means to “hold all else equal.” +In the image context, we can augment this derivative-based notion of interpretability via the use +of salience maps Gilpin et al. (2018). In particular, we can examine the following quantity in +observational inference to explain what the prediction of the treatment assignment is particularly +sensitive to in the image: +S = +� +� +� +� +C +� +c=1 +� +∂ � +Pr(T | M) +∂ M·,·,c +�2 +. +(4) +S is of the same height and width dimensions as the raw image, M, and M·,·,c denotes the image +slice of channel/feature c.1 +The quantity, S, represents the magnitude of the gradient of the predicted probability of treatment with +respect to the image, where the magnitudes are computed across all the channel/feature dimensions +1These features typically correspond to quantities such as reflectance, ultimately measured in energy per unit +area per wavelength such as Watts/(m2 · µm). +9 + +of the image. This salience map, therefore, provides some indication of the parts of the image that +are “important” in predicting the outcome, where that importance is calculated using derivatives +to quantify importance as mathematical sensitivity. These derivatives are calculated via automatic +differentiation, a computational tool to evaluate exact gradients even when no closed form is available +(Griewank and Walther, 2003). Because they use automatic differentiation, salience maps in satellite- +based observational inference can be computed with some, but not all, models for � +Pr(T | M). In +particular, we require that � +Pr(T | M) is continuously differentiable with respect to M, which is the +case for the most common class of image models such as convolutional networks (see Table 2 for a +list of models outlined by differentiability). +Table 2: Differentiability of candidate models in image-based observational inference. Salience +maps can be readily calculated for differentiable models. +Differentiable +Generalized linear models +Feed-forward neural networks +Convolutional models +Transformers +Recurrent neural networks +Non-differentiable +Tree-based models +Models involving greedy/discrete optimization +Another challenge related to interpretability in satellite image-based causal inference relates to the +Stable Unit Treatment Value Assumption (SUTVA). This assumption states that the treatment of any +scene s should not affect the treatment status or outcome of another unit s′. In other words, each +scene is i.i.d, as defined by our DAG. However, in spatial analysis, it may be harder to defend or even +define SUTVA. Units that are closer in space may affect each other via spillover effects (Breza, 2016). +For example, a policymaker allocating aid to one village may unintentionally affect outcomes in a +nearby village, which benefits from the nearby allocation of assistance. SUTVA violations, and other +forms of dependence (e.g., spatial clustering), can in principle be accounted for by specifying an +appropriate variance-covariance structure (Sinclair et al., 2012). However, this variance-covariance +structure may be difficult to capture in practice, so that there may be difficulties in interpreting or +explaining results from image-based observational causal inference. +A more interesting possibility lies in the prospect that satellite image data may actually contain +information about the latent interference structure present within the social system. As already noted, +satellite data is informative transportation networks (Nagne and Gawali, 2013) which may correlate +with patterns of social influence. Thus, while we can use block bootstrapping or other techniques +to address spatial dependence, future work should probe the degree to which satellite data itself is +informative about the underlying patterns of influence present within social reality. +To conclude this section on interpretability in satellite image-based observational inference, we offer +a few reflections on the centrality of resolution in this task. +First, we note that resolution is a key driver of the residual confounding, as discussed in Section 3.4. +We can only adjust for confounder objects in the image that can be resolved. Smaller confounder +objects, therefore, introduce residual bias, indicating how technological improvements to sensor +technology play a critical role in improving image-based causal inference methods. In short, the +resolution determines the kinds of explanations we can make regarding our ability to reconstruct +the treatment assignment mechanism. If there is a single pixel per scene, then that pixel can only +capture global information about things such as the abundance of greenspace, soil moisture, and other +quantities. If there are hundreds or thousands of pixels per scene, then more complex objects can +be detected such as houses, roads, and trees. Interpretability in image-based inference is therefore +inextricably tied to resolution. +Another related motivation for obtaining high-resolution remote sensing data lies in the possibility +that the use of satellite images can reduce researcher assumptions about how information from smaller +scales aggregates up to the scene scale. Because each image is defined at a more granular resolution +than the unit of analysis, we can use it to potentially reconstruct some unobserved confounders by +learning the function generating confounders from the image. +10 + +For example, assume that researchers seek to analyze the effect of a village-level treatment. From a +government census, they obtain mean income information for the village (s) and then perform an +analysis assuming Ys(0), Ys(1) ⊥ Ts|Xs, where Xs contains the income data, Ts is the treatment, +and Ys(t), the potential outcome under t ∈ {0, 1}. Yet, unless the mean income is the true confounder, +the analysis will still be biased, which would occur if, in fact, minimum income drove the decision +to allocate Ts. However, using satellite images for each scene, we seek to reconstruct the minimum +income signal based on our access to the higher-resolution data. +In other words, we can weaken the assumption used in many empirical analyses that the variables +measured at the scene level in fact contain the true confounders, when in reality highly non-linear +functions—that use more granular information—may have generated the confounding structure. +With image data, we, in principle, can hope to reconstruct some of those factors using advances in +image-based machine learning models effective in the prediction domain (e.g., Sun et al. (2013)). +Despite the potential of learning about how lower-level information aggregates up using high- +resolution satellite images, there are still other subtleties to consider. For example, if treatments +are defined at a high level of resolution (say, a house), but satellite data is defined at a lower level +of resolution (say, a neighborhood block), there are ambiguities in how observational inference +should be performed. Without higher-resolution data, we cannot adjust for home-level confounders, +but if we aggregate treatments to the block level, information is lost, and extra researcher degrees +of freedom are introduced in terms of how that aggregation should be done (whether in terms of +summing, averaging, or aerial interpolating to the block level). For pixel-level treatments, therefore, +extra caution is warranted due to the nuances of how resolution affects the confounders that can be +captured by an image model using satellite resources. +3.6 +Other Challenges and Opportunities of Satellite Image-based Observational Inference +In addition to model dependence of observational inference with satellite images, there is another, +perhaps more fundamental challenge. By its nature, satellite data is geo-referenced, and data on +treatment assignments need to be geo-referenced as well if it is to be integrated into this pipeline. This +linkage in some circumstances may be straightforward: a town receives an anti-poverty intervention, +or it does not, and that town can be geo-referenced using APIs such as Google Maps and Open- +StreetMap (Yeboah et al., 2021). However, in other circumstances, the linkage may be ambiguous: if +regions are the units receiving treatment, it is less clear what satellite information should be used in +modeling the treatment mechanism. In some situations, units are not geo-referenced at all: there may +be data privacy or other reasons why even the approximate location of experimental units is explicitly +not measured by researchers. Finally, it is important to note that, if disadvantaged units are less likely +to be geo-referenced to neighborhoods than more advantaged units, we will again be in a situation +where systematic missingness patterns bias causal estimates. These important limitations can also +present opportunities for future research. +4 +Experiments Illustrating Observational Causal Inference Dynamics +Under Model Misspecification and Varying Resolution +Although the identification results described in Section 3.4 are general, they are also theoretical and, +as described in Section 3.6, there are several practical considerations that will affect performance in +real data. In order to better understand these dynamics, we use simulation, since, in practice. true +causal targets are unknown. In these simulations, the image data is observed, but the confounding +features are not, and must be estimated, as is the case in real applications. +4.1 +Simulation Design +The simulation design centers on the scene level of analysis because, in practice, there seem to be +fewer situations where pixels (14.25 meter by 14.25 meter grid cells) are treated than the situation +where larger areas are treated (or where units are treated and we seek to adjust for confounding given +their entire neighborhood context). +In the simulation, confounding is generated by (1) applying a single convolution to our set of Landsat +images from Nigeria (operation by fl(·)), (2) finding the maximum similarity across an image to +11 + +the kernel filter (i.e. using the global maximum pooling operation), and (3) normalizing across +observations so that the resulting confounder has mean 0, standard deviation 1. The confounder then +enters equations for the treatment and outcome: +Uswh = fl(MsΠs(wh)) +Us = GN(max{Uswh : wh ∈ Πs}), +Pr(Ts|Ms) = logistic(βUs + ϵ(W ) +s +) +Ys = γUs + τ Ts + ϵ(Y ) +s +, +where GN(·) denotes normalization to mean 0 and variance 1, done to prevent all units from receiving +treatment. fl(·) denotes the linear kernel function, where the single kernel filter, l, is a diagonal +matrix and is shown in Figure 2. The set of images used in the simulation are drawn from the corpus +of Landsat satellite images that we later use in the application (see Section 5.1). The ϵ’s are Normally +distributed random error terms. +We vary the dimensions of the estimating convolutional filter, l, keeping the structure of the true +confounder-generating filter pattern fixed with a width of 9 (see Figure 2). By varying the estimating +filter dimensions, we alter the size of the neighborhood used in estimating the latent confounder, +allowing us to probe model misspecification, where there is a gap between the true data-generating +process and estimating model structure. +We also vary the resolution of the image used in estimation. This quantity varies from 1 (when the +observed image resolution is the same as used in confounder generation) to 0.12 (where the image +resolution observed in the estimation step is 12% that of the original; in other words, pixels are 88% +larger in width). By varying the resolution, we can probe how this unique feature of images affects +observational causal inference estimation in practice. +We compare two estimators. First, we examine the difference-in-means estimator, ˆτ0, defined as the +difference in conditional outcome means for the two treatment groups. Because of confounding, this +baseline quantity is biased for τ. +We also report results from an Inverse Propensity Weighting (IPW) estimator (Austin and Stuart, +2015), with image model estimation performed using a single-layer convolutional network model +which is trained using stochastic gradient descent with the binary cross-entropy loss function (which +is equivalent to the negative log-likelihood). The estimation formula for IPW is ˆπ(Ms) = �Pr(Ts = +1|Ms), �τ = 1 +n +�n +s=1 +� +TsYs +ˆπ(Xs) − (1−Ts)Ys +1−ˆπ(Xs) +� +for the scene analysis and is defined analogously at the +pixel level. We report results from the Hajek IPW estimator where the weights have been normalized, +reducing estimator variance at the cost of some finite sample bias (Skinner and Wakefield, 2017). +In our evaluation, we compare the true τ with the estimated values. We compute the absolute bias of +ˆτ and the Root Mean Squared Error (RMSE): +Absolute Bias = +��E[ˆτ − τ] +��; RMSE = +� +E[(ˆτ − τ)2], +(5) +where expectations are taken over the data-generating process and are approximated via Monte Carlo. +These evaluation metrics are then normalized using the MSE from the baseline estimator, ˆτ0, to +facilitate the interpretability of the results (so that Relative Absolute Bias and Relative RMSE are +reported). +4.2 +Simulation Results +In Figure 4, we show the dynamics of satellite image-based observational inference in this simulation. +As expected, we find that the absolute bias and RMSE are minimized when the kernel width used +in estimation is the same as the kernel width used to generate the true confounder and when the +resolution is the same as that used in generating the confounder. The bias and RMSE grow larger +when the estimating kernel width is greater than the kernel width used in confounder generation. +This finding is likely due to the fact that, when the neighborhood size used in estimation is larger +than that used to create the confounder, the additional parameters allow the model to overfit. The +bias and RMSE panels of Figure 4 look similar because the variance of estimation is relatively small, +especially when the resolution has been down-shifted. +12 + +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +Relative + Absolute Bias + + + + + +0.12 +0.12 +0.12 +0.12 +0.12 + + + + + +0.25 +0.25 +0.25 +0.25 +0.25 + + + + + +0.5 +0.5 +0.5 +0.5 +0.5 + + + + + +1 +1 +1 +1 +1 +3 +5 +9 +17 +33 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +Relative RMSE + + + + + +0.12 +0.12 +0.12 +0.12 +0.12 + + + + + +0.25 +0.25 +0.25 +0.25 +0.25 + + + + + +0.5 +0.5 +0.5 +0.5 +0.5 + + + + + +1 +1 +1 +1 +1 +3 +5 +9 +17 +33 +Estimating Kernel Width (# of Pixels) +Figure 4: Scene-level results varying the resolution and character of model misspecification. The +true confounder was generated with a kernel width of 9 and resolution scaling factor of 1. Resolution +scaling factors are numerically labeled and colored with grayscale. A resolution scaling factor of 0.5, +for example, indicates that the pixels in that analysis are twice as wide as in the original raw image. +Resolution also plays an important role in explaining the dynamics of observational inference with +satellite images. For all values of resolution, bias and RMSE of estimation are still favorable compared +to using the naive difference-in-means estimator without satellite-based neighborhood information. +In other words, low-resolution images still improve upon the simple difference-in-means estimator of +the treatment effect. +When the resolution of the image used in estimation is the same as that that actually generated +the confounder, performance in terms of bias and RMSE is optimal when model specification +is correct, but, with this higher resolution image, the analysis is also more sensitive to model +misspecification. When the resolution is down-shifted, model misspecification matters less; there is +less image information available so the dependence of the results on the image model is weakened. +We find similar results when we vary the degree of noise in the scene-level confounders, loosening +the determinism assumption discussed above (see Section A.2). +5 +Empirical Illustration: The Impact of Local Aid Programs on +Neighborhood-level Poverty in Nigeria +Having examined some of the opportunities and pitfalls of satellite-based observational inference, in +this section, we demonstrate it in the context of Nigeria, Africa’s largest economy and a country that +is projected to be the world’s second most populous by 2100 (Vollset et al., 2020). This context is +one in which identifying effective anti-poverty interventions carries substantial practical importance: +despite an average economic growth rate of around 3% since 2000, about 40 percent of the Nigerian +population live below the poverty line ($2 per day). In response, governments and NGOs have +deployed a variety of local aid programs to the country. However, the causal impact of these programs +is difficult to estimate (Roodman, 2008). +While geo-temporal data on poverty, Y , and interventions, T, are readily available, there is a lack of +geo-temporal data on potential confounders, U, at the local level (Daoud et al., 2017; Halleröd et al., +2013). While some of these confounders may be difficult to capture directly using images (such as +the quality of political institutions), there may be information present in remote sensing imagery +about other confounder objects related to infrastructure or agriculture (Schnebele et al., 2015; Steven +13 + +and Clark, 2013). We therefore use satellite images of Nigerian communities in order to estimate the +impact of local aid programs. +5.1 +Data Description +Our outcome data on poverty is drawn from the Demographic and Health Surveys (DHS), which +are conducted by a non-profit organization, ICF International, with funding from USAID, WHO, +and other international organizations (Rutstein et al., 2006). The DHS surveys are conducted at +the household level for randomly selected clusters that aggregate to geographically explicit scenes +of about 300 households for de-identification purposes. Our outcome measure is the International +Wealth Index (IWI) from 2018, which is a principal-components-derived summary of 12 variables +including households’ access to public services and possession of consumer products such as a phone. +Its scale is between 0 and 100, with higher values indicating greater wealth. +Our treatment data is drawn from a data set on international aid programs to Nigeria compiled by +AidData (AidData, 2016) used under an Open Data Commons License. The aid programs we examine +took place after 2003. The aid providers include entities such as the World Bank and WHO, as well +as domestic governments such as the United States. The programs we examine are diverse, focusing +on infrastructure (e.g., support for road development) and agriculture (e.g., support for small-scale +farmers), among other things. For the simplicity of our presentation, we take the presence of a +geographically-specific aid program within 7000 meters of a DHS point as our treatment. +Our pre-treatment image data is drawn from Landsat, the satellite imagery program operated by +NASA/USGS. We use the Orthorectified ETM+ pan-sharpened data derived from the raw satellite +imagery captured between 1998 and 2001; the raw data have been processed to contain minimal +cloud cover and to be correctly geo-referenced. Resolution is 14.25 meters; reflectance in the green, +near-infrared, and short-wave infrared bands is recorded. Treatment and control locations are shown +in the left panel of Figure 5. +Figure 5: The left panel identifies Nigeria as the context of interest in this observational study. The +right panel illustrates the location of treatment and control sites. Gray points are control locations; +black points are treatment locations. +5.2 +Observational Inference Image Model Design +Our image model for the treatment is built using three convolutional layers (32 filters each) with max +pooling. After the application of the filters, we project the channel dimension into a 3-dimensional +space to facilitate interpretability via a single image post-convolution. Batch normalization is used +on the channel dimensions and before the final projection layer. Weights are learned using ADAM +with Nesterov momentum with cosine learning rate decay (baseline rate = 0.005; Gotmare et al. +(2018)). We compare performance across a variety of filter sizes. We attempt to limit overfitting by +randomly reflecting each image dimension with probability 1/2 during training. We assess out-of- +sample performance using three random training/test splits, averaging over this sampling process +14 + +0.2 +0.3 +0.4 +0.5 +0.6 +Out−of−Sample + Loss +Loss +0.1 +0.12 +0.17 +0.17 +0.25 +0.25 +0.25 +0.5 +0.5 +0.5 +1 +1 +1 +3 +5 +7 +Baseline +Estimating Kernel Width +5 +10 +15 +20 +Effect Estimate +Estimate Value +3 +5 +7 +Baseline +0.1 +0.12 +0.17 +0.17 +0.25 +0.25 +0.25 +0.5 +0.5 +0.5 +1 +1 +1 +Figure 6: The left panel shows out-of-sample binary cross entropy loss compared to the baseline loss +when guessing the dominant class. The right panel shows the IPW estimate values across the range +of estimating kernel width values compared to the baseline difference in conditional means. +and reporting 95% confidence intervals from the three test set assessments. We vary the image +model structure by altering the convolutional filter width (which affects the size of the image patterns +analyzed in the image model). +We also assess stability of the results to resolution of the satellite images used. This task contains +some subtleties. It is always possible to lower the resolution of an image, averaging across pixel +grids. However, when we change the dimensionality of the input image, we may no longer be able +to employ the same image models as in the full dimensionality case. The reason for this is due to +the fact that each convolutional layer in the image model reduces the output dimensionality; if our +starting resolution is too small, the implied output dimensionality of some of the convolutional layers +would be negative. We keep the image model structure the same and analyze results across different +resolutions. As a consequence, when the image model cannot be fit with a given kernel width and +resolution combination, we drop that model from the set of models analyzed. +5.3 +Empirical Results +First, in the left panel of Figure 6, we assess the propensity model fit to the treatment data. We +find that the image model always improves on the baseline out-of-sample loss value obtained by +random guessing of the dominant class (the control class, comprising 71% of the data sample). +Performance is stable across values of the kernel width used in estimation. Moreover, resolution does +not greatly affect performance. This stability implies that more macroscopic aspects of the landscape +are ultimately what is driving the estimated treatment assignment mechanism, as opposed to the +lower-level features such as the grouping of houses of a certain type. +Next, in the right panel of Figure 6, we analyze the estimated treatment effects. We find that, across +the estimating kernel width range, adjusted estimates are positive but smaller in magnitude than +the baseline difference in means. This hints at the importance of confounder adjustment, as these +programs may be given to areas already primed for growth. +We also analyze the estimation model dynamics. In Figure 7, we visualize data for the three out-of- +sample control (left) and treated (right) units. We also display the salience maps for the predicted +15 + +Raw Image +Salience Map +Final Spatial Layer +Lat, Long: 12.49, 9.303 +xlim +xlim +Lat, Long: 12.29, 9.170 +xlim +xlim +Lat, Long: 12.91, 9.917 +xlim +xlim +Lat, Long: 10.50, 7.439 +xlim +xlim +Lat, Long: 6.469, 3.346 +xlim +xlim +Lat, Long: 6.614, 3.372 +xlim +xlim +Figure 7: The three left panels depict the raw data for control units, salience maps for the predicted +treatment probability, and output from the final spatially resolved layer in the image model. The three +right panels depict the same things for treated units. Note that the three bands of the satellite image +are not “red”, “green”, and “blue” (see main text), so in this visual representation, reddish pink +indicates soil moisture content. +treatment probabilities as well as the output from the final spatially resolved layer in the image model +when the kernel width is 7. To explore robustness, we also display in Section A.3 results from another +run with a kernel of 7 in the estimation model, as well as width kernel widths 3 and 5. +To take examples from the model output, a particularly low treatment probability site is estimated in +the remote desert city of Machina (pop. 62,000); a high probability site is from the city of Katsina +(pop. 429,000) near a large agricultural basin and with rich water resources (water soil content is +displayed as red in the RBG satellite images using non-visible band). The output of this model would +seem to resonate with the fact that many of the projects undertaken by global actors are specifically +designed to assist farmers and agriculture more generally. +Finally, we assess the performance of the estimated propensity scores on reducing covariate imbalance +between treated and control groups. The same absence of rich covariate information in places such as +sub-Saharan Africa that motivates this paper also makes this assessment task difficult. Still, we can +analyze differences in longitude/latitude between treated/control groups. We find a raw difference of +(-1.12, -1.13). After weighting, this difference decreases in magnitude to (-0.39,0.53), indicative of +improved counterfactual comparisons. +Overall, this application shows some of the nuances of satellite image-based observational inference— +how resolution affects the space of possible image models, how to think about interpreting the output +of the causal inference model in the image context, as well as how to validate the improvement in +counterfactual comparisons after employing the satellite information. +6 +Discussion and Future Work +In this paper, we characterize key challenges and opportunities of satellite image-based observational +causal inference. We formally show that, even though confounder aspects of a neighborhood may be +latent, we can adjust for them using the image information. There are several subtleties, however, +related to resolution and the degree to which the image truly proxies those confounders. We illustrate +some of these tradeoffs in simulation. We also apply the satellite image-based observational inference +approach in order to understand the causal effect of aid programs on reducing neighborhood-level +poverty in Africa’s largest economy. As previously stated, our approach is not limited to poverty. It +can be used for analyzing the effects of environmental factors (Shiba et al., 2021; Daoud et al., 2016) +to neighborhood-level change (Lin et al., 2022), and beyond. +16 + +The use of satellite images in observational inference approach has some inherent limitations. For +example, as described above, confounding may be due to objects that cannot be resolved in the +image data, and, as a result, bias reduction will not occur conditioning on the image information. In +this context, the collection and application of imagery with higher spatial, temporal, and spectral +resolution is a priority. Spatial and temporal resolution may both be achieved, for example, using +sensors mounted on ground-based infrastructure (e.g., Johnston et al. (2021)); spatial resolution and +extent could be optimized with drone- or airplane-based instruments (e.g., Gray et al. (2018)). Future +considerations should examine privacy and fairness issues with causal analyses based on passive +sensor technologies. +Second, while in some cases, such as if disparate individuals are treated, the scene-level unit of +analysis is clearly defined (e.g., the individual within a neighborhood context), in other contexts, the +scene-level unit of analysis is more ambiguous. The researcher therefore has choices about how to +define the scene (e.g., at the street, village, or region levels), a choice that could introduce systematic +bias into the analysis. This issue is known as the modifiable areal unit problem (Fotheringham +and Wong, 1991), and the approach described here is vulnerable to it as well. 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Publisher: SAGE Publications Inc. +24 + +A +Appendix +A.1 +Connections with the Causal Proxy Literature +Besides facilitating the use of causal inference for a social science audience, our work is related to the +literature on identification via proxies (Tchetgen et al., 2020) or drivers (Pearl, 2013a) of confounders. +For the former, Louizos et al., developed Causal Effect Variational Autoencoder (CEVAE) which uses +proxies to infer the distribution of the latent confounder and use this in adjustment. In contrast, our +approach adjusts for an observed variable—the image. We formalize key assumptions required for the +correctness of this method and provided a general framework for conducting causal inference using +images, where unlabeled objects in the image may affect both treatment and outcome (Castro et al., +2020). This image-based confounding bias might in some circumstances be equivalent to traditional +spatial interdependence, but differs insofar as the confounding bias is defined with reference to +unlabeled entities in the image, thereby injecting bias (Paciorek, 2010). Relying on our formalization +and model implementation, we analyze aid interventions (treatment) and poverty (outcome) in Africa— +something of policy relevance as policymakers often rely on satellite images for aid intervention +(Voigt et al., 2016; Bedi et al., 2007). +A.2 +Additional Simulation Results +A.2.1 +Probing Estimation Bias as the Determinism Assumption is Relaxed +We here explore how model misspecification affects estimation error. In particular, we probe how +relaxing the determinism assumption of Proposition 1 affects satellite-based observational inference. +In particular, we now let the unobserved confounder be a random function of the satellite image, as +depicted visually in Figure 3b. In particular, the confounder values are now +Uswh = fl(MsΠs(wh)) + ϵ(U) +swh, +(6) +where ϵ(U) +swh ∼ N(0, σ2 +U). We vary σ2 +U ∈ {1, 3, 5, 7}. We then apply the same data-generating process +to obtain scene-level treatments and outcomes. +We see in Figure A.1 how performance is affected by relaxing the determinism assumption. As +expected, we see that estimation bias grows as the unobserved confounding is increasingly determined +by the noise factor, ϵ(U) +swh. When the noise scale is at its maximum, bias is still no worse than the +simple difference in means baseline (i.e., relative bias/RMSE approaches 1). This fact is likely +because the noise injected into the confounding mechanism is itself exogenous. Nevertheless, having +established a theoretical baseline in this paper, future research should examine this noise-induced +confounding to image-based causal inference in greater detail, connecting this line of work with the +proxy literature. +1 + +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Relative + Absolute Bias +1 +3 +5 +7 +Relative RMSE +1 +3 +5 +7 +Non−image Unobserved Confounder Scale +Figure A.1: Bias and RMSE for the scene-level analysis as we vary the stochasticity present in the +confounding mechanism, holding the estimation kernel width fixed at 8. Gray circles indicate effect +estimate values using the true (in practice unobserved) treatment probabilities. +A.3 +Robustness to Model Specification in the Empirical Results +Raw Image +Salience Map +Final Spatial Layer +Lat, Long: 12.49, 9.303 +xlim +xlim +Lat, Long: 12.29, 9.170 +xlim +xlim +Lat, Long: 12.91, 9.917 +xlim +xlim +Lat, Long: 10.50, 7.439 +xlim +xlim +Lat, Long: 6.469, 3.346 +xlim +xlim +Lat, Long: 6.614, 3.372 +xlim +xlim +Figure A.2: Replicating Figure 7 with another training/test split. +2 + +Raw Image +Salience Map +Final Spatial Layer +Lat, Long: 9.635, 8.802 +xlim +xlim +Lat, Long: 12.34, 6.705 +xlim +xlim +Lat, Long: 12.58, 12.66 +xlim +xlim +Lat, Long: 5.094, 7.349 +xlim +xlim +Lat, Long: 8.481, 4.520 +xlim +xlim +Lat, Long: 6.633, 3.323 +xlim +xlim +Figure A.3: Replicating Figure 7 with an estimating kernel width of 5 instead of 7. +Raw Image +Salience Map +Final Spatial Layer +Lat, Long: 12.55, 9.643 +xlim +xlim +Lat, Long: 13.29, 5.163 +xlim +xlim +Lat, Long: 9.577, 11.42 +xlim +xlim +Lat, Long: 6.459, 7.489 +xlim +xlim +Lat, Long: 5.094, 7.349 +xlim +xlim +Lat, Long: 6.661, 3.288 +xlim +xlim +Figure A.4: Replicating Figure 7 with an estimating kernel width of 3 instead of 7. +3 + +A.4 +Implementation Details +We implement our computational analyses on an Apple M1 GPU using Metal-optimized TensorFlow +2.11 with GNU Parallel. Total compute time for the simulations is about 48 hours; total compute +time for the application results is about 24 hours. +4 + diff --git a/b9FPT4oBgHgl3EQfBTTg/content/tmp_files/load_file.txt b/b9FPT4oBgHgl3EQfBTTg/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ea562616be42dd37c66d8690c418096364cc9b0f --- /dev/null +++ b/b9FPT4oBgHgl3EQfBTTg/content/tmp_files/load_file.txt @@ -0,0 +1,1394 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf,len=1393 +page_content='Integrating Earth Observation Data into Causal Inference: Challenges and Opportunities Connor T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Jerzak Department of Government University of Texas at Austin Email: connor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='jerzak@austin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='utexas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='edu Website: ConnorJerzak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='com Fredrik Johansson Data Science and AI Division Chalmers University of Technology Email: fredrik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='johansson@chalmers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='se Website: fredjo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='com Adel Daoud Institute for Analytical Sociology Linköping University Email: adel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='daoud@liu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='se Website: AdelDaoud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='se AI and Global Development Lab: global-lab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='ai Abstract Observational studies require adjustment for confounding factors that are corre- lated with both the treatment and outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' In the setting where the observed variables are tabular quantities such as average income in a neighborhood, tools have been developed for addressing such confounding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' However, in many parts of the developing world, features about local communities may be scarce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' In this context, satellite imagery can play an important role, serving as a proxy for the confounding variables otherwise unobserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' In this paper, we study confounder adjustment in this non-tabular setting, where patterns or objects found in satellite images contribute to the confounder bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Using the evaluation of anti-poverty aid programs in Africa as our running example, we formalize the challenge of perform- ing causal adjustment with such unstructured data—what conditions are sufficient to identify causal effects, how to perform estimation, and how to quantify the ways in which certain aspects of the unstructured image object are most predictive of the treatment decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Via simulation, we also explore the sensitivity of satellite image-based observational inference to image resolution and to misspecification of the image-associated confounder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Finally, we apply these tools in estimating the effect of anti-poverty interventions in African communities from satellite imagery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Keywords: Earth observation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Causal inference;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Neighborhood dynamics Word count: 12,207 Note: This work largely subsumes Jerzak, Connor T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=', Fredrik Johansson, and Adel Daoud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' “Estimating Causal Effects Under Image Confounding Bias with an Application to Poverty in Africa.” arXiv preprint arXiv:2206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='06410 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Preprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='12985v1 [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='ML] 30 Jan 2023 1 Introduction The causal revolution in the social sciences has entered a new phase where scholars are increasingly combining traditional tabular data with novel data sources (Daoud and Dubhashi, 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Morgan and Winship, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Imai, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Pearl, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' And, as a response to increasing data digitization and availability, a growing literature has emerged more recently that seeks to provide valid estimation of treatment effects in the presence of high-dimensional confounders, where the number of variables is large relative to the number of observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' (Li and Pearl, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Mozer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Chernozhukov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Yoon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Wager and Athey, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Shalit et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Hill, 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Schneeweiss et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=', 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Belloni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Alexander et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Yet, merely adding more covariates describing each unit’s context does not itself address concerns about non-random missingness within that confounder data, about the difficulty in obtaining information, about historical interventions, and about the lack or unreliability of data in the most economically disadvantaged places (Jerven and Johnston, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Dinku, 2019)—precisely those in need of effective interventions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' In this article, we argue that satellite data and remote sensing information provide an important resource for expanding the reach of causal inquiry in otherwise data-scarce environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Satellite data is available for every corner of the globe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' This breadth of information is available on a historical basis stretching back to the 1970s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Moreover, unlike a sizable proportion of datasets which are a single snapshot of a social system at a particular point in time, many earth observation satellites return to every place on earth every two weeks or even more frequently—providing 26 or more temporal slices per year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' The temporally-resolved information contained in satellite data has been shown to be associated with variables of social science importance often characterized as neighborhood features—features such as the development of transportation networks (Nagne and Gawali, 2013), the degree of urbanness (Schneider et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=', 2009), health and material conditions (Daoud et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=', 2021), living standards (Jean et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Yeh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=', 2020), and a host of other neighborhood features (Sowmya and Trinder, 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' For these reasons, data from space-borne instruments may provide critical information for causal inference analyses, especially given the rapid proliferation of earth observation satellites from the hundreds into the thousands Tatem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' (2008) and sub-100 cm resolution now widely available on the latest generation of satellites (Hallas, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Despite the potential offered by satellite images in causal inference, there is a lack of methodological guidance for causal estimation when confounding is induced by patterns or objects observed in an image (Castro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' To help fill this need, we examine observational causal inference in the presence of confounding captured as latent image patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' A story-based intuition for we have in mind is the following: an actor examines a neighborhood, looking for certain aspects of that neighborhood (such as the presence of poverty) to guide the choice of intervention, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Those neighborhood aspects, U, are observed by the decision maker but unobserved by outside analysts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Some of that neighborhood-level information is also embedded in the satellite data representation (M) of that same neighborhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Thus, observed image patterns indicate the existence of real-world objects that provide information about the confounding factors associated with both treatment and outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' We may not directly observe the true latent variables about which the treatment decision was made, but we can readily observe, from earth observation resources, and adjust for inferred image patterns that correlate with the treatment, even if these patterns are difficult to adjust for directly (Voigt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Our focus in this paper on causal inference with earth observation data complements a social-science research trend, especially in sociology and political science, where scholars increasingly leverage visual data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' For example, such visual data are used in qualitatively analyzing photos (Pauwels, 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' O’Hara and Higgins, 2019), estimating image similarities (Zhang and Peng, 2022), and approximating the number of demonstrating people from news photos (Cruz and González-Villa, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Recently, the use of visual data also includes video data for analyzing social processes, such as police violence (Nassauer and Legewie, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' For larger visual data and in a quantitative design, scholars have to train algorithms “learning to see” objects of interest (Torres and Cantú, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Nonetheless, although these contributions are critical for research designs relying on image data, there is a need for deeper grounding in the causal inference literature (Daoud and Dubhashi, 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Morgan and Winship, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Imai, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Pearl, 2009), creating a knowledge gap about how to leverage images for causal inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Our article contributes to filling that gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' In what follows, we first in Section 3 describe several causal structures relevant to satellite image- based causal inference and discuss their implications for identification and estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' We focus on under what conditions the image alone is sufficient to adjust for the confounding introduced 2 by U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' This holds, for example, when U may be derived deterministically from M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' An important special case of image pattern confounding occurs when decision-makers make choices based on the (translation-invariant) existence of a pattern in the image, which motivates adjustment techniques based on convolutional models developed in the machine learning community (Goodfellow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' We also analyze the complementary case where the image confounder is itself the cause of the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' We next study in Section 4 finite-sample estimation of average treatment effects in the fully identified case by conducting a simulation in which confounders are derived from the observed image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' In this setting, we investigate the impact of model misspecification on estimates, as well as the role of image resolution—a key aspect of image data with no perfect analog in tabular, network, or text data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Finally, in Section 5, we demonstrate the use of the proposed estimation framework in an application in which we evaluate the effectiveness of international aid programs on neighborhood-level poverty by estimating treatment propensity using geo-referenced satellite data to proxy for confounding factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' 2 Related Work and Contribution Observational causal inference methods primarily evolved in the context of tabular covariates (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=', see Imbens and Rubin (2016)), where separate, human understandable features are used as covariates in adjusting for confounding via regression (Best and Wolf, 2013), weighting Jung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' (2020), or doubly robust (Funk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=', 2011) methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' While tabular covariates are readily interpretable and can be tailored by researchers at the time of data collection, they face several limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' For example, tabular covariates such as gender, income, and ethnicity are typically collected at the time of data collection by researchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Thus, they cannot capture historical confounders (such as income pre-data collection) without additional effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Moreover, they are subject to missingness or mismeasurement due to unit-level behavior: a unit may decide–or not—to answer a survey question or may mask their true opinions, both of which induce bias into the resulting causal analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' The causal text analysis literature has recently developed interesting insight into how text can supplement tabular covariates in observational causal inference, emphasizing how text information can provide insight into why treatments were assigned (Grimmer and Stewart, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Egami et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Keith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=', 2020), even if there is a risk posed by the open-ended quality of text that can lead to bias in observational analyses Daoud et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' While text does pose an opportunity for investigators, it is limited by availability pre-intervention: much text is gathered via web resources, so historical interventions can be difficult to model via text, not to mention the fact that some interventions might not have relevant text associated with them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' While text is a promising new data source for causal inference, it is not available universally for all spatially defined interventions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Social network data has also been increasingly emphasized in the causal inference setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' The typical use of this network data is not to adjust for confounding bias but instead to model how units respond to the treatment status of social connections, primarily to estimate spillover effects VanderWeele and An (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Network data must be gathered for every unit about every other unit: this kind of data source scales non-linearly—with a polynomial growth in the number of possible connections as the number of units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' As a consequence, network data has generally been difficult to obtain outside a few contexts such as trade relations, alliances, or online social networks, where social links are assumed to be online website links (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=', Lewis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' (2012)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Because of the computational complexity, the temporal resolution of network data used in social science research is generally refined at best on a yearly basis, as in international relations network analyses (see, for example, Lebo and Box-Steffensmeier (2008)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' In this context, satellite image data potentially addresses a number of the limitations of other data streams for observational causal inference (see Table 1 for a summary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Satellite image data— obtained from space-born instruments funded by NASA, the European Space Agency, and others—is high-dimensional and readily interpretable as a raw data source in that human observers can generally identify key features of the data, such as the presence of a city, forest, river, soil moisture content, and so forth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Satellite data are also available for all geo-referenced interventions—not just those about which a human observer decided to write a document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Moreover, while satellite data does have missingness due to cloud cover, experimental units cannot affect the availability of satellite images about their neighborhood, an inability that reduces the risk of bias due to systematic data missingness 3 (Kenward and Carpenter, 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Finally, remote sensing data also has a temporal resolution of two weeks or better for many leading satellite image providers—meaning that, in principle, one can examine the evolution of a neighborhood with 26 or more data slices per year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Table 1: Comparing satellite imagery data with other data types in the context of observational causal inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' ∗ The European Space Agency’s Sentinel earth observation satellites revisit locations at least once every five days;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Landsat revisits locations around once every 14 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='Tabular ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='Text ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='Network ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='Earth ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='Observation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='Origin & Character ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='Common origin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='Researcher ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='surveys/gov’t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='Web ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='resources ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='Online social ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='network ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='databases ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='Space-born ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='instruments ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='High-dimensional ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='Not usually ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='Yes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='Yes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='Yes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='Unstructured in raw ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='form ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='No ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='Yes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='Yes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='Yes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='Readily interpretable ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='as a raw data source ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='Yes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='Yes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='No ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='Yes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='Scaling with # of units ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='Linear ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='Linear ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='Polynomial ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='Linear ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='Availability ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='Available for all ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='geo-referenced ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='interventions ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='No ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='No ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='No ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='Yes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='Readily available ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='pre-intervention ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='No ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='No ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='No ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='Yes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='Typical temporal ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='resolution ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='Singular/1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='year ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='Singular ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='Singular ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='5-14 days∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='Subject to missingness ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='due to unit behavior ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='Yes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='Yes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='Yes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='No ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='The machine learning community has begun exploring the opportunities of images in the causal ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='image context (Castro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Reddy and Balasubramanian, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Du et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Related work has been done on estimating counterfactual outcomes under spatially defined counterfactual treatment strategies (Papadogeorgou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=', 2020), on accounting for spatial interdependence in causal effect estimation (Reich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=', 2021), and on balancing observed covariate representations using adversarial networks and image data (Kallus, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Other works address images as treatments (Kaddour et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=', 2021), counterfactual inference and interpretability (Pawlowski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Singla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=', 2020), and image-based treatment effect heterogeneity (Jerzak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=', 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' There are also several important works on how algorithms may discover causal structures from images in the causal discovery tradition (Chalupka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=', 2016b,a, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Schölkopf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Yi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Ding et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' What we can add to this literature is twofold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' First, we provide a systematic analysis on how to understand the latent confounding structure proxied for or induced by images in the causal inference setting, where latent objects in the image may affect treatment and outcome (Castro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Developing techniques for causal inference under image pattern confounding could open up new avenues for observational studies (Pawlowski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Singla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Kaddour et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' (See Section A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='1 for a more expansive connection developed with the proxy literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=') Second, this analysis also specifically provides guidance for social scientists interested in using earth observation resources to perform observational causal inference in neighborhood or developing contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' As mentioned, earth observation data provides a large amount of raw information about 4 potential confounding variables of importance in social science research: we here analyze the promise and pitfalls of this emerging analytic pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' To conclude this section, we also note that this work on integrating satellite images into causal pipelines in data-poor environments also has policy and practical resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Policymakers may consult maps to evaluate where to allocate aid to villages that otherwise may remain poor (Holmgren and Thuresson, 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Bedi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=', 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' For example, since 2000, policymakers frequently rely on raw satellite images to evaluate damage due to natural disasters or war (Voigt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Burke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Kino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Based on these images, they decide where to intervene in helping the poor (Borie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Thus, the use of satellite images in causal inference may help those in the policy world as well as evaluate anti-poverty programs, especially since areas experiencing poverty are also often affected by weak state capacity (Besley and Persson, 2010), meaning that the poorest places are often those about which we know least in the absence of remote sensing information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' 3 Characterizing Challenges and Opportunities of Satellite Image-based Observational Inference A major reason why satellite images can serve as a useful tool for observational causal inference is that the satellite images can proxy for confounders that explain why some places but not others received a treatment of some kind, such as an anti-poverty aid program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' With this motivation in mind, we study the identification and estimation of the average treatment effect (ATE) of T on a real-valued outcome of interest, Y , based on observational data (suppressing the unit subscripts of the various random variables except when important for expository purposes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' With Y (1) being the potential outcome (Rubin, 2005) under intervention with T = 1 and Y (0) for T = 0, ATE = E[Y (1) − Y (0)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' The ATE can represent, for example, the difference in average wealth in villages after anti-poverty interventions and after no intervention, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' In that setting, historical interventions can be thought of as being determined by a decision-maker using information about neighborhoods that are proxied by a satellite image representation, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Whether we think that the image patterns cause the confounder or the confounder dynamics cause the image pattern is of little consequence in practice: in the latter, the image is proxy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' in the former, the image is a driver;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' the same type of analysis will be done in either case (Pearl, 2013b) as described in the proxy literature (Kuroki and Pearl, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Louizos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' To understand the estimation dynamics of the ATE from observational images and adjust for induced confounding bias, we analyze the data-generating process next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='1 Baseline Model of Confounding Bias To understand similarities and differences of observational causal inference in the tabular case and in the satellite image context, first consider the causal graph in Figure 1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Note that we reintroduce unit-level subscripts in the Figure to emphasize some distinctions: s denotes a scene (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=', village), and w and h denote the width and height indices of a spatially resolved context like an image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' This figure depicts classical confounding: a treatment of interest (Tswh), such as an anti-poverty intervention, is associated with factors (such as the presence of mineral extraction sites) that affect both the treatment and the outcome (Yswh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Observed confounding variables are grouped in Xswh;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' unobserved confounders in Uswh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' In the general, s need not be spatially defined when working with non-satellite-derived images, like if the images were to be X-ray snapshots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' In that case, the scene index would refer to a patient or body part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' However, in the social science setting with satellite data, s will usually refer to some neighborhood or the neighborhood context around an observational unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='2 Pixel-based Image Confounding We now turn to the causal model in Figure 1a, which depicts the kind of confounding dealt with in much of observational research (Rosenbaum and Rubin, 1983).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' We extend this model to describe image-based confounding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' 5 Uswh Tswh Yswh Xswh (a) Baseline causal diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Uswh Msw′h′ Tswh Yswh Xswh w′, h′ ∈ Πs(wh) (b) Diagram illustrating image-based confounding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Figure 1: Diagram representing variables associated with a scene s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' In our running example, Uswh represents unobserved confounders, Xswh observed confounders, Tswh treatment and Yswh the outcome, all at location w, h in scene s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' In the right-hand diagram, latent confounders Uswg are determined by a neighborhood Πs(wh) of the location h, w in the image M representing scene s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' The arrow between Uswh and Msw′h′ is bi-directional to indicate that we are agnostic about the direction of causality (whether the image “causes” the confounder or the confounder “causes” the image).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' First, we define this model at the local level, where treatments are implemented at specific locations w, h (in, for example, the precision agriculture context where treatments are applied to small sections of land;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Liaghat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' (2010)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' To describe causal dependencies at the neighborhood level, we introduce the following notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Let Πs(wh) ∈ N2 denote the set of location indices involved in generating the confounder value from the neighborhood around swh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' For example, if the confounder at swh were generated using neighborhood information in a z × z region around swh, Πs(wh) = � w − ⌊z/2⌋, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=', w + ⌊z/2⌋ � × � h − ⌊z/2⌋, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=', h + ⌊z/2⌋ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' For example, if z = 2, Πs(wh) = {w − 1, w, w + 1} × {h − 1, h, h + 1}, where the × symbol here denotes the Cartesian product capturing all ordered pairs of the left and right set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' In other words, Πs(wh) simply characterizes the set of width and height indices centered around location wh in scene s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' With this notation, we can illustrate the confounding structure induced by image-based confounding at the neighborhood level in Figure 1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Figure 1b is a formulation of spatial interdependence, as the image-information for indices in Πs(wh) affects the confounder Uswh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Conditional on the value of the confounder, the treatment decision for each unit is made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' This confounding dynamic would be invoked if a decision maker who scans a scene looking for similarity of the neighborhood around swh to some mental image, defined, for example, by an image filter l, and makes a decision on this basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' We illustrate this process on satellite image data from Landsat, a U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Geological Survey and NASA satellite (see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='1 for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' We perform the illustration using a particular parametric model for the neighborhood-induced confounding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' The parametric model used for illustration is convolutional: we let a single convolutional filter in the form of a diagonal matrix represent an image pattern used by a decision maker to determine the treatment probability, as depicted in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' After applying fl to the raw image shown in the right panel of Figure 2, we obtain the resulting image- derived confounder values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' “Applying the filter to the image” here means calculating a similarity score at every place in the original image with the filter (the “image pattern”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' This score generates a new, latent image representing the similarity structure which, here, underlies the confounding dynamic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' The similarity score is calculated by summing up the result of multiplying the filter with the relevant image pixels (similar to how the leading term of the covariance between A and B would be calculated by taking the average of the product of A and B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' This simple example shows how the presence of objects or patterns in images (as represented by the diagonal line here) can generate confounder values in the context of satellite-based observational inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' 6 Raw Image After Convolution Figure 2: Left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' The kernel filter pattern used to generate Uswh in the right-hand image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Center, right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Illustration of image-based confounding using Landsat data for Nigeria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' The center panel depicts the raw reflectance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' the right panel depicts the transformed values after convolution with the filter, values which enter the model for treatment/outcome to generate confounding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='3 Scene-based Image Confounding While the pixel-based confounding structure may be relevant if small sections of land (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=', houses) receive treatment, in many social science applications, the image is defined at one resolution, but the treatment, outcome, and confounder potentially at another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' In the aid context, the treatment and outcome data are often defined at the neighborhood or village level, but the satellite data itself contains additional height and width dimensions (in a word, we can “peek inside” the village).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' To take another example, a policymaker may examine an entire village, looking for the maximum or average similarity to some target pattern: the village is the unit to which treatment is allocated, and the confounder is defined at that level but is created using more granular information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Satellite image-based observational inference can accommodate situations such as these, where the confounder, treatment, and outcome are defined at different scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' In particular, we can add the following to our causal model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Let Πs ∈ N2 denote the height and width indices (locations) used when aggregating up information to the final scene-based unit of analysis, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' The right panel of Figure 2 then illustrates a scene-based confounding structure generated by Πs, where Πs intuitively represents the image indices associated with the whole neighborhood context that was used in deciding upon treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' To further explain the role of Πs, consider how, if investigators are interested in a village, s, the width of the satellite image collected around s fixes the maximum size of Πs that can be accommodated in the estimation procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' If the image is, say, 1024 by 1024 pixels, then the cap on Πs is {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=', 1024} × {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=', 1024}, which specifies a square of 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='59 km in width using the Landsat images used throughout this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' In short, the maximum neighborhood around the village that can be used to recover the latent confounder is fixed by the width of the satellite image input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' In general, this scene-based causal model is significant because, while some studies are able to obtain resolved (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=', household-level) outcome data, this data may, in other cases, be costly or even impossible to obtain due to privacy reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' In these situations, treatment and outcome data can only be measured at the scene level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' We next turn to the question of identification in this image-confounding context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='4 Identification in Satellite Image-based Observational Inference We can confirm that under image-based confounding as formalized in Figures 1b and 3a, treatment effects may be identified by adjusting for the image M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Here, we suppress dependence on the indexes s, w, and h since the same arguments will apply if T, U, or Y are defined at the swh (pixel) or s (scene) levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' To begin, we assume that the confounder U is a deterministic function of M and return to the case where U has multiple causes later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' This is justified, for example, in applications where confounding is based on the existence of an object—either if the policymaker scanned M for the object prompting the policymaker to allocate a treatment in that area of interest, or if M is more generally associated with important confounder patterns/objects without error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' 7 Uswh Us Msw′h′ Ts Ys Xs w, h ∈ Πs w′, h′ ∈Πs(wh) (a) Image-based confounding at the scene level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Uswh Us Rs Msw′h′ Ts Ys Xs w, h ∈ Πs w′, h′ ∈Πs(wh) Objects not in image (b) Some confounding not observed in the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Figure 3 As U is determined fully by M, ruling out other potential noise sources, there exists a deterministic function f such that U = f(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' The aforementioned case of U being the (pooled) convolution of a 2D filter with the image M satisfies this assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Suppose the confounder U is deterministic given the image M, such that U = f(M), (with f unknown), and that the causal structure obeys either of Figures 1b & 3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Then, p(Y (t)) and therefore ATE of T on Y is identifiable from p(M, X, T, Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' For simplicity of exposition, we give the proof for the case without additional confounding variables X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' The proof generalizes readily to non-empty X, by marginalization and conditioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' The claim follows from U being a deterministic function of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' By the backdoor criterion applied to the graphs in Figures 1b & 3a, X, U is an adjustment set for the effect of T on Y , which implies the exchangeability of potential outcomes: Y (t) ⊥ T | X (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=', Hernán MA (2020)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' In the case of empty X, p(Y (t)) = � u p(Y |T = t, U = u)p(U = u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' (1) Since U is a deterministic, but not necessarily invertible function of M, U = f(M), we have that p(Y | T = t, U = u) = p(Y | T = t, M ∈ f −1(u)) and p(U = u) = � m∈f −1(u) p(M = m) (2) where f −1 is the inverse map of f, so that p(Y (t)) = � u � m∈f −1(u) p(Y |T = t, M = m)p(M = m) = � m p(Y |T = t, M = m)p(M = m) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Hence, M is also an adjustment set for T on Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' From a similar proof, we see that X, M is an adjustment set in the case of non-empty X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' From here, standard arguments (Rosenbaum and Rubin, 1983) show that E[Y (t)] = E � Y p(T = t) p(T = t | M = m) ���� T = t � (3) which justifies the use of inverse-propensity weighting with respect to M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' The argument of Proposition 1 rests on the assumption that the image contains all information about the latent confounder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' When treatment decisions are made based on object detection, this assumption would be satisfied if the image contains all objects that are relevant to the outcome and treatment decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' This is violated if, for example, unlabeled objects, depicted as Rs in Figure 3b, 8 are themselves the driver of the treatment decision but are not possible to reconstruct from the image data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' If the image data imperfectly depicts those objects, full identification is no longer possible, as there is a possibility of residual confounding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Specifically, the inverse map f −1 in Equation 2 is no longer uniquely or well defined as a set of images;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Rs must be adjusted for as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' In this imperfect case, the image becomes a driver of the confounding, and thus, has similar properties to proxies (Pearl, 2013b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Peña, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' We have shown how, under assumptions, the image is itself an adjustment set for estimating the effect of programs on outcomes in the context of image-based confounding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' However, non-parametric inference is difficult in the image context because no two images are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Thus, the probability of seeing identical treatment/control images is zero, violating overlap assumptions necessary for model-free inference (D’Amour, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Machine learning models for the image may seek to estimate U, forming latent representations for the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' In this lower-dimensional space, there is more likely to be empirical overlap between treatment/control, justifying the use of modeling approaches like the ones discussed in Section 4 and Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Thus, while adjusting directly for U would fulfill the overlap assumptions optimally, this is infeasible;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' when adjusting for M instead, a critical argument for our approach to work is that the propensities depend only on aspects of M that capture U, aspects assumed to be compressible to a lower dimensional representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' As such, the situation is more benign than if propensities depended freely on all patterns in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Nevertheless, results from parametric models can be sensitive to the details of model specification—something that can partially be addressed by robustness checks varying key parameters but cannot be conclusively settled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='5 Interpretation in Satellite Image-based Observational Inference Having discussed identification, we now turn to important questions around estimation and interpreta- tion in satellite image-based observational inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' With tabular data, interpreting results is generally straightforward for two reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' First, features in a tabular dataset are human interpretable: we have measurements on pre-treatment variables such as age and ethnicity in a tabular dataset, and those quantities can be readily communicated via language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Second, linear models predominate in much of observational inference—both in weighting methods (which often involve a logistic regression step) or in methods modeling the outcome (where OLS is often applied).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Linear models have a particularly simple structure where the relationship between the inputs and the predictions can be conveyed simply via regression coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' For example, in the OLS context (with covariate vector, X, and with no interaction/polynomial terms), we can readily communicate how one thing relates to another: ∂ �E [Y | X] ∂ Xd = βd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' With linearity, gradients are the same for all values of Xd, leading to the interpretation of OLS coefficients: a one unit change in the predictor Xd is associated with a βd unit change in the outcome, holding all else equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Interpretation (in the sense of understanding how covariate inputs relate to model outputs) is arguably straightforward—at least if we sidestep the subtle issue of what it truly means to “hold all else equal.” In the image context, we can augment this derivative-based notion of interpretability via the use of salience maps Gilpin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' In particular, we can examine the following quantity in observational inference to explain what the prediction of the treatment assignment is particularly sensitive to in the image: S = � � � � C � c=1 � ∂ � Pr(T | M) ∂ M·,·,c �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' (4) S is of the same height and width dimensions as the raw image, M, and M·,·,c denotes the image slice of channel/feature c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='1 The quantity, S, represents the magnitude of the gradient of the predicted probability of treatment with respect to the image, where the magnitudes are computed across all the channel/feature dimensions 1These features typically correspond to quantities such as reflectance, ultimately measured in energy per unit area per wavelength such as Watts/(m2 · µm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' 9 of the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' This salience map, therefore, provides some indication of the parts of the image that are “important” in predicting the outcome, where that importance is calculated using derivatives to quantify importance as mathematical sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' These derivatives are calculated via automatic differentiation, a computational tool to evaluate exact gradients even when no closed form is available (Griewank and Walther, 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Because they use automatic differentiation, salience maps in satellite- based observational inference can be computed with some, but not all, models for � Pr(T | M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' In particular, we require that � Pr(T | M) is continuously differentiable with respect to M, which is the case for the most common class of image models such as convolutional networks (see Table 2 for a list of models outlined by differentiability).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Table 2: Differentiability of candidate models in image-based observational inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Salience maps can be readily calculated for differentiable models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Differentiable Generalized linear models Feed-forward neural networks Convolutional models Transformers Recurrent neural networks Non-differentiable Tree-based models Models involving greedy/discrete optimization Another challenge related to interpretability in satellite image-based causal inference relates to the Stable Unit Treatment Value Assumption (SUTVA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' This assumption states that the treatment of any scene s should not affect the treatment status or outcome of another unit s′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' In other words, each scene is i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='d, as defined by our DAG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' However, in spatial analysis, it may be harder to defend or even define SUTVA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Units that are closer in space may affect each other via spillover effects (Breza, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' For example, a policymaker allocating aid to one village may unintentionally affect outcomes in a nearby village, which benefits from the nearby allocation of assistance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' SUTVA violations, and other forms of dependence (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=', spatial clustering), can in principle be accounted for by specifying an appropriate variance-covariance structure (Sinclair et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=', 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' However, this variance-covariance structure may be difficult to capture in practice, so that there may be difficulties in interpreting or explaining results from image-based observational causal inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' A more interesting possibility lies in the prospect that satellite image data may actually contain information about the latent interference structure present within the social system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' As already noted, satellite data is informative transportation networks (Nagne and Gawali, 2013) which may correlate with patterns of social influence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Thus, while we can use block bootstrapping or other techniques to address spatial dependence, future work should probe the degree to which satellite data itself is informative about the underlying patterns of influence present within social reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' To conclude this section on interpretability in satellite image-based observational inference, we offer a few reflections on the centrality of resolution in this task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' First, we note that resolution is a key driver of the residual confounding, as discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' We can only adjust for confounder objects in the image that can be resolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Smaller confounder objects, therefore, introduce residual bias, indicating how technological improvements to sensor technology play a critical role in improving image-based causal inference methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' In short, the resolution determines the kinds of explanations we can make regarding our ability to reconstruct the treatment assignment mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' If there is a single pixel per scene, then that pixel can only capture global information about things such as the abundance of greenspace, soil moisture, and other quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' If there are hundreds or thousands of pixels per scene, then more complex objects can be detected such as houses, roads, and trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Interpretability in image-based inference is therefore inextricably tied to resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Another related motivation for obtaining high-resolution remote sensing data lies in the possibility that the use of satellite images can reduce researcher assumptions about how information from smaller scales aggregates up to the scene scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Because each image is defined at a more granular resolution than the unit of analysis, we can use it to potentially reconstruct some unobserved confounders by learning the function generating confounders from the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' 10 For example, assume that researchers seek to analyze the effect of a village-level treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' From a government census, they obtain mean income information for the village (s) and then perform an analysis assuming Ys(0), Ys(1) ⊥ Ts|Xs, where Xs contains the income data, Ts is the treatment, and Ys(t), the potential outcome under t ∈ {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Yet, unless the mean income is the true confounder, the analysis will still be biased, which would occur if, in fact, minimum income drove the decision to allocate Ts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' However, using satellite images for each scene, we seek to reconstruct the minimum income signal based on our access to the higher-resolution data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' In other words, we can weaken the assumption used in many empirical analyses that the variables measured at the scene level in fact contain the true confounders, when in reality highly non-linear functions—that use more granular information—may have generated the confounding structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' With image data, we, in principle, can hope to reconstruct some of those factors using advances in image-based machine learning models effective in the prediction domain (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=', Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' (2013)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Despite the potential of learning about how lower-level information aggregates up using high- resolution satellite images, there are still other subtleties to consider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' For example, if treatments are defined at a high level of resolution (say, a house), but satellite data is defined at a lower level of resolution (say, a neighborhood block), there are ambiguities in how observational inference should be performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Without higher-resolution data, we cannot adjust for home-level confounders, but if we aggregate treatments to the block level, information is lost, and extra researcher degrees of freedom are introduced in terms of how that aggregation should be done (whether in terms of summing, averaging, or aerial interpolating to the block level).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' For pixel-level treatments, therefore, extra caution is warranted due to the nuances of how resolution affects the confounders that can be captured by an image model using satellite resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='6 Other Challenges and Opportunities of Satellite Image-based Observational Inference In addition to model dependence of observational inference with satellite images, there is another, perhaps more fundamental challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' By its nature, satellite data is geo-referenced, and data on treatment assignments need to be geo-referenced as well if it is to be integrated into this pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' This linkage in some circumstances may be straightforward: a town receives an anti-poverty intervention, or it does not, and that town can be geo-referenced using APIs such as Google Maps and Open- StreetMap (Yeboah et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' However, in other circumstances, the linkage may be ambiguous: if regions are the units receiving treatment, it is less clear what satellite information should be used in modeling the treatment mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' In some situations, units are not geo-referenced at all: there may be data privacy or other reasons why even the approximate location of experimental units is explicitly not measured by researchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Finally, it is important to note that, if disadvantaged units are less likely to be geo-referenced to neighborhoods than more advantaged units, we will again be in a situation where systematic missingness patterns bias causal estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' These important limitations can also present opportunities for future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' 4 Experiments Illustrating Observational Causal Inference Dynamics Under Model Misspecification and Varying Resolution Although the identification results described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='4 are general, they are also theoretical and, as described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='6, there are several practical considerations that will affect performance in real data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' In order to better understand these dynamics, we use simulation, since, in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' true causal targets are unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' In these simulations, the image data is observed, but the confounding features are not, and must be estimated, as is the case in real applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='1 Simulation Design The simulation design centers on the scene level of analysis because, in practice, there seem to be fewer situations where pixels (14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='25 meter by 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='25 meter grid cells) are treated than the situation where larger areas are treated (or where units are treated and we seek to adjust for confounding given their entire neighborhood context).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' In the simulation, confounding is generated by (1) applying a single convolution to our set of Landsat images from Nigeria (operation by fl(·)), (2) finding the maximum similarity across an image to 11 the kernel filter (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' using the global maximum pooling operation), and (3) normalizing across observations so that the resulting confounder has mean 0, standard deviation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' The confounder then enters equations for the treatment and outcome: Uswh = fl(MsΠs(wh)) Us = GN(max{Uswh : wh ∈ Πs}), Pr(Ts|Ms) = logistic(βUs + ϵ(W ) s ) Ys = γUs + τ Ts + ϵ(Y ) s , where GN(·) denotes normalization to mean 0 and variance 1, done to prevent all units from receiving treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' fl(·) denotes the linear kernel function, where the single kernel filter, l, is a diagonal matrix and is shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' The set of images used in the simulation are drawn from the corpus of Landsat satellite images that we later use in the application (see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' The ϵ’s are Normally distributed random error terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' We vary the dimensions of the estimating convolutional filter, l, keeping the structure of the true confounder-generating filter pattern fixed with a width of 9 (see Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' By varying the estimating filter dimensions, we alter the size of the neighborhood used in estimating the latent confounder, allowing us to probe model misspecification, where there is a gap between the true data-generating process and estimating model structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' We also vary the resolution of the image used in estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' This quantity varies from 1 (when the observed image resolution is the same as used in confounder generation) to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='12 (where the image resolution observed in the estimation step is 12% that of the original;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' in other words, pixels are 88% larger in width).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' By varying the resolution, we can probe how this unique feature of images affects observational causal inference estimation in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' We compare two estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' First, we examine the difference-in-means estimator, ˆτ0, defined as the difference in conditional outcome means for the two treatment groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Because of confounding, this baseline quantity is biased for τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' We also report results from an Inverse Propensity Weighting (IPW) estimator (Austin and Stuart, 2015), with image model estimation performed using a single-layer convolutional network model which is trained using stochastic gradient descent with the binary cross-entropy loss function (which is equivalent to the negative log-likelihood).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' The estimation formula for IPW is ˆπ(Ms) = �Pr(Ts = 1|Ms), �τ = 1 n �n s=1 � TsYs ˆπ(Xs) − (1−Ts)Ys 1−ˆπ(Xs) � for the scene analysis and is defined analogously at the pixel level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' We report results from the Hajek IPW estimator where the weights have been normalized, reducing estimator variance at the cost of some finite sample bias (Skinner and Wakefield, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' In our evaluation, we compare the true τ with the estimated values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' We compute the absolute bias of ˆτ and the Root Mean Squared Error (RMSE): Absolute Bias = ��E[ˆτ − τ] ��;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' RMSE = � E[(ˆτ − τ)2], (5) where expectations are taken over the data-generating process and are approximated via Monte Carlo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' These evaluation metrics are then normalized using the MSE from the baseline estimator, ˆτ0, to facilitate the interpretability of the results (so that Relative Absolute Bias and Relative RMSE are reported).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='2 Simulation Results In Figure 4, we show the dynamics of satellite image-based observational inference in this simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' As expected, we find that the absolute bias and RMSE are minimized when the kernel width used in estimation is the same as the kernel width used to generate the true confounder and when the resolution is the same as that used in generating the confounder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' The bias and RMSE grow larger when the estimating kernel width is greater than the kernel width used in confounder generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' This finding is likely due to the fact that, when the neighborhood size used in estimation is larger than that used to create the confounder, the additional parameters allow the model to overfit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' The bias and RMSE panels of Figure 4 look similar because the variance of estimation is relatively small, especially when the resolution has been down-shifted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='9 Relative Absolute Bias 0.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='5 1 1 1 1 1 3 5 9 17 33 Estimating Kernel Width (# of Pixels) Figure 4: Scene-level results varying the resolution and character of model misspecification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' The true confounder was generated with a kernel width of 9 and resolution scaling factor of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Resolution scaling factors are numerically labeled and colored with grayscale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' A resolution scaling factor of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='5, for example, indicates that the pixels in that analysis are twice as wide as in the original raw image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Resolution also plays an important role in explaining the dynamics of observational inference with satellite images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' For all values of resolution, bias and RMSE of estimation are still favorable compared to using the naive difference-in-means estimator without satellite-based neighborhood information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' In other words, low-resolution images still improve upon the simple difference-in-means estimator of the treatment effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' When the resolution of the image used in estimation is the same as that that actually generated the confounder, performance in terms of bias and RMSE is optimal when model specification is correct, but, with this higher resolution image, the analysis is also more sensitive to model misspecification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' When the resolution is down-shifted, model misspecification matters less;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' there is less image information available so the dependence of the results on the image model is weakened.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' We find similar results when we vary the degree of noise in the scene-level confounders, loosening the determinism assumption discussed above (see Section A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' 5 Empirical Illustration: The Impact of Local Aid Programs on Neighborhood-level Poverty in Nigeria Having examined some of the opportunities and pitfalls of satellite-based observational inference, in this section, we demonstrate it in the context of Nigeria, Africa’s largest economy and a country that is projected to be the world’s second most populous by 2100 (Vollset et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' This context is one in which identifying effective anti-poverty interventions carries substantial practical importance: despite an average economic growth rate of around 3% since 2000, about 40 percent of the Nigerian population live below the poverty line ($2 per day).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' In response, governments and NGOs have deployed a variety of local aid programs to the country.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' However, the causal impact of these programs is difficult to estimate (Roodman, 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' While geo-temporal data on poverty, Y , and interventions, T, are readily available, there is a lack of geo-temporal data on potential confounders, U, at the local level (Daoud et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Halleröd et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' While some of these confounders may be difficult to capture directly using images (such as the quality of political institutions), there may be information present in remote sensing imagery about other confounder objects related to infrastructure or agriculture (Schnebele et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Steven 13 and Clark, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' We therefore use satellite images of Nigerian communities in order to estimate the impact of local aid programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='1 Data Description Our outcome data on poverty is drawn from the Demographic and Health Surveys (DHS), which are conducted by a non-profit organization, ICF International, with funding from USAID, WHO, and other international organizations (Rutstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=', 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' The DHS surveys are conducted at the household level for randomly selected clusters that aggregate to geographically explicit scenes of about 300 households for de-identification purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Our outcome measure is the International Wealth Index (IWI) from 2018, which is a principal-components-derived summary of 12 variables including households’ access to public services and possession of consumer products such as a phone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Its scale is between 0 and 100, with higher values indicating greater wealth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Our treatment data is drawn from a data set on international aid programs to Nigeria compiled by AidData (AidData, 2016) used under an Open Data Commons License.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' The aid programs we examine took place after 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' The aid providers include entities such as the World Bank and WHO, as well as domestic governments such as the United States.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' The programs we examine are diverse, focusing on infrastructure (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=', support for road development) and agriculture (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=', support for small-scale farmers), among other things.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' For the simplicity of our presentation, we take the presence of a geographically-specific aid program within 7000 meters of a DHS point as our treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Our pre-treatment image data is drawn from Landsat, the satellite imagery program operated by NASA/USGS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' We use the Orthorectified ETM+ pan-sharpened data derived from the raw satellite imagery captured between 1998 and 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' the raw data have been processed to contain minimal cloud cover and to be correctly geo-referenced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Resolution is 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='25 meters;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' reflectance in the green, near-infrared, and short-wave infrared bands is recorded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Treatment and control locations are shown in the left panel of Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Figure 5: The left panel identifies Nigeria as the context of interest in this observational study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' The right panel illustrates the location of treatment and control sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Gray points are control locations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' black points are treatment locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='2 Observational Inference Image Model Design Our image model for the treatment is built using three convolutional layers (32 filters each) with max pooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' After the application of the filters, we project the channel dimension into a 3-dimensional space to facilitate interpretability via a single image post-convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Batch normalization is used on the channel dimensions and before the final projection layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Weights are learned using ADAM with Nesterov momentum with cosine learning rate decay (baseline rate = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Gotmare et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' (2018)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' We compare performance across a variety of filter sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' We attempt to limit overfitting by randomly reflecting each image dimension with probability 1/2 during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' We assess out-of- sample performance using three random training/test splits, averaging over this sampling process 14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='6 Out−of−Sample Loss Loss 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='5 1 1 1 3 5 7 Baseline Estimating Kernel Width 5 10 15 20 Effect Estimate Estimate Value 3 5 7 Baseline 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='5 1 1 1 Figure 6: The left panel shows out-of-sample binary cross entropy loss compared to the baseline loss when guessing the dominant class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' The right panel shows the IPW estimate values across the range of estimating kernel width values compared to the baseline difference in conditional means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' and reporting 95% confidence intervals from the three test set assessments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' We vary the image model structure by altering the convolutional filter width (which affects the size of the image patterns analyzed in the image model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' We also assess stability of the results to resolution of the satellite images used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' This task contains some subtleties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' It is always possible to lower the resolution of an image, averaging across pixel grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' However, when we change the dimensionality of the input image, we may no longer be able to employ the same image models as in the full dimensionality case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' The reason for this is due to the fact that each convolutional layer in the image model reduces the output dimensionality;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' if our starting resolution is too small, the implied output dimensionality of some of the convolutional layers would be negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' We keep the image model structure the same and analyze results across different resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' As a consequence, when the image model cannot be fit with a given kernel width and resolution combination, we drop that model from the set of models analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='3 Empirical Results First, in the left panel of Figure 6, we assess the propensity model fit to the treatment data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' We find that the image model always improves on the baseline out-of-sample loss value obtained by random guessing of the dominant class (the control class, comprising 71% of the data sample).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Performance is stable across values of the kernel width used in estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Moreover, resolution does not greatly affect performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' This stability implies that more macroscopic aspects of the landscape are ultimately what is driving the estimated treatment assignment mechanism, as opposed to the lower-level features such as the grouping of houses of a certain type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Next, in the right panel of Figure 6, we analyze the estimated treatment effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' We find that, across the estimating kernel width range, adjusted estimates are positive but smaller in magnitude than the baseline difference in means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' This hints at the importance of confounder adjustment, as these programs may be given to areas already primed for growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' We also analyze the estimation model dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' In Figure 7, we visualize data for the three out-of- sample control (left) and treated (right) units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' We also display the salience maps for the predicted 15 Raw Image Salience Map Final Spatial Layer Lat, Long: 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='49, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='303 xlim xlim Lat, Long: 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='29, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='170 xlim xlim Lat, Long: 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='91, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='917 xlim xlim Lat, Long: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='50, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='439 xlim xlim Lat, Long: 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='469, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='346 xlim xlim Lat, Long: 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='614, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='372 xlim xlim Figure 7: The three left panels depict the raw data for control units, salience maps for the predicted treatment probability, and output from the final spatially resolved layer in the image model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' The three right panels depict the same things for treated units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Note that the three bands of the satellite image are not “red”, “green”, and “blue” (see main text), so in this visual representation, reddish pink indicates soil moisture content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' treatment probabilities as well as the output from the final spatially resolved layer in the image model when the kernel width is 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' To explore robustness, we also display in Section A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='3 results from another run with a kernel of 7 in the estimation model, as well as width kernel widths 3 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' To take examples from the model output, a particularly low treatment probability site is estimated in the remote desert city of Machina (pop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' 62,000);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' a high probability site is from the city of Katsina (pop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' 429,000) near a large agricultural basin and with rich water resources (water soil content is displayed as red in the RBG satellite images using non-visible band).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' The output of this model would seem to resonate with the fact that many of the projects undertaken by global actors are specifically designed to assist farmers and agriculture more generally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Finally, we assess the performance of the estimated propensity scores on reducing covariate imbalance between treated and control groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' The same absence of rich covariate information in places such as sub-Saharan Africa that motivates this paper also makes this assessment task difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Still, we can analyze differences in longitude/latitude between treated/control groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' We find a raw difference of (-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='12, -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' After weighting, this difference decreases in magnitude to (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='39,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='53), indicative of improved counterfactual comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Overall, this application shows some of the nuances of satellite image-based observational inference— how resolution affects the space of possible image models, how to think about interpreting the output of the causal inference model in the image context, as well as how to validate the improvement in counterfactual comparisons after employing the satellite information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' 6 Discussion and Future Work In this paper, we characterize key challenges and opportunities of satellite image-based observational causal inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' We formally show that, even though confounder aspects of a neighborhood may be latent, we can adjust for them using the image information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' There are several subtleties, however, related to resolution and the degree to which the image truly proxies those confounders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' We illustrate some of these tradeoffs in simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' We also apply the satellite image-based observational inference approach in order to understand the causal effect of aid programs on reducing neighborhood-level poverty in Africa’s largest economy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' As previously stated, our approach is not limited to poverty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' It can be used for analyzing the effects of environmental factors (Shiba et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Daoud et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=', 2016) to neighborhood-level change (Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=', 2022), and beyond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' 16 The use of satellite images in observational inference approach has some inherent limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' For example, as described above, confounding may be due to objects that cannot be resolved in the image data, and, as a result, bias reduction will not occur conditioning on the image information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' In this context, the collection and application of imagery with higher spatial, temporal, and spectral resolution is a priority.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Spatial and temporal resolution may both be achieved, for example, using sensors mounted on ground-based infrastructure (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=', Johnston et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' (2021));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' spatial resolution and extent could be optimized with drone- or airplane-based instruments (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=', Gray et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' (2018)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Future considerations should examine privacy and fairness issues with causal analyses based on passive sensor technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Second, while in some cases, such as if disparate individuals are treated, the scene-level unit of analysis is clearly defined (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=', the individual within a neighborhood context), in other contexts, the scene-level unit of analysis is more ambiguous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' The researcher therefore has choices about how to define the scene (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=', at the street, village, or region levels), a choice that could introduce systematic bias into the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' This issue is known as the modifiable areal unit problem (Fotheringham and Wong, 1991), and the approach described here is vulnerable to it as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Future work should therefore focus on the development of image-confounding methods that have theoretical guarantees on the robustness of the results to the scale of action examined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' This issue of scale is also related to the question of capturing the treatment and outcome at different scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Third, machine-learning-based image models learn the image patterns that best predict treatment assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' More research is needed to connect these patterns, learned inductive, with the mental processes of real actors as they consult images in decision-making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' This path of research could forge links between cognitive science, machine learning, text analysis, and causal inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' □ References AidData.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Nigeria aims geocoded research release level 1 v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='2 geocoded dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' D’Amour Alexander, Peng Ding, Avi Feller, Lihua Lei, and Jasjeet Sekhon.' 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3a2021_3ai_3a2_3ap_3a644-654.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='htm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Publisher: Elsevier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Peter C Austin and Elizabeth A Stuart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Moving towards best practice when using inverse probability of treatment weighting (iptw) using the propensity score to estimate causal treatment effects in observational studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Statistics in medicine, 34(28):3661–3679, 2015.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Journal of the American Statistical Association, 113(523):1228–1242, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Godwin Yeboah, João Porto de Albuquerque, Rafael Troilo, Grant Tregonning, Shanaka Perera, Syed AK Ahmed, Motunrayo Ajisola, Ornob Alam, Navneet Aujla, Syed Iqbal Azam, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Analysis of openstreetmap data quality at different stages of a participatory mapping process: Evidence from slums in africa and asia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' ISPRS International Journal of Geo-Information, 10(4): 265, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Christopher Yeh, Anthony Perez, Anne Driscoll, George Azzari, Zhongyi Tang, David Lobell, Stefano Ermon, and Marshall Burke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Using publicly available satellite imagery and deep learning to understand economic well-being in africa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Nature communications, 11(1):2583, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Kexin Yi, Chuang Gan, Yunzhu Li, Pushmeet Kohli, Jiajun Wu, Antonio Torralba, and Joshua B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Tenenbaum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' CLEVRER: CoLlision Events for Video REpresentation and Reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Technical Report arXiv:1910.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='01442, arXiv, March 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' URL http://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='org/abs/1910.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='01442.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' arXiv:1910.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='01442 [cs] type: article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' 23 Jinsung Yoon, James Jordon, and Mihaela Van Der Schaar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Ganite: Estimation of individualized treatment effects using generative adversarial nets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' In International Conference on Learning Representations, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Han Zhang and Yilang Peng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Image Clustering: An Unsupervised Approach to Categorize Visual Data in Social Science Research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Sociological Methods & Research, page 00491241221082603, April 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' ISSN 0049-1241.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='1177/00491241221082603.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' URL https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' 1177/00491241221082603.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Publisher: SAGE Publications Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' 24 A Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='1 Connections with the Causal Proxy Literature Besides facilitating the use of causal inference for a social science audience, our work is related to the literature on identification via proxies (Tchetgen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=', 2020) or drivers (Pearl, 2013a) of confounders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' For the former, Louizos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=', developed Causal Effect Variational Autoencoder (CEVAE) which uses proxies to infer the distribution of the latent confounder and use this in adjustment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' In contrast, our approach adjusts for an observed variable—the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' We formalize key assumptions required for the correctness of this method and provided a general framework for conducting causal inference using images, where unlabeled objects in the image may affect both treatment and outcome (Castro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' This image-based confounding bias might in some circumstances be equivalent to traditional spatial interdependence, but differs insofar as the confounding bias is defined with reference to unlabeled entities in the image, thereby injecting bias (Paciorek, 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Relying on our formalization and model implementation, we analyze aid interventions (treatment) and poverty (outcome) in Africa— something of policy relevance as policymakers often rely on satellite images for aid intervention (Voigt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Bedi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=', 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='2 Additional Simulation Results A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='1 Probing Estimation Bias as the Determinism Assumption is Relaxed We here explore how model misspecification affects estimation error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' In particular, we probe how relaxing the determinism assumption of Proposition 1 affects satellite-based observational inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' In particular, we now let the unobserved confounder be a random function of the satellite image, as depicted visually in Figure 3b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' In particular, the confounder values are now Uswh = fl(MsΠs(wh)) + ϵ(U) swh, (6) where ϵ(U) swh ∼ N(0, σ2 U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' We vary σ2 U ∈ {1, 3, 5, 7}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' We then apply the same data-generating process to obtain scene-level treatments and outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' We see in Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='1 how performance is affected by relaxing the determinism assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' As expected, we see that estimation bias grows as the unobserved confounding is increasingly determined by the noise factor, ϵ(U) swh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' When the noise scale is at its maximum, bias is still no worse than the simple difference in means baseline (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=', relative bias/RMSE approaches 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' This fact is likely because the noise injected into the confounding mechanism is itself exogenous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Nevertheless, having established a theoretical baseline in this paper, future research should examine this noise-induced confounding to image-based causal inference in greater detail, connecting this line of work with the proxy literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='0 Relative Absolute Bias 1 3 5 7 Relative RMSE 1 3 5 7 Non−image Unobserved Confounder Scale Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='1: Bias and RMSE for the scene-level analysis as we vary the stochasticity present in the confounding mechanism, holding the estimation kernel width fixed at 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Gray circles indicate effect estimate values using the true (in practice unobserved) treatment probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='3 Robustness to Model Specification in the Empirical Results Raw Image Salience Map Final Spatial Layer Lat, Long: 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='49, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='303 xlim xlim Lat, Long: 12.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='469, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='346 xlim xlim Lat, Long: 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='614, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='372 xlim xlim Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='2: Replicating Figure 7 with another training/test split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' 2 Raw Image Salience Map Final Spatial Layer Lat, Long: 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='635, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='802 xlim xlim Lat, Long: 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='34, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='705 xlim xlim Lat, Long: 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='58, 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='66 xlim xlim Lat, Long: 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='094, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='349 xlim xlim Lat, Long: 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='481, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='520 xlim xlim Lat, Long: 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='633, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='323 xlim xlim Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='3: Replicating Figure 7 with an estimating kernel width of 5 instead of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Raw Image Salience Map Final Spatial Layer Lat, Long: 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='55, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='643 xlim xlim Lat, Long: 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='29, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='163 xlim xlim Lat, Long: 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='577, 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='42 xlim xlim Lat, Long: 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='459, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='489 xlim xlim Lat, Long: 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='094, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='349 xlim xlim Lat, Long: 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='661, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='288 xlim xlim Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='4: Replicating Figure 7 with an estimating kernel width of 3 instead of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' 3 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='4 Implementation Details We implement our computational analyses on an Apple M1 GPU using Metal-optimized TensorFlow 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content='11 with GNU Parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' Total compute time for the simulations is about 48 hours;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' total compute time for the application results is about 24 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} +page_content=' 4' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf'} diff --git a/bdAzT4oBgHgl3EQf2_4K/vector_store/index.faiss b/bdAzT4oBgHgl3EQf2_4K/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..f9adc70090b7d6cb5ff916087a8f6779e81d29ba --- /dev/null +++ b/bdAzT4oBgHgl3EQf2_4K/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:2dbdb829009ea400ddb70ecad968bfa4f4208ca94178d6579f22fe91e8fa2ac3 +size 10616877 diff --git a/cNE1T4oBgHgl3EQfdgR_/content/tmp_files/2301.03196v1.pdf.txt b/cNE1T4oBgHgl3EQfdgR_/content/tmp_files/2301.03196v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..3a1ff16d44453434e367aabd55e34dd2830465ec --- /dev/null +++ b/cNE1T4oBgHgl3EQfdgR_/content/tmp_files/2301.03196v1.pdf.txt @@ -0,0 +1,770 @@ +A near-optimal stochastic MIMO signal detection +with a mixture of t-distribution prior +Junichiro Hagiwara, Kazushi Matsumura, Hiroki Asumi, Yukiko Kasuga, Toshihiko Nishimura, +Takanori Sato, Yasutaka Ogawa, and Takeo Ohgane +Graduate School / Faculty of Information Science and Technology, Hokkaido University +Kita 14, Nishi 9, Kita-ku, Sapporo, Hokkaido, 060-0814 Japan +Email: {jhagiwara, kazu4.matsumura, asumi, kasuga, nishim, tksato, ogawa, ohgane}@m-icl.ist.hokudai.ac.jp +Abstract—Multiple-input multiple-output (MIMO) systems are +a promising key technology for future wireless communication. +However, improving their signal detection performance is still +challenging to further increase the wireless transmission effi- +ciency. To address this challenge, we propose to intentionally +extend the discrete signal detection problem in MIMO systems +to a continuous one and to utilize the Hamiltonian Monte Carlo +method, a type of efficient Markov chain Monte Carlo (MCMC). +We already presented the use of a mixture of normal distribution +for the prior distribution based on the same approach. This +paper proposes the application of a mixture of t-distribution +that further improves detection performance. We show that +the proposed method can achieve near-optimal signal detection +with a polynomial order computational complexity through +theoretical analysis and computer simulation. The proposed +high-performance and pragmatic MIMO signal detection should +significantly contribute to developing the 6th generation mobile +network. +Index Terms—MIMO, signal detection, MCMC, Hamiltonian +Monte Carlo, t-distribution +I. INTRODUCTION +Wireless communications have already become part of +society’s infrastructure, as evidenced by the widespread use +and growth of 4G and 5G mobile networks. For further +convenience, 6G research and development is now progressing +to make more effective use of radio resources. There are +several essential technologies for the effective use of radio +resources, of which multiple-input multiple-output (MIMO) is +highly anticipated. MIMO technology can improve wireless +transmission efficiency by using multiple antennas at trans- +mitting and receiving stations [1]. While MIMO with a few +antennas is practically used, research on massive MIMO with +numerous antennas has recently been actively considered [2]. +In MIMO systems, the more accurately the received signal +can be detected, the more efficient the transmission can be, so +how to improve the signal detection performance is essential. +Here, it is known that maximum likelihood decoding can +achieve ideal performance but is computationally impractical +because it examines all possible combinations of transmitted +symbols [3]. In contrast, linear detection methods such as +the minimum mean-square error method are computationally +less expensive but cannot achieve adequate detection perfor- +mance [3]. Therefore, the specific issue is improving the signal +detection performance as much as possible under realistic +computational complexity requirements. An approach to treat +signal detection from a stochastic perspective may be helpful +for such purposes. This is because it allows theoretical consid- +eration of the uncertainty of the problem and the application +of state-of-the-art probabilistic and statistical techniques. +Stochastic problem settings, especially methods based on +Bayesian approaches, have already been studied in various +ways [4], [5]. However, to the best of our knowledge, no +former study in MIMO signal detection applies a continuous +distribution to the prior distribution, except for our previous +proposals [6], [7]. For example, the mixed Gibbs sampling +(MGS) method [8] approximates the posterior distribution +with a large number of samples. This method sets a discrete +distribution for the prior distribution and uses an algorithm +that improves on Gibbs sampling [9], a type of Markov chain +Monte Carlo (MCMC). The expectation propagation (EP) +method [10] also sets a discrete distribution for the prior +distribution. This method uses the expectation propagation +algorithm [11] to find the parameters of the approximate +distribution for the posterior distribution. In addition, [12] +sets a mixture of truncated normal distribution for the prior +distribution. This method uses a variational Bayesian algo- +rithm to find the parameters of the approximate distribution +for the posterior distribution. All of these set up a discrete or +discontinuous distribution for the prior distribution based on +the firm assumption of discrete transmission symbols. They do +not take advantage of excellent algorithms that can be applied +to a continuous distribution without discontinuous points. +Therefore, we intentionally extend the discrete signal detec- +tion problem to a continuous one and utilize the Hamiltonian +Monte Carlo (HMC) method [13], an efficient MCMC applica- +ble to continuous problems. We already presented the use of a +mixture of normal distribution for the prior distribution based +on the same approach [6], [7]. This study then proposes the +application of a mixture of t-distribution that further improves +detection performance. We show that the proposed method +can achieve near-optimal signal detection with a polynomial +order computational complexity through theoretical analysis +and computer simulation. The proposed novel MIMO signal +detection would contribute to both practical and theoretical +aspects of future wireless communications. +This paper is structured as follows: Section II provides +a stochastic formulation of the problem. Section III briefly +describes the MGS and EP methods as the existing primary +arXiv:2301.03196v1 [cs.NI] 9 Jan 2023 + +methods. Section IV explains the details of the proposed +method. Section V discusses the results of the theoretical +analysis and computer simulations. Finally, we conclude in +section VI. +In the remainder of this paper, we use the following no- +tations: C denotes the field of complex numbers. Re(x) and +Im(x) indicate the real and imaginary parts of x, respectively. +ˆx indicates the estimates of x. ⌊x⌋ gives the largest integer +less than equal to x. 0 and I represent the zero vector and the +unit matrix, respectively. XT denotes the transpose of X. The +symbol ∼ means that the random variable on its left-hand side +follows the probability distribution on its right-hand side. The +symbol ∝ represents a proportional relationship. +II. PROBLEM FORMULATION +A. System model +Suppose full-stream transmission in a MIMO system with +N transmit antennas and M receive antennas. In this case, the +following relationship holds: +y = Hu + w, +w ∼ CN(0, σ2 +wI), (1) +where received symbol vector is y = [y1, . . . yM]T ∈ CM, +channel matrix is H ∈ CM×N, transmission symbol vector +is u = [u1, . . . , uN]T ∈ CN, noise vector is w = [w1, +. . . , wM]T ∈ CM, noise variance is σ2 +w, and CN represents a +circularly symmetric complex normal distribution. For ease of +handling, this study internally divides complex numbers into +real and imaginary parts as in the following: y → +� +Re(y) +Im(y) +� +, +H → +� +Re(H) −Im(H) +Im(H) +Re(H) +� +, u → +� +Re(u) +Im(u) +� +, and w → +� +Re(w) +Im(w) +� +. +Note that the expression of the formula hereafter remains the +same but is still valid by regarding N and M as 2N and +2M, respectively. This study assumes known H and σw. In +addition, the appearance of transmitted symbols is supposed +to be uniformly random across antennas due to scramblers. +The purpose of signal detection is to estimate u given y. +When a stochastic interpretation is introduced here, from +Bayes’ theorem, “posterior distribution ∝ likelihood × prior +distribution” holds as follows: +p(u | y) ∝ p(y | u)p(u). +(2) +Therefore, the purpose of stochastic signal detection is a point +estimate of the posterior distribution p(u | y). +B. Likelihood +(1) shows p(y | u) = N(y; Hu, σ2 +wI), where N represents +a real value normal distribution density. +C. Prior distribution +(2) implies that prior distribution corresponds to regulariza- +tion terms that correct the likelihood. In the context of signal +detection, this represents preferences at or around the trans- +mission signal point, which improves the estimation accuracy +of the posterior distribution. The original prior distribution in +valley of death +Fig. 1. Two examples of priors for BPSK. +signal detection is a discrete multinomial distribution in line +with discrete signal points as in (3) (see also Fig. 1 (a)): +p(u) = +2N +� +n=1 +1 +q {δ(un − a1) + · · · + δ(un − aq)}, +(3) +where q is the square root of the modulation order, a1, . . . , aq +is the real-valued coordinate of the transmission signal points, +and δ(x) is the unit probability mass at x. The MGS and EP +methods suppose such prior. +Assuming a continuous distribution for the prior distribution +can intentionally extend the problem to a continuous value +problem. For example, the following equation yields the +application of a mixture of normal distribution (see also Fig. 1 +(b)): +p(u) = +2N +� +n=1 +1 +q {N(un; a1, σ) + · · · + N(un; aq, σ)}, +(4) +where σ2 is a variance of a component’s normal distribution +and tuning matter. In our previous study [6], [7] that proposed +a mixture of normal distribution prior, the σ was set to the +optimal value minimizing the bit error rate (BER) through a +preliminary search. The mechanism for an optimal σ is as +follows. A large value of σ results in low search efficiency +because areas other than the transmission signal points are +examined unnecessarily. In contrast, a small value of σ also +results in poor search efficiency because there is little overlap +in the component’s distribution. The such little overlap makes +exploring other possible transmission signal points difficult +(“valley of death” in Fig. 1 (b)). +D. Posterior distribution +When the prior distribution is assumed as a mixture of +any distribution, the posterior distribution cannot be obtained +analytically in the closed form [14], regardless of whether the +component distribution is discrete or continuous. In such a +case, a numerical approximation algorithm is used to derive +the posterior distribution. Point estimates ˆu for the posterior +distribution can yield multiple candidates depending on the +approximation algorithm. Also, the ˆu may deviate from the +original transmission signal point. For example, a continuous + +a1 +a2 +(a) Discrete multinomial distribution +a1 +a2 +(b) Continuous mixture of normal distributiondistribution for the prior distribution causes such a situation +because areas other than discrete transmission signal points +are searched. Therefore, this study calculates the likelihood +p(y | ˜u) after ˆu is quantized to the nearest transmission signal +point ˜u and treats ˜u with the highest likelihood as the final +point estimates. +III. PREVIOUS WORK +A. MGS method [8] +The MGS method approximates the posterior distribution +with a large number of samples. For this purpose, Gibbs sam- +pling, a type of MCMC, is utilized to search more intensively +in areas with higher posterior probability densities to obtain +samples. Mixing the initialization of searching values with the +optimal probability 1/(2N) mitigates the stalling in the local +optima to improve search efficiency. This corresponds to a +virtual parallelization of Markov chains. [8] also proposes the +multiple restarts technique. In this technique, multiple MGS +methods are run with different initial values of the Markov +chain, and the result with the highest likelihood is selected +from among them. It is shown that a sufficient number of +multiple restarts can yield near-optimal performance. +The computational complexity of the MGS method is +O(MN) per step of the Markov chain because the com- +putation of the likelihood is the dominant influence on its +complexity. Let LMGS be the total steps in the Markov chain, +and the final computational complexity is O(LMGSMN). +B. EP method [10] +The EP method approximates the posterior distribution by +an uncorrelated multivariate normal distribution q(u). For this +purpose, the EP algorithm is used to find the parameters that +minimize the Kullback–Leibler divergence − +� +p(u | y) ln{ +q(u)/p(u | y)}du. Specifically, the mean and variance pa- +rameters are refined in an iterative process, and the finally +obtained mean parameter is used to estimate the transmitted +symbols. Let the total number of iterations be LEP, and [10] +reports that at most LEP = 10 is needed to reach the upper +limit of detection performance. +The computational complexity of the EP method is O(N 3) +per iteration because the inverse matrix operation of H is the +dominant influence on its complexity. When a total number +of iterations LEP = 10 is assumed, the final computational +complexity is O(10N 3). +IV. STOCHASTIC SIGNAL DETECTION WITH A MIXTURE OF +t-DISTRIBUTION PRIOR +A. Setting the prior distribution +This study proposes the application of a mixture of t- +distribution to the prior distribution as in (5): +p(u) = +2N +� +n=1 +1 +q {T (un; a1, σ, ν) + · · · + T (un; aq, σ, ν)}, (5) +where T represents a real value t-distribution density [14], +and scale parameter σ and the degrees of freedom ν are +Fig. 2. The normal and t-distributions with the same scale parameter σ. +tuning matter. Concerning these, this study applies optimal +values based on preliminary searches. The t-distribution has +a narrower peak and thicker tail than the normal distribution +with the same scale parameter σ (Fig. 2). Therefore, compared +to the normal distribution, the t-distribution can search more +intensively around the transmission signal points. In addition, +this distribution can actively search for other possible trans- +mission signal points by overcoming the so-called “valley +of death” between the transmission signal points. Thus, it is +expected to have better search properties than our previously +proposed mixture of normal distribution prior [6], [7]. +B. Approximation algorithm for the posterior distribution +When a continuous distribution is set for the prior distribu- +tion, the posterior distribution is also continuous because the +likelihood is a continuous normal distribution. An effective ap- +proximation algorithm for continuous problems can be applied +to posterior estimation in this case. Our previous study [7] +compared the results of applying Newton’s, automatic differ- +entiation variational inference [15], and HMC methods to the +approximation algorithm under the condition of the mixture of +normal distribution prior. The results confirm that the HMC +method mostly achieves superior signal detection performance +under the same computational complexity. This is because the +HMC algorithm implies that the searching value is always +randomly initialized at each step of the Markov chain, which +tends to avoid local optima. Therefore, this study applies the +HMC method to the approximation algorithm for the posterior +distribution. +Since the HMC method is a type of MCMC, as with +MGS, it utilizes the Markov chain mechanism. Thus, this +method searches more intensively for areas with higher pos- +terior probability densities to obtain samples that approxi- +mate the posterior distribution. However, the HMC method +can typically increase sampling efficiency over other MCMC +methods. This is because the HMC method ingeniously utilizes +a Hamiltonian mechanics framework. That is, this method +intentionally incorporates a quantity r = du/dτ (where τ +is virtual time) that corresponds to the momentum of u in +addition to the originally estimated variable u. A summary +of the HMC method is as follows. At first, the system’s +potential energy U and kinetic energy K are defined as +U(u) = − ln(p(u | y)) and K(r) = 1/2||r||2, respectively. +The Hamiltonian is introduced as +H(u, r) = U(u) + K(r), +(6) + +Normal distribution +t-distribution (v = 2)Algorithm 1 HMC method sampling +1: Initialize u at random +2: for l = 1, . . . , LHMC do +3: +Draw r from N(0, I) +4: +Numerically solve Hamilton’s equations (7) to obtain u′ and r′ +5: +Update u ← u′ with probability min[1, exp{H(u, r) − H(u′, r′)}] +6: +Regard the updated u as a sample from the posterior distribution +p(u | y) +7: end for +which represents the total energy of the system. Then, Hamil- +ton’s equations are expressed by the following two partial +differential equations: +du +dτ = ∂H(u, r) +∂r += r, +dr +dτ = −∂H(u, r) +∂u += −∂U(u) +∂u +. +(7) +Algorithm 1 shows the sampling with the HMC method. +The Hamiltonian is constantly based on (6). Thus, if the +momentum r changes significantly, the sample value u also +changes considerably. In addition, according to Algorithm 1, +most proposals u′ are accepted with probability one, except +for numerical error cases. Therefore, the sampling efficiency of +the HMC method is typically better than that of other MCMC +algorithms. +According to Algorithm 1, when (7) is numerically solved +during one step of the Markov chain, the log-posterior prob- +ability density derivative is actually evaluated L times. The +value L varies depending on the problem and conditions but +is typically assumed to be ten in this study [16]. Regarding +complexity, the dominant influence comes from calculating +the term (HTH)u. This term is included in the derivative for +the log-likelihood contained in the log-posterior probability +density. Recall that H is assumed as known, and (HTH) +only needs to be calculated once. Thus, the computational +complexity of the HMC method is O(LN 2) = O(10N 2) per +step of the Markov chain. Since LHMC is set to the total steps +in the Markov chain, the final computational complexity is +O(10LHMCN 2). +V. NUMERICAL RESULTS AND DISCUSSION +The main objective of this study is to show that our pro- +posal can achieve near-optimal MIMO signal detection with +polynomial-order computational complexity. For this purpose, +we confirm the computational complexity theoretically and +perform computer simulations on signal detection. In addition, +this section compares the proposed method with our previous +one using a mixture of normal distribution prior, as well as +with the MGS and EP methods representing existing works. +A. Settings in numerical analysis +Common assumptions: Assume typical and exhaustive condi- +tions, such as the number of antennas conscious for Massive +MIMO and the modulation order from low to high (Table I). +If the computer simulation results are error-free, we omit the +plot of the corresponding bit error rate. +TABLE I +COMMON ASSUMPTIONS IN NUMERICAL ANALYSIS +Item +Setting +Trials +5000 +Number of antennas +N = M = 96 +Modulation order +QPSK, 16QAM, and 64QAM +Average transmission power +1 +Fading +Quasi-static Rayleigh +Channel correlation +Kronecker model +(correlation coefficient ρ = 0 or 0.5) +Coding +Uncoded +TABLE II +PARAMETERS OF COMPONENT’S NORMAL AND t-DISTRIBUTIONS +Mixture of normal distribution +Mixture of t-distribution +σ +σ +ν +QPSK +0.2483 +0.5 × 0.2483 +1.8 +16QAM +0.1242 +0.5 × 0.1242 +1.8 +64QAM +0.0664 +0.8 × 0.0664 +2.5 +Proposed method setting: The parameters of the component’s +normal and t-distributions are set to the values that showed +good performance in the preliminary rough search (Table II). +Each Markov chain is assumed to have 2N steps, consistent +with the virtual parallelization of the MGS method. The +parallel number of Markov chains is set to ⌊1000/(2N)⌋, +the minimum number required to achieve adequate perfor- +mance [6]. Consequently, the total steps in the Markov chain, +LHMC, are equivalent to 1000. +Existing method setting: The total steps in the Markov chain +of the MGS method are set to LMGS = 1000, consistent with +LHMC. However, since the computational complexity of our +proposed method is ten times larger than the MGS method due +to L = 10, we apply ten multiple restarts of the MGS method +runs for a fair comparison. If the multiple restarts number +were increased further, there would be room for performance +improvement of the MGS method. This study still sets the +multiple restarts number at ten to compare the performance +under the same computational complexity. +The total number of EP method iterations is set to LEP = 10, +which is at most required number to obtain adequate perfor- +mance. +The single-input single-output transmission under additive +white Gaussian noise (SISO AWGN) performance corresponds +to the ideal one eliminating inter-antenna interference and fad- +ing effects. We show only its BER performance for indicating +the theoretical lower bound of MIMO signal detection. +B. Computational complexity +Given N = M, the computational complexity is O(10 × +1000N 2) for the proposed method, 10 × O(1000N 2) for the +MGS method, and O(10N 3) for the EP method, respectively. +All of these are polynomial orders. Comparing these in more +detail, while the proposed and MSG methods are the same +due to their matching conditions, the EP method is the least +computationally expensive since N = 96 < 1000. + +Fig. 3. Average BER vs. average received SNR for ρ = 0 and 0.5. +C. BER performance: a mixture of t-distribution prior vs. a +mixture of normal distribution prior +Overall trend (Fig. 3 (a) through (c)): The larger the modulation +order, the further the performance is away from the SISO +AWGN. This is because the larger the modulation order, the +more possible transmission signal points and the narrower the +distance between signal points, which increases the difficulty +of estimation. +Comparison with a mixture of normal distribution prior (Fig. 3 +(a) through (c)): For ρ = 0, while we can recognize areas +where the t-distribution is advantageous, there are also areas +where two performances are overlapped, making it difficult to +confirm the difference. Examine the results for ρ = 0.5 since +the performance difference is more apparent when difficult +signal detection. When we also consider the case of ρ = 0.5, +it is clear that the t-distribution is dominant for all modulation +orders. For example, for ρ = 0.5, the SNR gain from a mixture +of normal distribution prior at 10−3 BER is 0.8 dB for QPSK, +0.5 dB for 16QAM, and 0.3 dB for 64QAM. From the above, +we conclude that the mixture of t-distribution prior has better +signal detection performance than the mixture of normal dis- +tribution prior. This benefits from the t-distribution’s narrower +peak and thicker tail than the normal distribution. The former +allows for a more intensive search around the transmission +signal points, and the latter allows for an aggressive search +over the “valley of death” between the transmission signal +points. +Note that the degree of improvement from the mixture of +normal distribution prior decreases as the modulation order +increases, and the advantage is particularly small for 64QAM. +As the modulation order increases and the solution search +space grows, even aggressive search by t-distribution seems +less effective under a limited number of searches. +D. BER performance: proposed method with a mixture of t- +distribution prior vs. existing methods +Overall trend (Fig. 4 (a) through (c)): As mentioned above, +the larger the modulation order, the further the performance is +away from the SISO AWGN. +Comparison with existing methods for QPSK (Fig. 4 (a)): The +performance of all methods is close to SISO AWGN with +small differences. This is because signal detection is easy due +to few possible transmission signal points and a wide distance +between signal points. At low SNR, the proposed method is +slightly inferior to the MGS and EP methods. In noisy condi- +tions, searching areas other than the transmission signal points +seems counterproductive under a limited number of searches. +Namely, the original impulse-like prior distribution appears +adequate as in the MGS and EP methods. At moderate to +high SNR, the proposed method outperforms the EP method, +while its detection performance is almost the same as the MGS +method. For example, the SNR gain of the proposed method +at 10−3 BER is nearly 0 dB vs. the MGS method and 0.4 dB +vs. the EP method. +Comparison with existing methods for 16QAM (Fig. 4 (b)): +Compared to the QPSK case, the number of possible trans- +mission signal points increases, and the distance between +signal points becomes narrower. This tendency makes signal +detection more difficult and reveals the characteristics of the +method. The difference from the QPSK case is that the +detection performance of the proposed method outperforms +the MGS method at moderate to high SNR. Therefore, the +detection performance of the proposed method is the best at +moderate to high SNR, and the performance difference with +the existing method is larger than that for the QPSK case. For +example, the SNR gain of the proposed method at 10−3 BER +is 0.4 dB vs. the MGS method and 1.6 dB vs. the EP method. +Comparison with existing methods for 64QAM (Fig. 4 (c)): +Compared to the QPSK and 16QAM cases, signal detection is +the most difficult, and the characteristics of the method are the +most apparent. This is due to the largest number of possible +transmission signal points and the narrowest distance between +signal points. The difference from the QPSK and 16QAM +cases is that the detection performance of the proposed method +outperforms the MGS method at low SNR. While the detection +performance of the proposed method is best at moderate to +high SNR, as in the 16QAM case, the performance difference + +10° +10-1 +10-2 +BER +B +10-3 +Average +10-4 +10-5 +Mix. of normal (p = 0.0) +Mix. of t +(p = 0.0) +Mix. of normal (p = 0.5) +10-6 +-A-- +Mix. of t +(p = 0.5) +SISO AWGN +10-7 +6 +8 +1 +10 +12 +14 +Average received SNR [dB] +(a) QPSK100 +10-1 +10-2 +10-3 +10-4 +10-5 +Mix. of normal (p = 0.0) +Mix. of t +(p = 0.0) +..+. +Mix. of normal (p = 0.5) +10-6 +-A-- +Mix. of t +(p = 0.5) +SISO AWGN +10-7 +14 +16 +18 +20 +22 +Average received SNR [dB] +(b) 16QAM100 +10-1 +10-2 +10-3 +10-4 +10-5 +Mix. of normal (p = 0.0) +Mix. of t +(p = 0.0) +..+. +Mix. of normal (p = 0.5) +10-6 +-A-- +Mix. of t +(p = 0.5) +SISO AWGN +10-7 +22 +24 +26 +28 +30 +Average received SNR [dB] +(c) 64QAMFig. 4. Average BER vs. average received SNR for ρ = 0. +with the existing method is larger than that for the 16QAM +case. For example, the SNR gain of the proposed method at +10−3 BER is undeterminable large vs. the MGS method and +4.2 dB vs. the EP method. +Comparison with SISO AWGN (Fig. 4 (a) through (c)): We +consider that the proposed method can achieve near-optimal +performance. For example, the SNR degradation of the pro- +posed method at 10−3 BER is within 0.3 dB for QPSK, 1.3 +dB for 16QAM, and 2.9 dB for 64QAM. This good result +comes from the combination of the prior distribution setting +and an effective algorithm. Especially, the prior setting is close +enough to the impulse around the transmission signal point +and allows the active search to other possible signal points. +We believe our proposal would be a reliable and powerful +method. +VI. CONCLUSIONS +The proposed signal detection method utilizing the mix- +ture of t-distribution prior and the HMC algorithm should +achieve near-optimal performance with polynomial compu- +tational complexity. As for the detection performance, the +proposed method had a remarkable feature: the gain from +the typical existing method became very large as the mod- +ulation order increased. Considering the demand for more +speed and capacity in today’s wireless communications, the +use of higher-order modulation is an inevitable trend in the +future. We consider our proposed method beneficial from +such a viewpoint, too. However, some limitations were also +identified when compared to typical existing methods. First, +the computational complexity was still polynomial order and +equivalent to the MGS method but more than the EP method. +Next, BER performance might be slightly inferior to the MGS +and EP methods at low SNR. The SNR-dependent fine-tuning +of the component’s t-distribution parameter may improve BER +performance at low SNR, but details will be verified for +future work. From a mathematical viewpoint, our proposal +has extended the discrete problem to a continuous one and +examined the effect of prior distribution in detail. Therefore, +it could also be applied to similar sparse problems. +REFERENCES +[1] A. Goldsmith, Wireless Communications. +Cambridge University Press, +2005. +[2] L. Lu, G. Y. Li, A. L. Swindlehurst, A. Ashikhmin, and R. Zhang, “An +overview of massive MIMO: Benefits and challenges,” IEEE Journal of +Selected Topics in Signal Processing, vol. 8, no. 5, pp. 742–758, 2014. +[3] A. Chockalingam and B. Rajan, Large MIMO Systems. +Cambridge +University Press, 2014. +[4] S. Yang and L. Hanzo, “Fifty years of MIMO detection: The road +to large-scale MIMOs,” IEEE Communications Surveys & Tutorials, +vol. 17, no. 4, pp. 1941–1988, 2015. +[5] M. A. Albreem, M. Juntti, and S. Shahabuddin, “Massive MIMO +detection techniques: A survey,” IEEE Communications Surveys & +Tutorials, vol. 21, no. 4, pp. 3109–3132, 2019. +[6] K. Matsumura, J. Hagiwara, T. Nishimura, T. Ohgane, Y. Ogawa, and +T. Sato, “A novel MIMO signal detection method using Hamiltonian +Monte Carlo approach,” in 2021 24th International Symposium on +Wireless Personal Multimedia Communications (WPMC), 2021, pp. 1–6. +[7] H. Asumi, Y. Kasuga, K. Matsumura, J. Hagiwara, T. Nishimura, T. Sato, +Y. Ogawa, and T. Ohgane, “Comparison of performance and complexity +for different search methods in stochastic MIMO signal detection,” in +IEICE Technical Report RCS, vol. 122, no. 73, June 2022, pp. 150–155, +(in Japanese). +[8] T. Datta, N. A. Kumar, A. Chockalingam, and B. S. Rajan, “A novel +Monte-Carlo-sampling-based receiver for large-scale uplink multiuser +MIMO systems,” IEEE Transactions on Vehicular Technology, vol. 62, +no. 7, pp. 3019–3038, 2013. +[9] S. Geman and D. Geman, “Stochastic relaxation, Gibbs distributions, +and the Bayesian restoration of images,” IEEE Transactions on Pattern +Analysis and Machine Intelligence, vol. 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Bishop, Pattern Recognition and Machine Learning. +Springer, +2006. +[15] A. Kucukelbir, D. Tran, R. Ranganath, A. Gelman, and D. M. Blei, +“Automatic differentiation variational inference,” J. Mach. Learn. Res., +vol. 18, no. 1, pp. 430–474, Jan. 2017. + +100 +10-1 +10-2 +10-3 +10-4 +10~5 +EP +MGS +HMC +SISO AWGN +10-7 +22 +24 +26 +28 +30 +Average received SNR [dB] +(c) 64QAM100 +10-1 +10-2 +10-3 +10-4 +10-5 +EP +MGS +HMC +SISO AWGN +10-7 +14 +16 +18 +20 +22 +Average received SNR [dB] +(b) 16QAM100 +10-1 +10-2 +Average BER +10-3 +10 +10-5 +EP +MGS +10-6 +HMC +SISOAWGN +10-7 +6 +8 +10 +12 +14 +Average received SNR [dB] +(a) QPSK[16] A. Gelman, J. B. Carlin, H. S. Stern, D. B. Dunson, A. Vehtari, and +D. B. Rubin, Bayesian Data Analysis, 3rd ed. +CRC Press, 2013. + diff --git a/cNE1T4oBgHgl3EQfdgR_/content/tmp_files/load_file.txt b/cNE1T4oBgHgl3EQfdgR_/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..fffd7a2b9d34f4d381e56e61efefbed0b851dfd0 --- /dev/null +++ b/cNE1T4oBgHgl3EQfdgR_/content/tmp_files/load_file.txt @@ -0,0 +1,451 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf,len=450 +page_content='A near-optimal stochastic MIMO signal detection with a mixture of t-distribution prior Junichiro Hagiwara, Kazushi Matsumura, Hiroki Asumi, Yukiko Kasuga, Toshihiko Nishimura, Takanori Sato, Yasutaka Ogawa, and Takeo Ohgane Graduate School / Faculty of Information Science and Technology, Hokkaido University Kita 14, Nishi 9, Kita-ku, Sapporo, Hokkaido, 060-0814 Japan Email: {jhagiwara, kazu4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content='matsumura, asumi, kasuga, nishim, tksato, ogawa, ohgane}@m-icl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content='ist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content='hokudai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content='jp Abstract—Multiple-input multiple-output (MIMO) systems are a promising key technology for future wireless communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' However, improving their signal detection performance is still challenging to further increase the wireless transmission effi- ciency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' To address this challenge, we propose to intentionally extend the discrete signal detection problem in MIMO systems to a continuous one and to utilize the Hamiltonian Monte Carlo method, a type of efficient Markov chain Monte Carlo (MCMC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' We already presented the use of a mixture of normal distribution for the prior distribution based on the same approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' This paper proposes the application of a mixture of t-distribution that further improves detection performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' We show that the proposed method can achieve near-optimal signal detection with a polynomial order computational complexity through theoretical analysis and computer simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' The proposed high-performance and pragmatic MIMO signal detection should significantly contribute to developing the 6th generation mobile network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' Index Terms—MIMO, signal detection, MCMC, Hamiltonian Monte Carlo, t-distribution I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' INTRODUCTION Wireless communications have already become part of society’s infrastructure, as evidenced by the widespread use and growth of 4G and 5G mobile networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' For further convenience, 6G research and development is now progressing to make more effective use of radio resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' There are several essential technologies for the effective use of radio resources, of which multiple-input multiple-output (MIMO) is highly anticipated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' MIMO technology can improve wireless transmission efficiency by using multiple antennas at trans- mitting and receiving stations [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' While MIMO with a few antennas is practically used, research on massive MIMO with numerous antennas has recently been actively considered [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' In MIMO systems, the more accurately the received signal can be detected, the more efficient the transmission can be, so how to improve the signal detection performance is essential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' Here, it is known that maximum likelihood decoding can achieve ideal performance but is computationally impractical because it examines all possible combinations of transmitted symbols [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' In contrast, linear detection methods such as the minimum mean-square error method are computationally less expensive but cannot achieve adequate detection perfor- mance [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' Therefore, the specific issue is improving the signal detection performance as much as possible under realistic computational complexity requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' An approach to treat signal detection from a stochastic perspective may be helpful for such purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' This is because it allows theoretical consid- eration of the uncertainty of the problem and the application of state-of-the-art probabilistic and statistical techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' Stochastic problem settings, especially methods based on Bayesian approaches, have already been studied in various ways [4], [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' However, to the best of our knowledge, no former study in MIMO signal detection applies a continuous distribution to the prior distribution, except for our previous proposals [6], [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' For example, the mixed Gibbs sampling (MGS) method [8] approximates the posterior distribution with a large number of samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' This method sets a discrete distribution for the prior distribution and uses an algorithm that improves on Gibbs sampling [9], a type of Markov chain Monte Carlo (MCMC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' The expectation propagation (EP) method [10] also sets a discrete distribution for the prior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' This method uses the expectation propagation algorithm [11] to find the parameters of the approximate distribution for the posterior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' In addition, [12] sets a mixture of truncated normal distribution for the prior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' This method uses a variational Bayesian algo- rithm to find the parameters of the approximate distribution for the posterior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' All of these set up a discrete or discontinuous distribution for the prior distribution based on the firm assumption of discrete transmission symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' They do not take advantage of excellent algorithms that can be applied to a continuous distribution without discontinuous points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' Therefore, we intentionally extend the discrete signal detec- tion problem to a continuous one and utilize the Hamiltonian Monte Carlo (HMC) method [13], an efficient MCMC applica- ble to continuous problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' We already presented the use of a mixture of normal distribution for the prior distribution based on the same approach [6], [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' This study then proposes the application of a mixture of t-distribution that further improves detection performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' We show that the proposed method can achieve near-optimal signal detection with a polynomial order computational complexity through theoretical analysis and computer simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' The proposed novel MIMO signal detection would contribute to both practical and theoretical aspects of future wireless communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' This paper is structured as follows: Section II provides a stochastic formulation of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' Section III briefly describes the MGS and EP methods as the existing primary arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content='03196v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content='NI] 9 Jan 2023 methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' Section IV explains the details of the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' Section V discusses the results of the theoretical analysis and computer simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' Finally, we conclude in section VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' In the remainder of this paper, we use the following no- tations: C denotes the field of complex numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' Re(x) and Im(x) indicate the real and imaginary parts of x, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' ˆx indicates the estimates of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' ⌊x⌋ gives the largest integer less than equal to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' 0 and I represent the zero vector and the unit matrix, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' XT denotes the transpose of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' The symbol ∼ means that the random variable on its left-hand side follows the probability distribution on its right-hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' The symbol ∝ represents a proportional relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' PROBLEM FORMULATION A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' System model Suppose full-stream transmission in a MIMO system with N transmit antennas and M receive antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' In this case, the following relationship holds: y = Hu + w, w ∼ CN(0, σ2 wI), (1) where received symbol vector is y = [y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' yM]T ∈ CM, channel matrix is H ∈ CM×N, transmission symbol vector is u = [u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' , uN]T ∈ CN, noise vector is w = [w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' , wM]T ∈ CM, noise variance is σ2 w, and CN represents a circularly symmetric complex normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' For ease of handling, this study internally divides complex numbers into real and imaginary parts as in the following: y → � Re(y) Im(y) � , H → � Re(H) −Im(H) Im(H) Re(H) � , u → � Re(u) Im(u) � , and w → � Re(w) Im(w) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' Note that the expression of the formula hereafter remains the same but is still valid by regarding N and M as 2N and 2M, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' This study assumes known H and σw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' In addition, the appearance of transmitted symbols is supposed to be uniformly random across antennas due to scramblers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' The purpose of signal detection is to estimate u given y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' When a stochastic interpretation is introduced here, from Bayes’ theorem, “posterior distribution ∝ likelihood × prior distribution” holds as follows: p(u | y) ∝ p(y | u)p(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' (2) Therefore, the purpose of stochastic signal detection is a point estimate of the posterior distribution p(u | y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' Likelihood (1) shows p(y | u) = N(y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' Hu, σ2 wI), where N represents a real value normal distribution density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' Prior distribution (2) implies that prior distribution corresponds to regulariza- tion terms that correct the likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' In the context of signal detection, this represents preferences at or around the trans- mission signal point, which improves the estimation accuracy of the posterior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' The original prior distribution in valley of death Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' Two examples of priors for BPSK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' signal detection is a discrete multinomial distribution in line with discrete signal points as in (3) (see also Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' 1 (a)): p(u) = 2N � n=1 1 q {δ(un − a1) + · · · + δ(un − aq)}, (3) where q is the square root of the modulation order, a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' , aq is the real-valued coordinate of the transmission signal points, and δ(x) is the unit probability mass at x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' The MGS and EP methods suppose such prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' Assuming a continuous distribution for the prior distribution can intentionally extend the problem to a continuous value problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' For example, the following equation yields the application of a mixture of normal distribution (see also Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' 1 (b)): p(u) = 2N � n=1 1 q {N(un;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' a1, σ) + · · · + N(un;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' aq, σ)}, (4) where σ2 is a variance of a component’s normal distribution and tuning matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' In our previous study [6], [7] that proposed a mixture of normal distribution prior, the σ was set to the optimal value minimizing the bit error rate (BER) through a preliminary search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' The mechanism for an optimal σ is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' A large value of σ results in low search efficiency because areas other than the transmission signal points are examined unnecessarily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' In contrast, a small value of σ also results in poor search efficiency because there is little overlap in the component’s distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' The such little overlap makes exploring other possible transmission signal points difficult (“valley of death” in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' 1 (b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' Posterior distribution When the prior distribution is assumed as a mixture of any distribution, the posterior distribution cannot be obtained analytically in the closed form [14], regardless of whether the component distribution is discrete or continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' In such a case, a numerical approximation algorithm is used to derive the posterior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' Point estimates ˆu for the posterior distribution can yield multiple candidates depending on the approximation algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' Also, the ˆu may deviate from the original transmission signal point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' For example, a continuous a1 a2 (a) Discrete multinomial distribution a1 a2 (b) Continuous mixture of normal distributiondistribution for the prior distribution causes such a situation because areas other than discrete transmission signal points are searched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' Therefore, this study calculates the likelihood p(y | ˜u) after ˆu is quantized to the nearest transmission signal point ˜u and treats ˜u with the highest likelihood as the final point estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' PREVIOUS WORK A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' MGS method [8] The MGS method approximates the posterior distribution with a large number of samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' For this purpose, Gibbs sam- pling, a type of MCMC, is utilized to search more intensively in areas with higher posterior probability densities to obtain samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' Mixing the initialization of searching values with the optimal probability 1/(2N) mitigates the stalling in the local optima to improve search efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' This corresponds to a virtual parallelization of Markov chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' [8] also proposes the multiple restarts technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' In this technique, multiple MGS methods are run with different initial values of the Markov chain, and the result with the highest likelihood is selected from among them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' It is shown that a sufficient number of multiple restarts can yield near-optimal performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' The computational complexity of the MGS method is O(MN) per step of the Markov chain because the com- putation of the likelihood is the dominant influence on its complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' Let LMGS be the total steps in the Markov chain, and the final computational complexity is O(LMGSMN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' EP method [10] The EP method approximates the posterior distribution by an uncorrelated multivariate normal distribution q(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' For this purpose, the EP algorithm is used to find the parameters that minimize the Kullback–Leibler divergence − � p(u | y) ln{ q(u)/p(u | y)}du.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' Specifically, the mean and variance pa- rameters are refined in an iterative process, and the finally obtained mean parameter is used to estimate the transmitted symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' Let the total number of iterations be LEP, and [10] reports that at most LEP = 10 is needed to reach the upper limit of detection performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' The computational complexity of the EP method is O(N 3) per iteration because the inverse matrix operation of H is the dominant influence on its complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' When a total number of iterations LEP = 10 is assumed, the final computational complexity is O(10N 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' STOCHASTIC SIGNAL DETECTION WITH A MIXTURE OF t-DISTRIBUTION PRIOR A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' Setting the prior distribution This study proposes the application of a mixture of t- distribution to the prior distribution as in (5): p(u) = 2N � n=1 1 q {T (un;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' a1, σ, ν) + · · · + T (un;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' aq, σ, ν)}, (5) where T represents a real value t-distribution density [14], and scale parameter σ and the degrees of freedom ν are Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' The normal and t-distributions with the same scale parameter σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' tuning matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' Concerning these, this study applies optimal values based on preliminary searches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' The t-distribution has a narrower peak and thicker tail than the normal distribution with the same scale parameter σ (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' Therefore, compared to the normal distribution, the t-distribution can search more intensively around the transmission signal points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' In addition, this distribution can actively search for other possible trans- mission signal points by overcoming the so-called “valley of death” between the transmission signal points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' Thus, it is expected to have better search properties than our previously proposed mixture of normal distribution prior [6], [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' Approximation algorithm for the posterior distribution When a continuous distribution is set for the prior distribu- tion, the posterior distribution is also continuous because the likelihood is a continuous normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' An effective ap- proximation algorithm for continuous problems can be applied to posterior estimation in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' Our previous study [7] compared the results of applying Newton’s, automatic differ- entiation variational inference [15], and HMC methods to the approximation algorithm under the condition of the mixture of normal distribution prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' The results confirm that the HMC method mostly achieves superior signal detection performance under the same computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' This is because the HMC algorithm implies that the searching value is always randomly initialized at each step of the Markov chain, which tends to avoid local optima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' Therefore, this study applies the HMC method to the approximation algorithm for the posterior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' Since the HMC method is a type of MCMC, as with MGS, it utilizes the Markov chain mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' Thus, this method searches more intensively for areas with higher pos- terior probability densities to obtain samples that approxi- mate the posterior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' However, the HMC method can typically increase sampling efficiency over other MCMC methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' This is because the HMC method ingeniously utilizes a Hamiltonian mechanics framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' That is, this method intentionally incorporates a quantity r = du/dτ (where τ is virtual time) that corresponds to the momentum of u in addition to the originally estimated variable u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' A summary of the HMC method is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' At first, the system’s potential energy U and kinetic energy K are defined as U(u) = − ln(p(u | y)) and K(r) = 1/2||r||2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' The Hamiltonian is introduced as H(u, r) = U(u) + K(r), (6) Normal distribution t-distribution (v = 2)Algorithm 1 HMC method sampling 1: Initialize u at random 2: for l = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' , LHMC do 3: Draw r from N(0, I) 4: Numerically solve Hamilton’s equations (7) to obtain u′ and r′ 5: Update u ← u′ with probability min[1, exp{H(u, r) − H(u′, r′)}] 6: Regard the updated u as a sample from the posterior distribution p(u | y) 7: end for which represents the total energy of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' Then, Hamil- ton’s equations are expressed by the following two partial differential equations: du dτ = ∂H(u, r) ∂r = r, dr dτ = −∂H(u, r) ∂u = −∂U(u) ∂u .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' (7) Algorithm 1 shows the sampling with the HMC method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' The Hamiltonian is constantly based on (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' Thus, if the momentum r changes significantly, the sample value u also changes considerably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' In addition, according to Algorithm 1, most proposals u′ are accepted with probability one, except for numerical error cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' Therefore, the sampling efficiency of the HMC method is typically better than that of other MCMC algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' According to Algorithm 1, when (7) is numerically solved during one step of the Markov chain, the log-posterior prob- ability density derivative is actually evaluated L times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' The value L varies depending on the problem and conditions but is typically assumed to be ten in this study [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' Regarding complexity, the dominant influence comes from calculating the term (HTH)u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' This term is included in the derivative for the log-likelihood contained in the log-posterior probability density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' Recall that H is assumed as known, and (HTH) only needs to be calculated once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' Thus, the computational complexity of the HMC method is O(LN 2) = O(10N 2) per step of the Markov chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' Since LHMC is set to the total steps in the Markov chain, the final computational complexity is O(10LHMCN 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' NUMERICAL RESULTS AND DISCUSSION The main objective of this study is to show that our pro- posal can achieve near-optimal MIMO signal detection with polynomial-order computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' For this purpose, we confirm the computational complexity theoretically and perform computer simulations on signal detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' In addition, this section compares the proposed method with our previous one using a mixture of normal distribution prior, as well as with the MGS and EP methods representing existing works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' Settings in numerical analysis Common assumptions: Assume typical and exhaustive condi- tions, such as the number of antennas conscious for Massive MIMO and the modulation order from low to high (Table I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' If the computer simulation results are error-free, we omit the plot of the corresponding bit error rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' TABLE I COMMON ASSUMPTIONS IN NUMERICAL ANALYSIS Item Setting Trials 5000 Number of antennas N = M = 96 Modulation order QPSK, 16QAM, and 64QAM Average transmission power 1 Fading Quasi-static Rayleigh Channel correlation Kronecker model (correlation coefficient ρ = 0 or 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content='5) Coding Uncoded TABLE II PARAMETERS OF COMPONENT’S NORMAL AND t-DISTRIBUTIONS Mixture of normal distribution Mixture of t-distribution σ σ ν QPSK 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content='2483 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content='5 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content='2483 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content='8 16QAM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content='1242 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content='5 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content='1242 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content='8 64QAM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content='0664 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content='8 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content='0664 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content='5 Proposed method setting: The parameters of the component’s normal and t-distributions are set to the values that showed good performance in the preliminary rough search (Table II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' Each Markov chain is assumed to have 2N steps, consistent with the virtual parallelization of the MGS method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' The parallel number of Markov chains is set to ⌊1000/(2N)⌋, the minimum number required to achieve adequate perfor- mance [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' Consequently, the total steps in the Markov chain, LHMC, are equivalent to 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' Existing method setting: The total steps in the Markov chain of the MGS method are set to LMGS = 1000, consistent with LHMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' However, since the computational complexity of our proposed method is ten times larger than the MGS method due to L = 10, we apply ten multiple restarts of the MGS method runs for a fair comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' If the multiple restarts number were increased further, there would be room for performance improvement of the MGS method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' This study still sets the multiple restarts number at ten to compare the performance under the same computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' The total number of EP method iterations is set to LEP = 10, which is at most required number to obtain adequate perfor- mance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' The single-input single-output transmission under additive white Gaussian noise (SISO AWGN) performance corresponds to the ideal one eliminating inter-antenna interference and fad- ing effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' We show only its BER performance for indicating the theoretical lower bound of MIMO signal detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' Computational complexity Given N = M, the computational complexity is O(10 × 1000N 2) for the proposed method, 10 × O(1000N 2) for the MGS method, and O(10N 3) for the EP method, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' All of these are polynomial orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' Comparing these in more detail, while the proposed and MSG methods are the same due to their matching conditions, the EP method is the least computationally expensive since N = 96 < 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' Average BER vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' average received SNR for ρ = 0 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' BER performance: a mixture of t-distribution prior vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' a mixture of normal distribution prior Overall trend (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' 3 (a) through (c)): The larger the modulation order, the further the performance is away from the SISO AWGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' This is because the larger the modulation order, the more possible transmission signal points and the narrower the distance between signal points, which increases the difficulty of estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' Comparison with a mixture of normal distribution prior (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' 3 (a) through (c)): For ρ = 0, while we can recognize areas where the t-distribution is advantageous, there are also areas where two performances are overlapped, making it difficult to confirm the difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' Examine the results for ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content='5 since the performance difference is more apparent when difficult signal detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' When we also consider the case of ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content='5, it is clear that the t-distribution is dominant for all modulation orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' For example, for ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content='5, the SNR gain from a mixture of normal distribution prior at 10−3 BER is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content='8 dB for QPSK, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content='5 dB for 16QAM, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content='3 dB for 64QAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' From the above, we conclude that the mixture of t-distribution prior has better signal detection performance than the mixture of normal dis- tribution prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' This benefits from the t-distribution’s narrower peak and thicker tail than the normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' The former allows for a more intensive search around the transmission signal points, and the latter allows for an aggressive search over the “valley of death” between the transmission signal points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' Note that the degree of improvement from the mixture of normal distribution prior decreases as the modulation order increases, and the advantage is particularly small for 64QAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' As the modulation order increases and the solution search space grows, even aggressive search by t-distribution seems less effective under a limited number of searches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' BER performance: proposed method with a mixture of t- distribution prior vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' existing methods Overall trend (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' 4 (a) through (c)): As mentioned above, the larger the modulation order, the further the performance is away from the SISO AWGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' Comparison with existing methods for QPSK (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' 4 (a)): The performance of all methods is close to SISO AWGN with small differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' This is because signal detection is easy due to few possible transmission signal points and a wide distance between signal points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' At low SNR, the proposed method is slightly inferior to the MGS and EP methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' In noisy condi- tions, searching areas other than the transmission signal points seems counterproductive under a limited number of searches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' Namely, the original impulse-like prior distribution appears adequate as in the MGS and EP methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' At moderate to high SNR, the proposed method outperforms the EP method, while its detection performance is almost the same as the MGS method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' For example, the SNR gain of the proposed method at 10−3 BER is nearly 0 dB vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' the MGS method and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content='4 dB vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' the EP method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' Comparison with existing methods for 16QAM (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' 4 (b)): Compared to the QPSK case, the number of possible trans- mission signal points increases, and the distance between signal points becomes narrower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' This tendency makes signal detection more difficult and reveals the characteristics of the method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' The difference from the QPSK case is that the detection performance of the proposed method outperforms the MGS method at moderate to high SNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' Therefore, the detection performance of the proposed method is the best at moderate to high SNR, and the performance difference with the existing method is larger than that for the QPSK case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' For example, the SNR gain of the proposed method at 10−3 BER is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content='4 dB vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' the MGS method and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content='6 dB vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' the EP method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' Comparison with existing methods for 64QAM (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' 4 (c)): Compared to the QPSK and 16QAM cases, signal detection is the most difficult, and the characteristics of the method are the most apparent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' This is due to the largest number of possible transmission signal points and the narrowest distance between signal points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' The difference from the QPSK and 16QAM cases is that the detection performance of the proposed method outperforms the MGS method at low SNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' While the detection performance of the proposed method is best at moderate to high SNR, as in the 16QAM case, the performance difference 10° 10-1 10-2 BER B 10-3 Average 10-4 10-5 Mix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' of normal (p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content='0) Mix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' of t (p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content='0) Mix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' of normal (p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content='5) 10-6 A-- Mix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' of t (p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content='5) SISO AWGN 10-7 6 8 1 10 12 14 Average received SNR [dB] (a) QPSK100 10-1 10-2 10-3 10-4 10-5 Mix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' of normal (p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content='0) Mix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' of t (p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content='0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content='.+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' Mix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' of normal (p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content='5) 10-6 A-- Mix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' of t (p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content='5) SISO AWGN 10-7 14 16 18 20 22 Average received SNR [dB] (b) 16QAM100 10-1 10-2 10-3 10-4 10-5 Mix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' of normal (p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content='0) Mix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' of t (p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content='0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content='.+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' Mix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' of normal (p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content='5) 10-6 A-- Mix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' of t (p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content='5) SISO AWGN 10-7 22 24 26 28 30 Average received SNR [dB] (c) 64QAMFig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' Average BER vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' average received SNR for ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' with the existing method is larger than that for the 16QAM case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' For example, the SNR gain of the proposed method at 10−3 BER is undeterminable large vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' the MGS method and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content='2 dB vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' the EP method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' Comparison with SISO AWGN (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' 4 (a) through (c)): We consider that the proposed method can achieve near-optimal performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' For example, the SNR degradation of the pro- posed method at 10−3 BER is within 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content='3 dB for QPSK, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content='3 dB for 16QAM, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content='9 dB for 64QAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' This good result comes from the combination of the prior distribution setting and an effective algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' Especially, the prior setting is close enough to the impulse around the transmission signal point and allows the active search to other possible signal points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' We believe our proposal would be a reliable and powerful method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' CONCLUSIONS The proposed signal detection method utilizing the mix- ture of t-distribution prior and the HMC algorithm should achieve near-optimal performance with polynomial compu- tational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' As for the detection performance, the proposed method had a remarkable feature: the gain from the typical existing method became very large as the mod- ulation order increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' Considering the demand for more speed and capacity in today’s wireless communications, the use of higher-order modulation is an inevitable trend in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' We consider our proposed method beneficial from such a viewpoint, too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' However, some limitations were also identified when compared to typical existing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' First, the computational complexity was still polynomial order and equivalent to the MGS method but more than the EP method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' Next, BER performance might be slightly inferior to the MGS and EP methods at low SNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' The SNR-dependent fine-tuning of the component’s t-distribution parameter may improve BER performance at low SNR, but details will be verified for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' From a mathematical viewpoint, our proposal has extended the discrete problem to a continuous one and examined the effect of prior distribution in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' Therefore, it could also be applied to similar sparse problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' REFERENCES [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' Goldsmith, Wireless Communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' Cambridge University Press, 2005.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' Learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' 18, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' 430–474, Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' 100 10-1 10-2 10-3 10-4 10~5 EP MGS HMC SISO AWGN 10-7 22 24 26 28 30 Average received SNR [dB] (c) 64QAM100 10-1 10-2 10-3 10-4 10-5 EP MGS HMC SISO AWGN 10-7 14 16 18 20 22 Average received SNR [dB] (b) 16QAM100 10-1 10-2 Average BER 10-3 10 10-5 EP MGS 10-6 HMC SISOAWGN 10-7 6 8 10 12 14 Average received SNR [dB] (a) QPSK[16] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' Gelman, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' Carlin, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' Stern, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' Dunson, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' Vehtari, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' Rubin, Bayesian Data Analysis, 3rd ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} +page_content=' CRC Press, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE1T4oBgHgl3EQfdgR_/content/2301.03196v1.pdf'} diff --git a/cdE4T4oBgHgl3EQfpA0g/content/tmp_files/2301.05188v1.pdf.txt b/cdE4T4oBgHgl3EQfpA0g/content/tmp_files/2301.05188v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..c98b34ba2363b0500df1eb3428b2c86c4834bb3a --- /dev/null +++ b/cdE4T4oBgHgl3EQfpA0g/content/tmp_files/2301.05188v1.pdf.txt @@ -0,0 +1,755 @@ + + + +MULTI-EPOCH RADIO SOURCE STRUCTURE ANALYSIS OF 11 +CALIBRATORS AT 2.3 AND 8.4 GHZ IN THE SOUTH + +A PREPRINT + +Sanway Chatterjee +Rasulpur, Memari, Purba Bardhaman +West Bengal, PIN 713151, India +iamsanway@gmail.com +Sayan Basu +Wits Centre for Astrophysics, School of Physics +University of the Witwatersrand +Private Bag 3, 2050, Johannesburg, South Africa + +Daniel MacMillan +NVI, Inc., NASA Goddard Space Flight Center +Code 61A, Greenbelt, MD, USA + + +January 12, 2023 + +ABSTRACT +We present the source structure analysis of 11 calibrator sources below −40° south at 2.3 (S-band) +and 8.4 GHz (X-band). We used multi-epoch very long baseline interferometry source maps +available in the radio fundamental catalog to analyse jet-structure variability and also used fluxes +from the Goddard Space Flight Center database to see whether these two complement each other or +not. Also, total fluxes from the maps were plotted with the fluxes from the database. The S/X-band +light curve analysis provides a more clear picture of the structural variability at the S/X-band also +indicates the possibility of the "core-shift" phenomenon. We found jet-like structures in the majority +of the sources in the sample. + +Keywords : +Radio astronomy – quasars – interferometry – source structure + +1 Introduction +Extragalactic radio sources used in radio catalogs such as the International Celestial Reference Frame (ICRF; Ma et al., +1998) or Radio Fundamental Catalog1 (RFC), are generally active galactic nuclei (AGN) with an active supermassive +black hole at the centre. Extragalactic radio sources, in general, exhibit time- and frequency-dependent source structures. The +structure of these sources varies with time, it is therefore important to model their structure at multiple epochs in +order to define a time-dependent source model. Quasars, being the brightest in the AGN subclasses and also being +located at far distances from the earth (e.g., the most distant quasar yet identified is J0313-1806 at redshift z = 7.642; +Wang et al., 2021), show almost no proper motion on the sky. Therefore, being the brightest and appearing as a point- +like source in the sky with no proper motion, they are considered to be good candidates as calibrators. +High angular resolution maps from Very Long Baseline Interferometry (VLBI) observations provide us an opportunity +to detect source structures at milliarcsecond (mas) scales. The VLBI positions of quasars are also used to define +and maintain the accuracy of the ICRF. The present catalog (ICRF-3; Charlot et al., 2020) currently contains VLBI +positions of 4,536 radio sources (mainly quasars). Among these observed sources, 2,615 (57.65%) are in the Northern +Hemisphere and 1,921 (42.35%) are in the Southern Hemisphere. The very long baseline array calibrator surveys (VCS; +Beasley et al., 2002, Fomalont et al., 2003, Petrov et al., 2005, Petrov et al., 2006, Kovalev et al., 2007, Petrov et al., +2008, Petrov, 2016) have been used to increase the number of calibrators in the Northern Hemisphere. However, the + +1http://astrogeo.org/rfc/ + +Radio source structure analysis +A PREPRINT +2 + + + +B1950 Name +J2000 Name +Optical ID +z +RA (hh mm ss) +DEC (deg mm ss) +St(S) +St(X) +Sp(S) +Sp(X) +Latest epoch +0048-427 +J0051-4226 +QSO +1.749 +00 51 09.501827 +-42 26 33.29329 +0.405 +0.860 +0.380 +0.616 +2021.01.27 +0104-408 +J0106-4034 +QSO +0.584 +01 06 45.107971 +-40 34 19.96031 +0.884 +1.154 +0.868 +1.097 +2018.08.10 +0332-403 +J0334-4008 +QSO +1.445 +03 34 13.654488 +-40 08 25.39791 +1.285 +0.961 +1.257 +0.975 +2021.05.19 +0537-441 +J0538-4405 +QSO +0.894 +05 38 50.361557 +-44 05 08.93893 +2.034 +1.813 +1.406 +1.535 +2021.05.19 +1104-445 +J1107-4449 +QSO +1.598 +11 07 08.694118 +-44 49 07.61837 +1.603 +1.179 +1.360 +0.675 +2021.01.27 +1349-439 +J1352-4412 +QSO +0.050 +13 52 56.534938 +-44 12 40.38769 +0.224 +0.389 +0.179 +0.353 +2021.05.19 +1424-418 +J1427-4206 +QSO +1.522 +14 27 56.297561 +-42 06 19.43769 +0.893 +1.090 +0.499 +1.067 +2021.01.27 +1451-400 +J1454-4012 +QSO +1.810 +14 54 32.912361 +-40 12 32.51452 +0.367 +0.541 +0.085 +0.377 +2018.07.31 +2052-474 +J2056-4714 +QSO +1.489 +20 56 16.359815 +-47 14 47.62776 +1.896 +1.881 +1.518 +1.447 +2017.09.06 +2106-413 +J2109-4110 +QSO +1.406 +21 09 33.188592 +-41 10 20.60545 +0.903 +0.581 +0.439 +0.247 +2018.05.19 +2333-415 +J2336-4115 +QSO +1.058 +23 36 33.985083 +-41 15 21.98402 +0.468 +0.676 +0.305 +0.530 +2018.08.10 +Table 1: The physical properties of the selected sources. Right ascension (RA) and declination (Dec) are shown with the most recent +positions in the RFC database. Optical identification and redshift (z) are taken from the NASA/IPAC database. The parameter St(S) +is the total flux in S-band; St(X) is the total flux in X-band; Sp(S) is the peak flux in S-band; Sp(X) is the peak flux in X-band. + +long baseline array calibrator survey (LCS; Petrov et al., 2011, Petrov et al., 2019), which is dedicated to increase the +number of sources as well as to study the VLBI positions, contributed significantly in the south to increase the number +of calibrators. Sources selected from the ICRF, are also being selected from the RFC that contains VLBI positions +of a total of 20,250 radio sources, where 11,462 sources (56.60%) are in the Northern Hemisphere and 8,788 sources +(43.40%) are in the Southern Hemisphere. Despite all these surveys, no dedicated initiatives have been taken yet to +study source structure in the Southern Hemisphere routinely. +Being motivated by this problem, we tried to analyze the radio source structure using the available VLBI source maps +from the RFC and to complement the analysis of these maps with the fluxes from the Goddard Space Flight Center +(GSFC) database. The GSFC database contains the fluxes of the observed sources from all the available VLBI geodetic +and astrometric observing sessions (generally 24 hours in duration) from the past several decades. Available multi-epoch +VLBI source maps of the selected sources from the RFC have been used to see whether their flux density variability +agrees with the flux variability from the light-curves generated using the fluxes available in the GSFC database. + +2 Observation and Methodology +We have selected sources in the Southern Hemisphere in the declination zone [−40°, −90°] which were observed at +2.3 (S-band) and 8.4 GHz (X-band). Since we are trying to analyze the radio source structure in the calibrator sources +at multi-epoch observations, we selected 11 sources which have been observed in more than 10 epochs (Table 1). +To construct light curves and analyse flux variability, we have used the flux densities of the selected sources available +in the RFC database. In addition to the RFC database, we used VLBI fluxes of the selected sources from the GSFC +database. The database contains S/X-band fluxes of ICRF sources observed in geodetic and astrometric VLBI sessions +around the world. Finally, we have used the available multi-epoch VLBI images to see if the flux density variability +detected from the light curve appears in the source maps. + +3 Results and Analysis +One of the characteristics of a calibrator source is its stability in flux density (no or very little flux density variation) +in time- and frequency domain. An ideal calibrator appears to be a compact or point-like source over all projected +baselines. +In this section, we present results and a detailed analysis of the sources from our sample. For that purpose, we use a +metric, flux variability index as well as we constructed light-curves using flux densities available on GSFC database +and flux densities obtained from the available VLBI source maps in the RFC database (rfc_2022c). Lastly, we analyse +the flux density variability trend (if any) between the flux density from the database and source structures in the VLBI +source maps to see whether modelling of source structure can actually be useful to detect structure variability. + +3.1 Flux Variability Index +The flux variability index is a statistical measure that is indicative of how the series of fluxes of a given source are +scattered around the mean flux. To analyse flux density variability in our sample, we used the GSFC database. This +metric is used to compare the flux dispersion of different sources. Unlike the standard deviation, which is always to +be considered in the context of the mean value, the flux variability index provides a relatively simple tool to compare + +Radio source structure analysis +A PREPRINT +3 + + + + + +Figure 1: Skyplot of all the sources available in the RFC catalog using the Aitoff-hammer projection. The red dots denote the +sources above -40 degrees in declination. The blue dots denote the sources below -40 degree which are comparatively much less +dense. The black diamonds represent the selected sources used in our analysis. + + +different flux data. Using the flux data, we calculate the mean flux density, averaged over all epochs in which the +sources were observed (Table 2, column 5 and 8). Flux variability index is the ratio of the standard deviation and the +mean of the total flux densities. A value of 0.0 indicates no variation over time. Mathematically, the standard formula +of the flux variability index is expressed as: + 𝐹𝐼 = +σ +𝑆̅ (1) +Where FI is flux variability index, σ and S̅ are the standard deviation and the mean of all the flux densities respectively +of each source over observing epochs. Using equation (1), we calculate the flux variability indices of all the sources +in our sample at S/X-band (Table 2, column 6 and 9). +We present the flux variability index distribution at S-band and X-band in Figure 2 and 3 respectively. At X-band, six +sources have a variability index of <0.5. However, at S-band, nine sources have a variability index of <0.5. We also +want to see what is the mean and median of the index >0.5. Our results show that selected sources have lower flux +density variability at S-band compared to X-band. + +3.2 Light Curve Analysis +Apart from flux density variability index, we also analysed light curves at S/X-band to understand the trend of flux +density variation. The flux densities from the RFC database and GSFC database are used to construct the light-curves. +Since the GSFC fluxes are collected on a daily basis from various VLBI observations, a light curve with both the fluxes +(from VLBI RFC maps and from the GSFC flux database) can provide us with more information on source structure +variability. In our analysis, we have constructed light curves of the selected sources at S/X-band. The total flux (sum of +all CLEAN components) from the RFC maps of the selected sources are plotted over epochs of their observation to +compare with the variation of the light curves. +Among the selected 11 sources, five sources have flux data for more than 20 years and six sources have flux data for +less than 20 years. In most cases, we notice that X-band light curves temporally lead the S-band curves. This time-lag +is likely caused by the “core-shift” effect (Shabala et al., 2014). For the source 0537-441, presented here, S- and X- +band light curves clearly show that there is a time-lag (Figure 4 and 5). +For example, the source 0537-441 (J0538-4405) has been routinely observed between 2000 and 2020 in 1965 sessions. +The source exhibits clear variability in flux-density, and it has multi-epoch VLBI maps to understand the source structure +along with the flux-density variation. The source has a flux variability index of 0.6, which indicates variations in fluxes +over epochs. Light curve analysis is a useful way to understand these variations. Also, it is useful to quantify the time +delay between the flux variations at different frequencies. Figure 4 shows X-band flux density variability over a period +of 21 years. In the figure, along with the GSFC fluxes, we plot total fluxes obtained from VLBI maps. + +Sourcesabove-40° +Sourcesbelow-40° +SelectedsourcesRadio source structure analysis +A PREPRINT +4 + + + + + + + +Figure 2: Distribution of flux variability index FI at S-band. +Figure 3: Distribution of flux variability index FI at X-band. + + +Observing the X-band light-curve indicates that the flux-density varies significantly between MJD 52236 to MJD 52931, +where we see flux density changes by a factor of 5.13. Source maps in (Figure 4(a)) exhibit jet-structure that agrees +with the flux variation. The total flux-density reaches the highest values around MJD 55300, then a rapid decrease in +the flux-density is observed. Again, the flux increases and another peak is observed at MJD 56000. After that, another +decrease occurs where the source-maps agree with the variability of the flux-density. The maps show jet-structures +at MJD 56203 and MJD 56266 (Figure 4(b)). At MJD 57700, we notice a further decrease, which is consistent with +the source-maps with jet-structures (Figure 4(c)). We can detect source structure variability by looking at short-term +flux variations. We also note that this opposes the flux variability index that was determined over the whole light curve +series. +In the S-band light curve (Figure 5), variations in total flux density are observed. It can be seen that the S-band light +curve lags the X-band light curve by about one year. Between MJD 52300 to MJD 53000, the flux density falls down +which is juxtaposed with the source-maps. At first, jet-structure appears, which soon disappears and results in a point- +like structure at MJD 52479 (Figure 5(a)). After MJD 52500, the flux density again increases and jet-structure +reappears (Figure 5(b)). From MJD 57600 to MJD 58000, the flux density increases. The source-maps of 2017 show +that the source exhibits jet structures throughout the year (Figure 5(c)). The light-curve shows that the flux-density +decreases at MJD 58731 and after that it increases. In 2020, the source-maps show a point-like structure and then in +2021, jet-structures. In March 2021, the source-map again shows a point-like structure. + +3.3 Source Structure Analysis +After analyzing flux variability index and light curve analysis, we present structure analysis of each source using +available contour maps. Multi-epoch VLBI source maps from RFC database have been considered for this purpose. + +3.3.1 0048-427 (J0051-4226) +The source 0048-427 has 19 VLBI maps available between 2002 and 2020. The mean fluxes of the source at S/X-band +are 0.52 Jy and 0.65 Jy respectively, and the flux variability indices are 0.34 and 0.55 respectively. The S-band light +curve of this source shows no rapid variations throughout the epochs, the S-band fluxes have a standard deviation of +0.18 Jy. The X-band light curve also shows no variation until MJD 58000. +We have nine VLBI maps of the source in 2017. Available source maps indicate jet-structure variability in the source. +The X-band maps indicate the appearance and disappearance of jet-structure over a period of three months. However, at S- +band, the source appears to be compact in nature in the majority of epochs. + +3.3.2 0104-408 (J0106-4034) +The source 0104-408 has 68 VLBI source-maps available from 1994 to 2018. Mean fluxes are 0.92 Jy and 1.88 Jy and +the flux variability indices are 0.4 and 0.44 at S/X-bands. The light curves of the source for both S and X bands show +similar variations. But a time delay is observed between the light-curves, the X-band light-curve is ahead of the S-band. + +4 +ofSources +m +Number +2 +1 +0 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +Flux Variability Index4 +ofSources +m +Number +2 +1 +0 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +Flux Variability IndexRadio source structure analysis +A PREPRINT +5 + + + + +Figure 4: Light curve of the source 0537-441 at X-band. Orange dots show fluxes of VLBI images from RFC, and blue points are +VLBI flux data from GSFC. (a) Jet-structure appeared at MJD 52899, (b) Jet-structure at MJD 56266 at X-band, (c) Jet-structure +appeared at MJD 57977. All the VLBI maps are available on rfc_2022c. + + +Figure 5: Light curve of the source 0537-441 at S-band. Orange dots show fluxes of VLBI images from RFC, and blue points are +VLBI flux data from GSFC. (a) Compact structure appeared at MJD 52479, (b) Jet-structure at MJD 52889, (c) Jet-structure at MJD +58006. All the VLBI maps are available on rfc_2022c. + +105384405 +6C +(a) +GSFC +25 +Astrogeo +Right escension (man) relative to 05:38:50.3615 +2012:12.05 +J0538~4405 +Freq:8.6GHz +20 +≤15 +(b) +10 +5 +20. +10 +Ny/beas +2017.08.12 +0 +J0538-4405 +Preq:B.7GHz +50000 +52000 +54000 +56000 +58000 +Epoch (MJD) +() +Right ascension (mas) relative to 05:38:50.38162002.07.2 +10538-4405 +6C6 +(a) +25 +GSFC +Astrogeo +Right ascension (mas) relative te 05:31:50.3815 +Penkirv +1.8miy/beam +2003.07.09 +J0538-4405 +Freq:2.3-GHz +20 +6C6'00 +8 +15 +(b) +10 +5- +Right_ascenslon (mes)relative te 05:30:50.3515 +80 +2017.10.09 +J053B4405 +0 +Freq:2.3CHz +50000 +52000 +54000 +56000 +58000 +:05:08.939 +Epoch (MJD) +(c) +54Radio source structure analysis +A PREPRINT +6 + + + + +X-band data +S-band data +B1950 Name +J2000 Name +z +Epochs +Mean +Std Dev +FI +Mean +Std Dev +FI +0048-427 +J0051-4226 +1.749 +22 +0.658 +0.366 +0.556 +0.528 +0.182 +0.345 +0104-408 +J0106-4034 +0.584 +68 +1.883 +0.838 +0.445 +0.923 +0.376 +0.407 +0332-403 +J0334-4008 +1.445 +28 +1.638 +0.601 +0.366 +1.548 +0.404 +0.261 +0537-441 +J0538-4405 +0.894 +59 +5.358 +3.482 +0.649 +4.446 +2.774 +0.623 +1104-445 +J1107-4449 +1.598 +15 +2.126 +0.932 +0.438 +1.466 +0.40 +0.272 +1349-439 +J1352-4412 +0.050 +10 +0.254 +0.088 +0.345 +0.208 +0.064 +0.308 +1424-418 +J1427-4206 +1.522 +52 +4.775 +3.341 +0.699 +2.609 +1.267 +0.485 +1451-400 +J1454-4012 +1.810 +30 +0.289 +0.113 +0.392 +0.374 +0.164 +0.439 +2052-474 +J2056-4714 +1.489 +28 +3.112 +2.271 +0.729 +1.607 +0.586 +0.364 +2106-413 +J2109-4110 +1.406 +18 +0.395 +0.317 +0.803 +0.591 +0.314 +0.532 +2333-415 +J2336-4115 +1.058 +13 +0.305 +0.125 +0.408 +0.256 +0.074 +0.290 +Table 2: Col 1 and Col 2 represent the B1950 and J2000 names of the sources. Col 3 denotes the redshift of the sources. Col 4 +shows the number of epochs in which observations were done. Col 5,6 and 7 represent the mean flux, standard deviation of fluxes +and flux variability index respectively of the respective sources at X-band. Col 8,9 and 10 represent the mean flux, standard deviation +of fluxes and flux variability index respectively of the respective sources at S-band. + + +The VLBI maps over all the epochs indicate the appearance and disappearance of jet-structure over a period of four +months. In between 2007 and 2012, the jet structure appears every three to six months but in the year 2017, the change +in the jet-structure is more frequent about 15 days. Both the S/X-band images show the same behaviour which agrees +with the light curves. + +3.3.3 0332-403 (J0334-4008) +The source 0332-403 has mean fluxes and flux variability indices of 1.54 Jy and 0.26 at S-band and 1.63 Jy and 0.36 +at X-band. It has 28 VLBI maps between 2005 and 2021. In 2017, the source has 15 images in a row which evince +jet-structures. +From the light curves, it is seen that the source has large deviations in its mean flux. At the same time, a different +scenario is noticed at S/X-bands. It is interesting that at MJD 55000, the total flux-density at X-band decreases and the +corresponding flux-density at the S-band increases. However, between MJD 55600 and MJD 56100, the X-band flux- +density increases but the same at the S-band decreases. After MJD 57800, both the S and X-band flux-densities present +the same behavior, they both start falling. Physically, this corresponds to the emergence of new jet-components. This +agrees with the source maps of the corresponding epochs, they show jet-structures at both S/X-bands. After MJD 58400, +the X-band flux-density rises more rapidly in comparison to the S-band flux-density. + +3.3.4 0537-441 (J0538-4405) +For this source, there are 56 VLBI contour maps between 1995 and 2021. We have calculated mean flux-densities as +4.44 Jy and 5.35 Jy and flux variability indices as 0.62 and 0.64 at S/X-band respectively. +The analysis of the light curves at S- and X-band of this source is discussed in section 3.2. Between 1995 and 2018, +jet-components can be seen on the source-maps of the source. At MJD 59037, the map shows no sign of jet-components, +it shows point-like structure. The jet-structure again appears after four months at MJD 59297. +At S-band, between 1998 and 2003, the source exhibit jet-structures. At MJD 51840, it becomes point-like, this structure +reappears after five months. The source-maps between MJD 52038 and MJD 52211 also show point-like structure of +the source. Though we don’t have enough maps in 2007 and 2008, but the point-like structures appear at MJD 54439 +and MJD 54817. In 2017, the source has ten VLBI source-maps, which show that the source emits jet throughout the +year. The behavior of the source in 2020 is discussed in section 3.2. + +3.3.5 1104-445 (J1107-4449) +This source has mean flux-densities of 1.46 Jy and 2.12 Jy and flux variability indices 0.27 and 0.43 for S- and X-band +respectively. The light-curves show that both the S/X flux-densities decreases first, but the rate of decreases is higher in +case of X-band. At X-band, the flux-density starts rising from MJD 55666, but at S-band it starts rising from MJD +56666. Here the X-band flux density leads that of the S-band. After MJD 58000, the X-band flux-density increases with +a greater first derivative. + +Radio source structure analysis +A PREPRINT +7 + + +This RFC does not have enough source maps to analyze the change in the source structure. It has two observations each +year, so we cannot make any obvious conclusion amid these observations. But if we analyze the available source-maps +at X-band after MJD 58000, we can find that the source exhibits jet-structures. This is in accordance to the rapid +increase of flux-density in this period. + +3.3.6 1349-439 (J1352-4412) +Between 2001 and 2021, the source 1349-439 has only nine contour maps. We have calculated the mean flux-densities +as 0.20 Jy and 0.25 Jy and flux variability indices as 0.30 and 0.34 for S- and X-band respectively of the source. The +variation of flux-density is much less about the mean for both S/X-bands. Both the light-curves mostly follow the same +pattern. Between MJD 51786 and 54570, the flux-density decreases and then they start increasing. After MJD 57500, +both the flux-densities rise, but the X-band density reaches a higher value. We don’t have enough source-maps for this source +too to analyze the source structure. +The radio fundamental catalog has two source-maps in 2021 with a gap of two months, which shows that the source has +minimal jet-structures and at MJD 59353, the source appears to be point-like at X-band. + +3.3.7 1424-418 (J1427-4206) +This source has mean flux-densities of 2.60 Jy and 4.77 Jy and flux variability indices 0.48 and 0.69 for S- and X-band +respectively. The light curves for both S/X-bands follow the same pattern, but the flux-density peaks reach higher values +at X-band in comparison to S-band. The highest peak of the flux-density at X-band has a value of 33.57 Jy, whereas at +S-band it has a value of 13.77 Jy. The X-band light-curve leads the S-band. After MJD 52000, both the fluxes increase, +the X-band flux-density shows a greater rate of increase. Between MJD 53621 and 54000, interestingly the X-band +density drops, but that for the S-band density moves up. After MJD 56000, both the flux-densities show rapid rise, +where both of them reach their highest values. +The source has 54 contour maps between 1994 and 2021. In the period from 1998 to 2003, we have seen the variation +of flux-density. Consequently, in this period, the S/X source maps show jet-structures. Moreover, between MJD 54600 +and 55070 where, the flux-densities increases a little and between MJD 56650 and 56850, where the flux-density rises +rapidly, all the source-maps in these periods show jet-structure. Also, after MJD 59000, the maps exhibit jets and the +light curves show declination in flux-densities. + +3.3.8 1451-400 (J1454-4012) +This source has mean fluxes of 0.37 and 0.28 Jy, flux variability indices of 0.43 and 0.39 at S- and X-band respectively. +This source has twenty-nine source-maps between 1999 and 2018. Between MJD 53000 and 54000, the S-band flux- +density increases, whereas the X-band flux-density decreases. Thereafter, between MJD 54000 and MJD 56000, the +behavior of the light-curves turns over, the S-band density falls and that of X-band rises. At MJD 58000, both S/X flux- +densities increase and the corresponding source maps show jet-like structures. +Between MJD 51500 and MJD 53000, the source-maps show jet-structures, but the source-map at MJD 52766 shows +that the source exhibits point-like structure at S-band. At X-band, the source seems to be compact at MJD 51574. This +source structure reappears at MJD 52290. At MJD 58000, both the S/X flux-densities rise and the corresponding source- +maps show jet-structures. + +3.3.9 2052-474 (J2056-4714) +The source has 27 contour-maps from 1999 to 2017. We have calculated the mean flux-densities of this source as 1.60 +Jy and 3.11 Jy and flux variability indices as 0.36 and 0.72 at S and X-band respectively. Similar to the source discussed +in section 3.3.7 (source 1424-418), the peaks in the light-curve also have higher values at X-band than the S-band. The +X-band light-curve also leads the S-band curve. Between MJD 51200 and 53000, the flux-densities fall, then they start +increasing simultaneously. Both the S/X plot reach their highest value at MJD 55440. +Between 1999 and 2003, we have 17 maps which show jet-structures, except the maps at MJD 51938 and MJD 52038, +where the source is compact for both S/X-band. After 2003, the radio fundamental catalog does not have enough +source-maps to identify the nature of the source structure precisely. In accordance to the light-curves, between MJD +56000 and 57000, the flux-density falls. In 2013, we have source-maps at two adjacent epochs (MJD 56497 and 56546) +in which jet components evolve. + +Radio source structure analysis +A PREPRINT +8 + + +− +3.3.10 2106-413 (J2109-4110) +This source has mean fluxes of 0.59 Jy and 0.39 Jy, flux variability indices 0.53 and 0.80 for S- and X-band respectively. +From the light curves, it is seen that between MJD 52000 and 56000, the flux-densities decrease. This indicates that jet +components may appear in the source structure, but unfortunately, we don’t have enough VLBI images to substantiate. +From MJD 56000 to 57000, the flux-densities rise and then after MJD 57000, they again decrease and the corresponding +source-maps shows jet-structures. + +3.3.11 2333-415 (J2336-4115) +The source 2333-415 has 13 contour-maps between 2012 and 2018. it has mean fluxes of 0.25 Jy and 0.30 Jy and flux +variability indices 0.29 and 0.40 at S/X-band respectively. The S-band light-curve shows that the flux-density increases +up to MJD 57000 and then decrease. At X-band, however, the flux-density increases after MJD 57000. Both the S/X- +band source-maps appear to be jet-structure. + +4 Summary and Conclusion +We presented radio source structure analysis for 11 calibrator sources at the S- and X-band. All the selected sources are +below 40° declination and were observed in multi-epoch VLBI observing sessions. To analyse the source structure +variability, we mainly relied on the available source maps to detect any visible changes in the structure and also on the +light-curves that were constructed using the GSFC fluxes. Firstly, we carefully went through source maps to detect any +kind of variability in the source structure at mas scales. Then we used the light-curve to see the fluctuations in flux. +Here, the available multi-epoch fluxes in the GSFC database were used. Finally, we extracted total fluxes (the sum of +all the CLEAN components in an RFC VLBI map) and plotted those along with the GSFC fluxes. This gives a clear +perspective of whether the detected structure in VLBI maps (thus change in source flux) agrees with the flux variation +detected in the GSFC flux-based light-curve. We also used a metric, flux variability index, to quantify the scale of the +structural variability. In this epoch-based analysis of the sources at S/X-band, we found all the selected sources exhibit +jet-like structures at some or all epochs. However, based on the multi-epoch VLBI source maps, light-curves, and flux +variability index analysis we analyzed the magnitude of source structure that can be used to quantify whether a source is +suitable as a calibrator or not. Sources with extended jet structures, random fluctuations in the light curve, and higher +flux variability index (>0.3) have been considered as not suitable for calibrators and should be observed for more +analysis. Therefore, at S-band, we found three sources to be suitable candidates as calibrators. The rest of the sources +may be used as calibrators, but we recommend more rigorous source structure analysis of these sources. At X-band, +all the sources have flux variability index greater than 0.3 which indicates that these sources have jet-like structures +consistent over all the epochs. There are six sources having flux variability index between 0.3 and 0.5. The rest of the +sources have flux variability index higher than 0.5. The flux variability indices also agree with the light curve analysis and +also jet-like structures in the source maps. Overall, we recommend all the selected sources to be monitored regularly to +analyze their suitability as calibrators. + +5 Acknowledgements +This work has made use of the Radio Fundamental Catalog database. The work also has made use of NASA’s +Astrophysics Data System Bibliographic Services and the NASA/IPAC Extragalactic Database (NED) which is operated by +the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and +Space Administration. + +References +Beasley, A., Gordon, D., Peck, A., Petrov, L., MacMillan, D., Fomalont, E., and Ma, C. (2002). The vlba calibrator +survey—vcs1. The Astrophysical Journal Supplement Series, 141(1):13. +Charlot, P., Jacobs, C. S., Gordon, D., Lambert, S., de Witt, A., Böhm, J., Fey, A. L., Heinkelmann, R., Skurikhina, E., Titov, +O., Arias, E. F., Bolotin, S., Bourda, G., Ma, C., Malkin, Z., Nothnagel, A., Mayer, D., MacMillan, D. S., Nilsson, +T., and Gaume, R. (2020). The third realization of the International Celestial Reference Frame by very long baseline +interferometry. , 644:A159. +Fomalont, E., Petrov, L., MacMillan, D., Gordon, D., and Ma, C. (2003). The second vlba calibrator survey: Vcs2. The +Astronomical Journal, 126(5):2562. + +Radio source structure analysis +A PREPRINT +9 + + +Kovalev, Y. Y., Petrov, L., Fomalont, E., and Gordon, D. (2007). The fifth vlba calibrator survey: Vcs5. The Astronomical +Journal, 133(4):1236. +Ma, C., Arias, E., Eubanks, T., Fey, A., Gontier, A.-M., Jacobs, C., Sovers, O., Archinal, B., and Charlot, P. (1998). The +international celestial reference frame as realized by very long baseline interferometry. The Astronomical Journal, +116(1):516. +Petrov, L. (2016). Vlba calibrator survey 9 (vcs-9). arXiv preprint arXiv:1610.04951. +Petrov, L., de Witt, A., Sadler, E. M., Phillips, C., and Horiuchi, S. (2019). The second lba calibrator survey of southern +compact extragalactic radio sources–lcs2. Monthly Notices of the Royal Astronomical Society, 485(1):88–101. +Petrov, L., Kovalev, Y. Y., Fomalont, E., and Gordon, D. (2005). The third vlba calibrator survey: Vcs3. The +Astronomical Journal, 129(2):1163. +Petrov, L., Kovalev, Y. Y., Fomalont, E., and Gordon, D. (2006). The fourth vlba calibrator survey: Vcs4. The +Astronomical Journal, 131(3):1872. +Petrov, L., Kovalev, Y. Y., Fomalont, E., and Gordon, D. (2008). The sixth vlba calibrator survey: Vcs6. The +Astronomical Journal, 136(2):580. +Petrov, L., Phillips, C., Bertarini, A., Murphy, T., and Sadler, E. M. (2011). The lba calibrator survey of southern +compact extragalactic radio sources–lcs1. Monthly Notices of the Royal Astronomical Society, 414(3):2528–2539. +Shabala, S.S. et al. (2014), The effects of frequency-dependent quasar variability on the celestial reference frame. +Journal of Geodesy, 88:575-586. +Wang, F., Yang, J., Fan, X., Hennawi, J. F., Barth, A. J., Banados, E., Bian, F., Boutsia, K., Connor, T., Davies, F. B., +et al. (2021). A luminous quasar at redshift 7.642. The Astrophysical Journal Letters, 907(1):L1. + diff --git a/cdE4T4oBgHgl3EQfpA0g/content/tmp_files/load_file.txt b/cdE4T4oBgHgl3EQfpA0g/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ec65086c7bd2507a5cccc588c34cc5ad40911d7e --- /dev/null +++ b/cdE4T4oBgHgl3EQfpA0g/content/tmp_files/load_file.txt @@ -0,0 +1,692 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf,len=691 +page_content='MULTI-EPOCH RADIO SOURCE STRUCTURE ANALYSIS OF 11 CALIBRATORS AT 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='3 AND 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='4 GHZ IN THE SOUTH A PREPRINT Sanway Chatterjee Rasulpur, Memari, Purba Bardhaman West Bengal, PIN 713151, India iamsanway@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='com Sayan Basu Wits Centre for Astrophysics, School of Physics University of the Witwatersrand Private Bag 3, 2050, Johannesburg, South Africa Daniel MacMillan NVI, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=', NASA Goddard Space Flight Center Code 61A, Greenbelt, MD, USA January 12, 2023 ABSTRACT We present the source structure analysis of 11 calibrator sources below −40° south at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='3 (S-band) and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='4 GHz (X-band).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' We used multi-epoch very long baseline interferometry source maps available in the radio fundamental catalog to analyse jet-structure variability and also used fluxes from the Goddard Space Flight Center database to see whether these two complement each other or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' Also, total fluxes from the maps were plotted with the fluxes from the database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' The S/X-band light curve analysis provides a more clear picture of the structural variability at the S/X-band also indicates the possibility of the "core-shift" phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' We found jet-like structures in the majority of the sources in the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' Keywords : Radio astronomy – quasars – interferometry – source structure 1 Introduction Extragalactic radio sources used in radio catalogs such as the International Celestial Reference Frame (ICRF;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' Ma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=', 1998) or Radio Fundamental Catalog1 (RFC), are generally active galactic nuclei (AGN) with an active supermassive black hole at the centre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' Extragalactic radio sources, in general, exhibit time- and frequency-dependent source structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' The structure of these sources varies with time, it is therefore important to model their structure at multiple epochs in order to define a time-dependent source model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' Quasars, being the brightest in the AGN subclasses and also being located at far distances from the earth (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=', the most distant quasar yet identified is J0313-1806 at redshift z = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='642;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=', 2021), show almost no proper motion on the sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' Therefore, being the brightest and appearing as a point- like source in the sky with no proper motion, they are considered to be good candidates as calibrators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' High angular resolution maps from Very Long Baseline Interferometry (VLBI) observations provide us an opportunity to detect source structures at milliarcsecond (mas) scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' The VLBI positions of quasars are also used to define and maintain the accuracy of the ICRF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' The present catalog (ICRF-3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' Charlot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=', 2020) currently contains VLBI positions of 4,536 radio sources (mainly quasars).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' Among these observed sources, 2,615 (57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='65%) are in the Northern Hemisphere and 1,921 (42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='35%) are in the Southern Hemisphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' The very long baseline array calibrator surveys (VCS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' Beasley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=', 2002, Fomalont et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=', 2003, Petrov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=', 2005, Petrov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=', 2006, Kovalev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=', 2007, Petrov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=', 2008, Petrov, 2016) have been used to increase the number of calibrators in the Northern Hemisphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' However, the 1http://astrogeo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='org/rfc/ Radio source structure analysis A PREPRINT 2 B1950 Name J2000 Name Optical ID z RA (hh mm ss) DEC (deg mm ss) St(S) St(X) Sp(S) Sp(X) Latest epoch 0048-427 J0051-4226 QSO 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='749 00 51 09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='501827 -42 26 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='29329 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='10 Table 1: The physical properties of the selected sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' Right ascension (RA) and declination (Dec) are shown with the most recent positions in the RFC database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' Optical identification and redshift (z) are taken from the NASA/IPAC database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' The parameter St(S) is the total flux in S-band;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' St(X) is the total flux in X-band;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' Sp(S) is the peak flux in S-band;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' Sp(X) is the peak flux in X-band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' long baseline array calibrator survey (LCS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' Petrov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=', 2011, Petrov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=', 2019), which is dedicated to increase the number of sources as well as to study the VLBI positions, contributed significantly in the south to increase the number of calibrators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' Sources selected from the ICRF, are also being selected from the RFC that contains VLBI positions of a total of 20,250 radio sources, where 11,462 sources (56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='60%) are in the Northern Hemisphere and 8,788 sources (43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='40%) are in the Southern Hemisphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' Despite all these surveys, no dedicated initiatives have been taken yet to study source structure in the Southern Hemisphere routinely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' Being motivated by this problem, we tried to analyze the radio source structure using the available VLBI source maps from the RFC and to complement the analysis of these maps with the fluxes from the Goddard Space Flight Center (GSFC) database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' The GSFC database contains the fluxes of the observed sources from all the available VLBI geodetic and astrometric observing sessions (generally 24 hours in duration) from the past several decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' Available multi-epoch VLBI source maps of the selected sources from the RFC have been used to see whether their flux density variability agrees with the flux variability from the light-curves generated using the fluxes available in the GSFC database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' 2 Observation and Methodology We have selected sources in the Southern Hemisphere in the declination zone [−40°, −90°] which were observed at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='3 (S-band) and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='4 GHz (X-band).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' Since we are trying to analyze the radio source structure in the calibrator sources at multi-epoch observations, we selected 11 sources which have been observed in more than 10 epochs (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' To construct light curves and analyse flux variability, we have used the flux densities of the selected sources available in the RFC database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' In addition to the RFC database, we used VLBI fluxes of the selected sources from the GSFC database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' The database contains S/X-band fluxes of ICRF sources observed in geodetic and astrometric VLBI sessions around the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' Finally, we have used the available multi-epoch VLBI images to see if the flux density variability detected from the light curve appears in the source maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' 3 Results and Analysis One of the characteristics of a calibrator source is its stability in flux density (no or very little flux density variation) in time- and frequency domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' An ideal calibrator appears to be a compact or point-like source over all projected baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' In this section, we present results and a detailed analysis of the sources from our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' For that purpose, we use a metric, flux variability index as well as we constructed light-curves using flux densities available on GSFC database and flux densities obtained from the available VLBI source maps in the RFC database (rfc_2022c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' Lastly, we analyse the flux density variability trend (if any) between the flux density from the database and source structures in the VLBI source maps to see whether modelling of source structure can actually be useful to detect structure variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='1 Flux Variability Index The flux variability index is a statistical measure that is indicative of how the series of fluxes of a given source are scattered around the mean flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' To analyse flux density variability in our sample, we used the GSFC database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' This metric is used to compare the flux dispersion of different sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' Unlike the standard deviation, which is always to be considered in the context of the mean value, the flux variability index provides a relatively simple tool to compare Radio source structure analysis A PREPRINT 3 Figure 1: Skyplot of all the sources available in the RFC catalog using the Aitoff-hammer projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' The red dots denote the sources above -40 degrees in declination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' The blue dots denote the sources below -40 degree which are comparatively much less dense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' The black diamonds represent the selected sources used in our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' different flux data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' Using the flux data, we calculate the mean flux density, averaged over all epochs in which the sources were observed (Table 2, column 5 and 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' Flux variability index is the ratio of the standard deviation and the mean of the total flux densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' A value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='0 indicates no variation over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' Mathematically, the standard formula of the flux variability index is expressed as: 𝐹𝐼 = σ 𝑆̅ (1) Where FI is flux variability index, σ and S̅ are the standard deviation and the mean of all the flux densities respectively of each source over observing epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' Using equation (1), we calculate the flux variability indices of all the sources in our sample at S/X-band (Table 2, column 6 and 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' We present the flux variability index distribution at S-band and X-band in Figure 2 and 3 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' At X-band, six sources have a variability index of <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' However, at S-band, nine sources have a variability index of <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' We also want to see what is the mean and median of the index >0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' Our results show that selected sources have lower flux density variability at S-band compared to X-band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='2 Light Curve Analysis Apart from flux density variability index, we also analysed light curves at S/X-band to understand the trend of flux density variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' The flux densities from the RFC database and GSFC database are used to construct the light-curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' Since the GSFC fluxes are collected on a daily basis from various VLBI observations, a light curve with both the fluxes (from VLBI RFC maps and from the GSFC flux database) can provide us with more information on source structure variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' In our analysis, we have constructed light curves of the selected sources at S/X-band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' The total flux (sum of all CLEAN components) from the RFC maps of the selected sources are plotted over epochs of their observation to compare with the variation of the light curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' Among the selected 11 sources, five sources have flux data for more than 20 years and six sources have flux data for less than 20 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' In most cases, we notice that X-band light curves temporally lead the S-band curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' This time-lag is likely caused by the “core-shift” effect (Shabala et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' For the source 0537-441, presented here, S- and X- band light curves clearly show that there is a time-lag (Figure 4 and 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' For example, the source 0537-441 (J0538-4405) has been routinely observed between 2000 and 2020 in 1965 sessions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' The source exhibits clear variability in flux-density, and it has multi-epoch VLBI maps to understand the source structure along with the flux-density variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' The source has a flux variability index of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='6, which indicates variations in fluxes over epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' Light curve analysis is a useful way to understand these variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' Also, it is useful to quantify the time delay between the flux variations at different frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' Figure 4 shows X-band flux density variability over a period of 21 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' In the figure, along with the GSFC fluxes, we plot total fluxes obtained from VLBI maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' Sourcesabove 40° Sourcesbelow 40° SelectedsourcesRadio source structure analysis A PREPRINT 4 Figure 2: Distribution of flux variability index FI at S-band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' Figure 3: Distribution of flux variability index FI at X-band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' Observing the X-band light-curve indicates that the flux-density varies significantly between MJD 52236 to MJD 52931, where we see flux density changes by a factor of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' Source maps in (Figure 4(a)) exhibit jet-structure that agrees with the flux variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' The total flux-density reaches the highest values around MJD 55300, then a rapid decrease in the flux-density is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' Again, the flux increases and another peak is observed at MJD 56000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' After that, another decrease occurs where the source-maps agree with the variability of the flux-density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' The maps show jet-structures at MJD 56203 and MJD 56266 (Figure 4(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' At MJD 57700, we notice a further decrease, which is consistent with the source-maps with jet-structures (Figure 4(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' We can detect source structure variability by looking at short-term flux variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' We also note that this opposes the flux variability index that was determined over the whole light curve series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' In the S-band light curve (Figure 5), variations in total flux density are observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' It can be seen that the S-band light curve lags the X-band light curve by about one year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' Between MJD 52300 to MJD 53000, the flux density falls down which is juxtaposed with the source-maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' At first, jet-structure appears, which soon disappears and results in a point- like structure at MJD 52479 (Figure 5(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' After MJD 52500, the flux density again increases and jet-structure reappears (Figure 5(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' From MJD 57600 to MJD 58000, the flux density increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' The source-maps of 2017 show that the source exhibits jet structures throughout the year (Figure 5(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' The light-curve shows that the flux-density decreases at MJD 58731 and after that it increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' In 2020, the source-maps show a point-like structure and then in 2021, jet-structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' In March 2021, the source-map again shows a point-like structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='3 Source Structure Analysis After analyzing flux variability index and light curve analysis, we present structure analysis of each source using available contour maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' Multi-epoch VLBI source maps from RFC database have been considered for this purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='1 0048-427 (J0051-4226) The source 0048-427 has 19 VLBI maps available between 2002 and 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' The mean fluxes of the source at S/X-band are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='52 Jy and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='65 Jy respectively, and the flux variability indices are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='34 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='55 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' The S-band light curve of this source shows no rapid variations throughout the epochs, the S-band fluxes have a standard deviation of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='18 Jy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' The X-band light curve also shows no variation until MJD 58000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' We have nine VLBI maps of the source in 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' Available source maps indicate jet-structure variability in the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' The X-band maps indicate the appearance and disappearance of jet-structure over a period of three months.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' However, at S- band, the source appears to be compact in nature in the majority of epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='2 0104-408 (J0106-4034) The source 0104-408 has 68 VLBI source-maps available from 1994 to 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' Mean fluxes are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='92 Jy and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='88 Jy and the flux variability indices are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='4 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='44 at S/X-bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' The light curves of the source for both S and X bands show similar variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' But a time delay is observed between the light-curves, the X-band light-curve is ahead of the S-band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' 4 ofSources m Number 2 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='8 Flux Variability Index4 ofSources m Number 2 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='8 Flux Variability IndexRadio source structure analysis A PREPRINT 5 Figure 4: Light curve of the source 0537-441 at X-band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' Orange dots show fluxes of VLBI images from RFC, and blue points are VLBI flux data from GSFC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' (a) Jet-structure appeared at MJD 52899, (b) Jet-structure at MJD 56266 at X-band, (c) Jet-structure appeared at MJD 57977.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' All the VLBI maps are available on rfc_2022c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' Figure 5: Light curve of the source 0537-441 at S-band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' Orange dots show fluxes of VLBI images from RFC, and blue points are VLBI flux data from GSFC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' (a) Compact structure appeared at MJD 52479, (b) Jet-structure at MJD 52889, (c) Jet-structure at MJD 58006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' All the VLBI maps are available on rfc_2022c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' 105384405 6C (a) GSFC 25 Astrogeo Right escension (man) relative to 05:38:50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='3615 2012:12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='05 J0538~4405 Freq:8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='6GHz 20 ≤15 (b) 10 5 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' 10 Ny/beas 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='12 0 J0538-4405 Preq:B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='7GHz 50000 52000 54000 56000 58000 Epoch (MJD) () Right ascension (mas) relative to 05:38:50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='38162002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='2 10538-4405 6C6 (a) 25 GSFC Astrogeo Right ascension (mas) relative te 05:31:50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='3815 Penkirv 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='8miy/beam 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='09 J0538-4405 Freq:2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content="3-GHz 20 6C6'00 8 15 (b) 10 5- Right_ascenslon (mes)relative te 05:30:50." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='3515 80 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='09 J053B4405 0 Freq:2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='3CHz 50000 52000 54000 56000 58000 :05:08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='939 Epoch (MJD) (c) 54Radio source structure analysis A PREPRINT 6 X-band data S-band data B1950 Name J2000 Name z Epochs Mean Std Dev FI Mean Std Dev FI 0048-427 J0051-4226 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='749 22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='658 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='366 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='556 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='528 0.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='392 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='374 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='164 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='439 2052-474 J2056-4714 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='489 28 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='112 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='271 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='729 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='607 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='586 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='364 2106-413 J2109-4110 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='406 18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='395 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='317 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='803 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='591 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='314 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='532 2333-415 J2336-4115 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='058 13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='305 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='408 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='256 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='074 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='290 Table 2: Col 1 and Col 2 represent the B1950 and J2000 names of the sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' Col 3 denotes the redshift of the sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' Col 4 shows the number of epochs in which observations were done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' Col 5,6 and 7 represent the mean flux, standard deviation of fluxes and flux variability index respectively of the respective sources at X-band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' Col 8,9 and 10 represent the mean flux, standard deviation of fluxes and flux variability index respectively of the respective sources at S-band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' The VLBI maps over all the epochs indicate the appearance and disappearance of jet-structure over a period of four months.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' In between 2007 and 2012, the jet structure appears every three to six months but in the year 2017, the change in the jet-structure is more frequent about 15 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' Both the S/X-band images show the same behaviour which agrees with the light curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='3 0332-403 (J0334-4008) The source 0332-403 has mean fluxes and flux variability indices of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='54 Jy and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='26 at S-band and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='63 Jy and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='36 at X-band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' It has 28 VLBI maps between 2005 and 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' In 2017, the source has 15 images in a row which evince jet-structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' From the light curves, it is seen that the source has large deviations in its mean flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' At the same time, a different scenario is noticed at S/X-bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' It is interesting that at MJD 55000, the total flux-density at X-band decreases and the corresponding flux-density at the S-band increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' However, between MJD 55600 and MJD 56100, the X-band flux- density increases but the same at the S-band decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' After MJD 57800, both the S and X-band flux-densities present the same behavior, they both start falling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' Physically, this corresponds to the emergence of new jet-components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' This agrees with the source maps of the corresponding epochs, they show jet-structures at both S/X-bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' After MJD 58400, the X-band flux-density rises more rapidly in comparison to the S-band flux-density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='4 0537-441 (J0538-4405) For this source, there are 56 VLBI contour maps between 1995 and 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' We have calculated mean flux-densities as 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='44 Jy and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='35 Jy and flux variability indices as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='62 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='64 at S/X-band respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' The analysis of the light curves at S- and X-band of this source is discussed in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' Between 1995 and 2018, jet-components can be seen on the source-maps of the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' At MJD 59037, the map shows no sign of jet-components, it shows point-like structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' The jet-structure again appears after four months at MJD 59297.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' At S-band, between 1998 and 2003, the source exhibit jet-structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' At MJD 51840, it becomes point-like, this structure reappears after five months.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' The source-maps between MJD 52038 and MJD 52211 also show point-like structure of the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' Though we don’t have enough maps in 2007 and 2008, but the point-like structures appear at MJD 54439 and MJD 54817.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' In 2017, the source has ten VLBI source-maps, which show that the source emits jet throughout the year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' The behavior of the source in 2020 is discussed in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='5 1104-445 (J1107-4449) This source has mean flux-densities of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='46 Jy and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='12 Jy and flux variability indices 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='27 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='43 for S- and X-band respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' The light-curves show that both the S/X flux-densities decreases first, but the rate of decreases is higher in case of X-band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' At X-band, the flux-density starts rising from MJD 55666, but at S-band it starts rising from MJD 56666.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' Here the X-band flux density leads that of the S-band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' After MJD 58000, the X-band flux-density increases with a greater first derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' Radio source structure analysis A PREPRINT 7 This RFC does not have enough source maps to analyze the change in the source structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' It has two observations each year, so we cannot make any obvious conclusion amid these observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' But if we analyze the available source-maps at X-band after MJD 58000, we can find that the source exhibits jet-structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' This is in accordance to the rapid increase of flux-density in this period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='6 1349-439 (J1352-4412) Between 2001 and 2021, the source 1349-439 has only nine contour maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' We have calculated the mean flux-densities as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='20 Jy and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='25 Jy and flux variability indices as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='30 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='34 for S- and X-band respectively of the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' The variation of flux-density is much less about the mean for both S/X-bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' Both the light-curves mostly follow the same pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' Between MJD 51786 and 54570, the flux-density decreases and then they start increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' After MJD 57500, both the flux-densities rise, but the X-band density reaches a higher value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' We don’t have enough source-maps for this source too to analyze the source structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' The radio fundamental catalog has two source-maps in 2021 with a gap of two months, which shows that the source has minimal jet-structures and at MJD 59353, the source appears to be point-like at X-band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='7 1424-418 (J1427-4206) This source has mean flux-densities of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='60 Jy and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='77 Jy and flux variability indices 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='48 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='69 for S- and X-band respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' The light curves for both S/X-bands follow the same pattern, but the flux-density peaks reach higher values at X-band in comparison to S-band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' The highest peak of the flux-density at X-band has a value of 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='57 Jy, whereas at S-band it has a value of 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='77 Jy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' The X-band light-curve leads the S-band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' After MJD 52000, both the fluxes increase, the X-band flux-density shows a greater rate of increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' Between MJD 53621 and 54000, interestingly the X-band density drops, but that for the S-band density moves up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' After MJD 56000, both the flux-densities show rapid rise, where both of them reach their highest values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' The source has 54 contour maps between 1994 and 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' In the period from 1998 to 2003, we have seen the variation of flux-density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' Consequently, in this period, the S/X source maps show jet-structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' Moreover, between MJD 54600 and 55070 where, the flux-densities increases a little and between MJD 56650 and 56850, where the flux-density rises rapidly, all the source-maps in these periods show jet-structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' Also, after MJD 59000, the maps exhibit jets and the light curves show declination in flux-densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='8 1451-400 (J1454-4012) This source has mean fluxes of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='37 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='28 Jy, flux variability indices of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='43 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='39 at S- and X-band respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' This source has twenty-nine source-maps between 1999 and 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' Between MJD 53000 and 54000, the S-band flux- density increases, whereas the X-band flux-density decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' Thereafter, between MJD 54000 and MJD 56000, the behavior of the light-curves turns over, the S-band density falls and that of X-band rises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' At MJD 58000, both S/X flux- densities increase and the corresponding source maps show jet-like structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' Between MJD 51500 and MJD 53000, the source-maps show jet-structures, but the source-map at MJD 52766 shows that the source exhibits point-like structure at S-band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' At X-band, the source seems to be compact at MJD 51574.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' This source structure reappears at MJD 52290.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' At MJD 58000, both the S/X flux-densities rise and the corresponding source- maps show jet-structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='9 2052-474 (J2056-4714) The source has 27 contour-maps from 1999 to 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' We have calculated the mean flux-densities of this source as 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='60 Jy and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='11 Jy and flux variability indices as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='36 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='72 at S and X-band respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' Similar to the source discussed in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='7 (source 1424-418), the peaks in the light-curve also have higher values at X-band than the S-band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' The X-band light-curve also leads the S-band curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' Between MJD 51200 and 53000, the flux-densities fall, then they start increasing simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' Both the S/X plot reach their highest value at MJD 55440.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' Between 1999 and 2003, we have 17 maps which show jet-structures, except the maps at MJD 51938 and MJD 52038, where the source is compact for both S/X-band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' After 2003, the radio fundamental catalog does not have enough source-maps to identify the nature of the source structure precisely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' In accordance to the light-curves, between MJD 56000 and 57000, the flux-density falls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' In 2013, we have source-maps at two adjacent epochs (MJD 56497 and 56546) in which jet components evolve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' Radio source structure analysis A PREPRINT 8 − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='10 2106-413 (J2109-4110) This source has mean fluxes of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='59 Jy and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='39 Jy, flux variability indices 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='53 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='80 for S- and X-band respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' From the light curves, it is seen that between MJD 52000 and 56000, the flux-densities decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' This indicates that jet components may appear in the source structure, but unfortunately, we don’t have enough VLBI images to substantiate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' From MJD 56000 to 57000, the flux-densities rise and then after MJD 57000, they again decrease and the corresponding source-maps shows jet-structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='11 2333-415 (J2336-4115) The source 2333-415 has 13 contour-maps between 2012 and 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' it has mean fluxes of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='25 Jy and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='30 Jy and flux variability indices 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='29 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='40 at S/X-band respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' The S-band light-curve shows that the flux-density increases up to MJD 57000 and then decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' At X-band, however, the flux-density increases after MJD 57000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' Both the S/X- band source-maps appear to be jet-structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' 4 Summary and Conclusion We presented radio source structure analysis for 11 calibrator sources at the S- and X-band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' All the selected sources are below 40° declination and were observed in multi-epoch VLBI observing sessions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' To analyse the source structure variability, we mainly relied on the available source maps to detect any visible changes in the structure and also on the light-curves that were constructed using the GSFC fluxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' Firstly, we carefully went through source maps to detect any kind of variability in the source structure at mas scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' Then we used the light-curve to see the fluctuations in flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' Here, the available multi-epoch fluxes in the GSFC database were used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' Finally, we extracted total fluxes (the sum of all the CLEAN components in an RFC VLBI map) and plotted those along with the GSFC fluxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' This gives a clear perspective of whether the detected structure in VLBI maps (thus change in source flux) agrees with the flux variation detected in the GSFC flux-based light-curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' We also used a metric, flux variability index, to quantify the scale of the structural variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' In this epoch-based analysis of the sources at S/X-band, we found all the selected sources exhibit jet-like structures at some or all epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' However, based on the multi-epoch VLBI source maps, light-curves, and flux variability index analysis we analyzed the magnitude of source structure that can be used to quantify whether a source is suitable as a calibrator or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' Sources with extended jet structures, random fluctuations in the light curve, and higher flux variability index (>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='3) have been considered as not suitable for calibrators and should be observed for more analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' Therefore, at S-band, we found three sources to be suitable candidates as calibrators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' The rest of the sources may be used as calibrators, but we recommend more rigorous source structure analysis of these sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' At X-band, all the sources have flux variability index greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='3 which indicates that these sources have jet-like structures consistent over all the epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' There are six sources having flux variability index between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='3 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' The rest of the sources have flux variability index higher than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' The flux variability indices also agree with the light curve analysis and also jet-like structures in the source maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' Overall, we recommend all the selected sources to be monitored regularly to analyze their suitability as calibrators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' 5 Acknowledgements This work has made use of the Radio Fundamental Catalog database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' The work also has made use of NASA’s Astrophysics Data System Bibliographic Services and the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' References Beasley, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=', Gordon, D.' metadata={'source': 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+page_content=' The third realization of the International Celestial Reference Frame by very long baseline interferometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' , 644:A159.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' Fomalont, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=', Petrov, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=', MacMillan, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=', Gordon, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=', and Ma, 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=', Archinal, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=', and Charlot, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' The international celestial reference frame as realized by very long baseline interferometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' The Astronomical Journal, 116(1):516.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' Petrov, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' Vlba calibrator survey 9 (vcs-9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' arXiv preprint arXiv:1610.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content='04951.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' Petrov, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=', de Witt, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=', Sadler, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=', Phillips, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=', and Horiuchi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' The second lba calibrator survey of southern compact extragalactic radio sources–lcs2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' Monthly Notices of the Royal Astronomical Society, 485(1):88–101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' Petrov, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=', Kovalev, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=', Fomalont, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=', and Gordon, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' The third vlba calibrator survey: Vcs3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' The Astronomical Journal, 129(2):1163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' Petrov, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=', Kovalev, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=', Fomalont, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=', and Gordon, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' The fourth vlba calibrator survey: Vcs4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' The Astronomical Journal, 131(3):1872.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' Petrov, L.' metadata={'source': 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+page_content=' The Astronomical Journal, 136(2):580.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' Petrov, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=', Phillips, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=', Bertarini, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=', Murphy, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=', and Sadler, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} +page_content=' M.' metadata={'source': 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+page_content=' The Astrophysical Journal Letters, 907(1):L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE4T4oBgHgl3EQfpA0g/content/2301.05188v1.pdf'} diff --git a/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf b/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..172c8c3b9f0305bf77072f955b21d0c6e6d6cac9 --- /dev/null +++ b/cdFST4oBgHgl3EQfDjjb/content/2301.13711v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:ebfebf5a9d525f76d21e9aa16d2d75f46940ceb165e6b25d275cdf10a3250795 +size 2054095 diff --git a/cdFST4oBgHgl3EQfDjjb/vector_store/index.faiss b/cdFST4oBgHgl3EQfDjjb/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..fd5ebf6fbf5136ffd6af9d27182789cd154a778d --- /dev/null +++ b/cdFST4oBgHgl3EQfDjjb/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid 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sha256:6efbaa83cb5ff43e579dfc8f825041e15fc69d6787c606c3d6d9f255ea3ce4b2 +size 144792 diff --git a/hNE3T4oBgHgl3EQf4AvA/content/tmp_files/2301.04769v1.pdf.txt b/hNE3T4oBgHgl3EQf4AvA/content/tmp_files/2301.04769v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..739812811de3cd5f0c42cc451d3adc4fc975c381 --- /dev/null +++ b/hNE3T4oBgHgl3EQf4AvA/content/tmp_files/2301.04769v1.pdf.txt @@ -0,0 +1,1538 @@ +arXiv:2301.04769v1 [cond-mat.mes-hall] 12 Jan 2023 +Semiconductor thermal and electrical properties decoupled by +localized phonon resonances∗ +Bryan T. Spann†‡a, Joel C. Weber‡a, Matt D. Brubakera, Todd E. Harveya, Lina Yangb, +Hossein Honarvar§c, Chia-Nien Tsaic, Andrew C. Treglia¶d, M. Leed, Mahmoud I. +Hussein‖c,d, and Kris A. Bertness∗∗a +aPhysical Measurement Laboratory, National Institute of Standards and Technology +(NIST), Boulder, CO 80302 USA +bSchool of Aerospace Engineering, Beijing Institute of Technology, Beijing 100081, China +cAnn and H.J. Smead Department of Aerospace Engineering Sciences, University of +Colorado Boulder, Boulder, Colorado 80303, USA +dDepartment of Physics, University of Colorado Boulder, Boulder, Colorado 80302, USA +Abstract +Thermoelectric materials convert heat into electricity through thermally driven charge trans- +port in solids, or vice versa for cooling. To be competitive with conventional energy-generation +technologies, a thermoelectric material must possess the properties of both an electrical conduc- +tor and a thermal insulator. However, these properties are normally mutually exclusive because +of the interconnection of the scattering mechanisms for charge carriers and phonons. Recent +theoretical investigations on sub-device scales have revealed that silicon membranes covered by +nanopillars exhibit a multitude of local phonon resonances, spanning the full spectrum, that +couple with the heat-carrying phonons in the membrane and collectively cause a reduction in +the in-plane thermal conductivity−while, in principle, not affecting the electrical properties be- +cause the nanopillars are external to the pathway of voltage generation and charge transport. +Here this effect is demonstrated experimentally for the first time by investigating device-scale +suspended silicon membranes with GaN nanopillars grown on the surface. +The nanopillars +cause up to 21 % reduction in the thermal conductivity while the electrical conductivity and +the Seebeck coefficient remain unaffected, thus demonstrating an unprecedented decoupling in +the semiconductor’s thermoelectric properties. The measured thermal conductivity behavior for +coalesced nanopillars and corresponding lattice-dynamics calculations provide further evidence +that the reductions are mechanistically tied to the phonon resonances. This finding breaks a +longstanding trade-off between competing properties in thermoelectricity and paves the way for +engineered high-efficiency solid-state energy recovery and cooling. +∗Contribution of an agency of the U.S. government; not subject to copyright. +†Current affiliation: Lockheed Martin Space, Advanced Technology Center, Louisville, CO 80027 +‡Equal contributors +§Current affiliation: ConcertAI, Cambridge, MA 02138, USA +¶Current affiliation: Department of Physics, Colorado State University, Fort Collins CO 80523 +‖Corresponding author: Mahmoud I. Hussein (mih@colorado.edu) +∗∗Corresponding author: Kris A. Bertness (kris.bertness@nist.gov) +1 + +1 +Introduction +Thermoelectric (TE) materials enable electrical power generation, refrigeration, and heating, all in +the solid state. Since no moving mechanical components, fluid systems, or chemical reactions are +involved, TE devices provide good reliability, stability, and overall practicality [1]. On the other +hand, their low efficiency, around 3-6 % in commercial devices, is a significant obstacle that im- +pedes competitive wide-scale use as a replacement to traditional electrical power generation and +fluid-based refrigeration/heat pump technologies [2]. The performance of a TE material under a +temperature gradient is based on a well-defined figure of merit, ZT = [(S)2σ/k)]T, where S is +the Seebeck coefficient, σ is the electrical conductivity, k is the thermal conductivity, and T is the +average temperature between the hot and cold sides of the material. The main challenge facing +TE material performance is the tight coupling between these properties among nearly all classes of +inorganic and organic materials; in particular, there is an inherent trade-off between exhibiting a +low k while simultaneously possessing a high σ and a high S−a combination of attributes needed +for a significant increase in ZT. This trade-off has stood as a key limitation to TE technological +development and proliferation since the early days of discovery of the Seebeck [3] and Peltier [4] +effects close to two hundred years ago [5]. +Increase of ZT by thermal conductivity reduction is a widely pursued strategy. Central to this +path are phonon confinement [6] and the key scattering mechanisms available for impeding phonon +transport; these include phonon-phonon scattering (which increases with temperature) [7], bound- +ary scattering (such as rough boundaries) [8–10], and scattering by impurities and internal barri- +ers [11] (Fig. 1a,b). With the advent of nanotechnology, nanostructuring has enabled precise access +and control of the internal microstructure of existing materials, especially semiconductors. A pre- +vailing approach is the introduction of obstacles, such as holes, inclusions, and interfaces, within +the interior of the TE medium to enhance phonon scattering and reduce k [12, 13]. However, in +addition to scattering the phonons, the motion of charge carriers is likely to be impeded by the +same obstacles. While it is possible to tune the separation distances between scattering centers to +selectively scatter phonons with longer mean free paths (MFPs) and minimize electron/hole scat- +tering at shorter MFPs, the problem remains constrained because the allowable range of selective +scattering is limited by the inherent overlap in the intrinsic phonon and charge carrier MFP dis- +tributions of the material [14, 15]. This in turn negates the possibility of a true decoupling of the +phononic and electronic properties and subsequent realization of a substantial increase in ZT. +2 +Results and Discussion +Departing from this constraining trade-off strategy, here we demonstrate decoupling of the ther- +mal conductivity reduction from the remaining TE properties by mechanistic means. This is done +by forming a thin, suspended membrane of Si with a random arrangement of closely-packed GaN +nanopillars standing on its top surface, i.e., exterior to the membrane nominal cross-section. The +membrane thickness and nanopillar spacing are selected to fall within the range of the phonon +MFP distribution for Si, which is estimated to average at around 200-300 nm at room temper- +ature [16, 17]. Similarly, the nanopillar feature sizes, i.e., the height and width, are selected to +fall within the MFP distribution of GaN; ab initio calculations predict that most of the thermal +conductivity of GaN arises from phonons with MFPs greater than 200 nm [18]. The dominant +portion of the heat transported in this nanostructured material is carried by traveling phonons, +because generally the electronic contribution to the thermal conductivity in silicon is negligible +even with heavy doping [19, 20]. The atoms making up the nanopillars, on the other hand, gener- +2 + +Transport +L >Λ +k drop by phonon scattering +Base +membrane +L +Nanopillar +L +Λ +k drop by phonon-vibron couplings +L +L +Nanop +L +Λ +≤ +L +Λ +k drop by boundary scattering +L +drop by boundary scattering +Vibron: +Phonon: +Average +phonon MFP: +Λ +≤ +Electron: +L >Λ +k drop by obstacle scattering +L +Λ +k drop by obstacle scattering + drop by obstacle scattering +≤ +Average +electron MFP: +Λ +p +e +Phonon-vibron +coupling +Internal +scatterers: +a +c +p +p +p +p +p +Membrane +Membrane +Bulk +Bulk +NPM +b +No obstacles +Transport +Figure 1: Prime mechanisms of thermal conductivity reduction in a semiconductor for +TE conversion. (a) bulk and reduced-dimension configurations where the key factors for k reduc- +tion are phonon-phonon scattering (top), and phonon confinement and scattering off rough surfaces +(bottom), respectively; (b) corresponding configurations where optimized internal scattering obsta- +cles such as holes, inclusions, and interfaces dominate the scattering (contemporary approach); +(c) NPM configuration in the form of a nanopillared membrane where the prime mechanism of k +reduction is resonance hybridization and the resulting phonon group velocity reductions and mode +localizations (current approach). +ate vibrons, or wavenumber-independent phonon resonances. These two types of waves, the trav- +elling and the standing, couple (Fig. 1c) and cause a substantial portion of the energy of the +heat-carrying phonons to modally localize in the nanopillars. In addition, the coupling causes the +base-membrane phonon group velocities to drop significantly. These two effects lead to a reduction +in the lattice thermal conductivity along the membrane portion and form the basis of the notion +of a nanophononic metamaterial (NPM) [21–27]. This mechanism of phonon hybridizations and +resonance localizations−which, in principle, takes place across the full phonon spectrum−is inde- +pendent of the mechanisms of voltage generation and electrical charge transport and is therefore +not expected to affect the Seebeck coefficient or the electrical conductivity. +Previous theoretical investigations using molecular dynamics (MD) simulations have shown the +presence of phonon-vibron couplings [23] and predicted up to two orders of magnitude reduction in +the thermal conductivity [25]. However, these studies were done on model sizes on the order of 10-20 +nm for the base membrane thickness due to computational limitations. Small nanostructures are +more amenable to coherent wave effects; the key challenge is sustaining these effects at larger scales +closer to the average MFP [27–29]. Here, we demonstrate the first experimental evidence for both +the thermal conductivity reduction by nanoresonators (designated as the NPM effect [21]) and the +decoupling with the electrical properties, S and σ. Importantly, this demonstration is accomplished +with device-scale structures, with the smallest dimension being the membrane thickness of 200 +3 + +SFigure 2: Nanofabricated samples of GaN-on-Si NPMs and corresponding lattice dy- +namics properties (a) Schematic of the NPM unit cell, (b) SEM image of GaN nanopillars on a Si +membrane. (c) Optical microscope image of a suspended membrane, which appears lighter due to its +partial transparency in the visible spectrum. The nanopillars produced a textured appearance, and +(d) schematic of the Raman thermometry measurement geometry. (e) Conventional unit cell of Si; +primitive unit cell of GaN; (f) atomic displacements for a bare membrane mode indicating intense +motion (left) and for a corresponding NPM mode indicating localized motion in the nanopillars and +minimal motion in the base membrane (right). (g) Phonon band structure, and (h) group velocity +(left) and mode participation (right) distributions of Si membrane with (red) or without (blue) +GaN nanopillars standing on the surface. The resonance hybridization (phonon-vibron) coupling +phenomenon is illustrated in the circular inset in (g). +nm. The thermal conductivity along the base membranes decreases as the nanopillars increase in +height, consistent with NPM theory [25]. Electrical conductivity and Seebeck coefficient measure- +ments on the same structures show that the nanopillars do not degrade the electrical properties. We +also show that the behavior of the thermal conductivity for coalesced nanopillars provides evidence +4 + +0.2 +a +Wave vector, K +LD +0.8 +Frequency, w (THz) +0.6 +0.4 +0.2 +NPMmode +0 +0 +5 +10 +0 +0.5 +Group velocity, V. Mode participatic +nel g +(nm/ps) +ratio, P,= 0.5431 nm += 0.5186 nm +aA = 13.58 nm +a +a. =.0.3186 nm +0.2759 nm +a.Al= 5.43 nm +Membrane mode +0.3186 nm +Modes at center of +circled hybridization zone in patry +sport +Silicon +handler +s +nc Membrane +d Raman thermome +Side view +Top view +Suspended +Probe +米 +membrane +beam +Trans +UV +Transport +Oxide +heatin +layer +beam +750mm +g +LD +Lattice dynamic +a A=3.50 nm +aAby=3.59 nm +Membrane +0.8 +equency, w (THz) +NPM +33.19-265.52 nm +Hybridization +0.6 +zone +0.4a +Molecular +b +NPM +-50-130 nm +beam epitaxy +0.77-5.7 um +GaN +nanopillar +Transport +≥~200 nm +Si base +Transport +500 r +membrane +~80-160 nm +e +f +Atomic models +Si +Ga +Nthat the reductions are primarily due to phonon resonances and not boundary scattering. +The thermal conductivity test structures are illustrated in Fig. 2. The GaN nanopillars were +grown on silicon-on-insulator (SOI) substrates via plasma-assisted molecular-beam epitaxy (MBE), +see Fig. 2b. The GaN nanopillars formed spontaneously at high growth temperature and high N:Ga +flux ratio [30]. Specimen sets with varying nanopillar height were grown with the expectation that +taller, more massive nanopillars would produce more vibrons and therefore a greater reduction in +the thermal conductivity [25]. The samples are of two types, Set A in which GaN nanopillar growth +was initiated directly on the Si after a brief nitridation step, and Set B in which a 8-nm AlN +buffer was grown prior to nanopillar growth. As described in more detail in the Appendix, the sets +differ in their electrical conductivity variation with nanopillar height because of different degrees +of diffusion of Ga and Al into the membrane during high-temperature nanopillar growth. Set A +displays an increase in electrical conductivity as a function of nanopillar height, while Set B displays +approximately constant electrical conductivity. Suspended membranes were formed by etching from +the backside of the substrate to the buried oxide layer, then removing the oxide layer. The as- +purchased SOI device layer thickness of 200 nm thus becomes the final membrane thickness. We +note that SOI substrates with such thin device layers are only available with very light p-type +doping, and therefore the electrical conductivity of these structures is not optimal for high ZT. This +limitation is not fundamental and does not interfere with the novelty of the mechanism for thermal +conductivity reduction. The membranes were heated with a strongly absorbed ultraviolet (UV) +laser beam incident from the unpatterned lower side, and the specimen temperature was measured +at the center of the hot spot using a green laser beam incident from the top side (Fig. 2d). The +temperature was determined by the shift in frequency of the Si Raman peak appearing near 520 +cm−1 at room temperature. Raman thermometry is a non-contact technique that has been widely +used to measure the thermal conductivity of a variety of thin membranes [10, 31–34]. +Following the development given in Ref. [31] for bare (unpillared) membranes, the lateral +thermal transport is governed by a radial heat equation with a source heating term. We find +∆T(r) = T(r) − Tamb = Pabsln(r/R)β(r)/(2πdiki) , where Pabs is the absorbed power from the +heating laser with beam radius r0, r is the radial distance from the center of the laser spot, Tamb +is the ambient temperature, R is the radius of the membrane (to the boundary where it attaches +to the silicon wafer), and di and ki are the effective conductive thickness and thermal conductivity, +respectively, where i represents either a bare membrane “Mem” or a nanopillared-covered membrane +“NPM”. As described in the Appendix, the radial temperature variation is small within the probe +beam diameter, and thus the measured temperature difference relative to Tamb can be equated to +∆T(0), for which ln(r/R)β(r) becomes ln(R/r0) + γ/2, where γ is the Euler constant = 0.57721 +to five significant digits. +In this study, we are primarily interested in the effects of the added +nanopillars on the surface of the Si membranes. In order to single out the nanopillar effects on +the thermal conductivity, we differentiate the previous equation with respect to the absorbed laser +power and take the ratio of this differential expression for the specimens with nanopillars and the +specimens with bare membranes, i.e., +∂∆TMem/∂Pabs +∂∆TNPM/∂Pabs += kNPMdNPM +kMemdMem +. +(1) +In our measurements, the power of the 325 nm beam was varied and the slope of the tempera- +ture versus absorbed power was used to derive the relative ratio of the thermal conductivities. We +convert the relative changes in k to estimates of absolute thermal conductivity by multiplying a +typical thermal conductivity of 200-nm thick Si membranes, 60 W/m·K [34, 35], by the ratio of +the inverse slope of each sample to the average value of the inverse slope for membranes without +nanopillars, 0.0251 mW/K. Although dNPM is greater on average than dMem because of the pres- +5 + +1 +2 +3 +4 +5 +6 +0 +1 +2 +3 +4 +5 +6 +0 +40 +50 +60 +70 +80 +1600 +800 +400 +1200 +0 +Thermal conductivity, k (W/m K) +Seebeck coefficient, S ( V/K) +Normlized figure of merit, ZT* +Electrical conductivity, �� (S/m) +0 +1 +2 +3 +4 +0 +100 +200 +300 +400 +500 +Without buffer layer (Set A) +With AlN buffer layer (Set B) +Without buffer layer (Set A) +With AlN buffer layer (Set B) +Without buffer layer (Set A) +With AlN buffer layer (Set B) +With AlN buffer layer +a +b +c +d +10 +30 +50 +70 +90 +0 +100 +200 +300 +Nanopillar height, h (nm) +MD Simulations +Thermal conductivity, k (W/m K) +. +Excluded due +to coalescence +Excluded due +to coalescence +1 +2 +3 +4 +5 +6 +0 +1 +2 +3 +4 +5 +6 +0 +Nanopillar height, h ( m) +� +Nanopillar height, h ( m) +� +� +� +Nanopillar height, h ( m) +� +Nanopillar height, h ( m) +� +. +Figure 3: Measurements of TE properties of GaN-on-Si NPMs with varying nanopillar +height (a) Thermal conductivity, (b) electrical conductivity, (c) Seebeck coefficient, and (d) ZT ∗ +figure of merit normalized with respect to bare membrane value. In (a), thermal conductivity +predictions by MD simulations for smaller (by a factor of ∼15) but proportionally-sized models are +shown in green; arrows point to relevant axes. The AlN buffer layer (Set B) minimized diffusion +of GaN into the Si membrane that dominated electrical properties in Set A. Data points circled in +blue represent samples with coalesced nanopillars and were excluded from the curve fittings. Solid +(dashed) curves represent phenomenon influenced (uninfluenced) by the nanopillar vibrons. +ence of the nanopillars, we make the assumption that these two thicknesses are equal and cancel in +Eq. (1). This assumption tends to underestimate the thermal conductivity reduction by the NPM +effect. In the SI, we discuss how surface roughness variation and heat loss to the environment are +not consequential in our experiments. +As can be seen in Fig. 3a, the thermal conductivity for the specimens displays a significant re- +duction as the height of the nanopillars increases, with a maximum reduction of 21 % ± 0.4 %. The +source of this reduction is explained by examining the phonon band structure of the NPM unit +cell. For our models, we consider a representative unit cell with a Si base width of 85 nm and +thickness of 200 nm, supporting a GaN nanopillar with a square cross-section, a width of 55 nm, +and a height targeted to vary from 0.5 to 4 µm. A corresponding atomic model was created with +all dimensions ∼15 times smaller for feasible computation (see Figs. 2e,f and Methods). As shown +in Fig. 2g, the nanopillars fundamentally transform the membrane band structure by adding a +population of localized modes that appear as horizontal lines spanning the Brillouin zone; these +represent the resonance/vibron modes that couple with the underlying membrane phonon disper- +sion modes throughout the spectrum (the NPM effect). The localizations manifest physically as +illustrated in the atomic motion close-up inserts in Fig. 2f. The outcome is strong reductions in +the phonon group velocities vg and their mode participation ratios pr which quantify the extent +6 + +Figure 4: Nanopillar coalescence: Evidence of NPM effect. SEM images of specimen with +(a) least coalescence and (b) greatest coalescence, both in tilt view 45◦. Top views of (a) and (b) +are shown in (c) and (d), respectively. (e) Plot of average tip area versus nanopillar height showing +that coalescence increased with nanopillar height. Atomic model of unit cell (f) without coalescence +and (g) with coalescence (base membrane brown, nanopillar purple), and corresponding (h) phonon +band structure and group-velocity distribution. The average group velocity for NPM normalized +with respect to corresponding bare membrane is shown to increase by 53 % with coalescence. +of mode localization in the NPM unit cell; see definitions in Methods. These two factors directly +contribute to reducing the in-plane thermal conductivity [25]. Equilibrium MD simulations were +also conducted on the same atomic-scale NPM model, followed by application of the Green-Kubo +method, producing a trend similar to the experimental trend of a reduction in k with nanopillar +height (see green curve in Fig. 3a and Methods). The MD results indicate a reduction of nearly 92 +%, which is higher than the experimental reduction because of the smaller features sizes compared +to the phonon MFP distributions of Si and GaN. This similarity in trends shows that the NPM +effect describes the data we observe experimentally. +Unlike strategies of introducing defects that also slow electronic carrier transport, we see no +negative impact on the electrical conductivity of the specimens (Fig. 3b), while both sets display +similar reductions in k. As Figs. 3b and 3c show, the σ and S values for Set B are unaffected by +the presence of increasingly taller nanopillars, while k is reduced for all but the severely coalesced +specimen.The low value of σ, around 200 S/m, is due to the low doping in the samples (see Meth- +ods and Appendix). This data rules out the possibility of scattering-induced reductions in carrier +7 + +Gr= 0.15 +G. +=023 +Gr +X +0 +2 +4 +6 +ctor, K +Group velocity, V. (nm/ps)Average tip area, Atip (μm* +E +0.05 +B- Without buffer layer (Set A) +0.3 +m +O- With AIN buffer layer (Set B +0.04 +requency, +0.25 +0.2 +0.03 +0.15 +0.02 +0.1 +0.01 +0.05 +0田 +0 +0 +1 +2 +3 +4 +5 +6 +Nanopillar height, h (μum) +Wave veTheory: LD + Not coalesced +g +Coalesced +.5558 nm +tm +Cross section A-A +16.5952 nm +3.2586 nm +um +3.5046 nm +h +5.431 nm +0.5 +NPM +0.45 +(not coalesced + NPM +0.4 +(coalesced) +N +0.35 +- MernbraneExperiment: SEM +Not coalesced +h = 0.77 um Coalesced +h = 5.7 +a +200 nm +nm +Not coalesced +h = 0.77 μm Coalesced +h = 5.7 +d +200 nm +200 nm +0.06 +Biiffemobility or density from the presence of the nanopillar forest. +As explained previously, Set A +shows an increase in electrical conductivity that we attribute to coincidental Ga diffusion and not +to improvement in mobility. The Seebeck coefficients for Set A show the typical decrease as carrier +concentrations increase [19, 36, 37]. Thus we have clearly shown that the thermal properties and +electrical properties of the nanopillared membranes have been decoupled. +Under ideal circumstances, theory predicts that having larger nanopillars attached to the Si +membranes should reduce the thermal conductivity by increasing the number of vibrons available +for coupling with the base-membrane phonons. As can be seen in Fig. 3a, the initial decreases +in k with increasing nanopillar height reverse themselves for Set B, with the NPM effect extin- +guished at a nanopillar height of 5.7 µm. This reversal is explained by an unavoidable coalescence +of neighboring nanopillars as the nanopillar height increases. We observe that the coalescence occurs +predominantly near the tips rather than at the roots. A comparison of two extreme cases is given +in Figs. 4a-d. We quantify the coalescence by calculating an average tip area using standard image +analysis techniques; the complete image set is available in the Appendix. The tip areas plotted +in Fig. 4e show that most of the specimens in this study display some degree of coalescence, and +the effect is significantly (∼ 3×) stronger for the tallest nanopillars in Set B. The nullification of +the observed NPM thermal conductivity reduction by coalescence is also seen in quasiharmonic +lattice dynamics calculations, as shown in Figs. 4f-i. The phonon band structure shows that vibron +states (horizontal black lines) move to higher frequencies when the nanopillars touch at the tips +and thus reduce the NPM effect at the lower frequency regime which is dominant in the thermal +transport [21]. Furthermore, an increase in the average group velocities across the spectrum is ob- +served due to having less isolated nanoresonators. These changes cause an increase in k relative +to nanopillars with unconnected tips, which provides further proof that the thermal conductivity +reduction is due to the NPM effect and not scattering of phonons from the nanopillar roots. More +broadly, the results offer an experimental demonstration of the role of wave effects in thermal +transport in nanostructures with feature sizes on the order of a few hundred nanometers, at room +temperature. This finding establishes a unique analogy with acoustics, given that the introduction +of substructures to induce intrinsic local resonances has been widely utilized in the form of acoustic +metamaterials [38]; here the concept is experimentally realized−for the first time−at the nanoscale +for influencing the thermal conductivity. +3 +Conclusions +The ultimate target of decoupling TE properties is to enable a route for increasing ZT. In Fig. 3d, +we see that the NPM effect has increased the relative ZT by a factor of 2.7, raising the absolute value +from 0.42×10−3 for the bare membrane to 1.12×10−3. The theory predicts that significantly larger +enhancements are possible in more ideal specimens with larger ratio of nanopillar-to-membrane +volume [25, 26]. Our results demonstrate that these gains are obtained by the NPM effect without +degradation in the electrical properties of membranes. By increasing doping in the base membrane, +the numerator in the ZT expression will also increase to provide significant additional gains in +the ZT absolute value. Furthermore, these results have been demonstrated in base membranes +with robust dimensions and in a material that is technologically advanced and inexpensive. The +enhancement through nanostructure-induced resonances would apply to other semiconductors as +well, including common TE materials [39], provided the phonon MFP distribution has significant +overlap with the nanostructure features. Together these results point to a long-sought solution to +the problem of maximizing TE material performance by breaking the coupling between the thermal +8 + +and electrical properties. +4 +Methods Section +Nanopillar synthesis +MBE growth: GaN nanopillars were grown by catalyst-free MBE with a plasma-assisted nitrogen +source onto the Si(100) device layer prior to membrane etching and release. The SOI substrates +(SEH America∗) had device, buried oxide, and carrier layer thicknesses of 200 nm, 380 nm, and +675 µm, respectively. The device layer was lightly boron doped with a resistivity of 28 Ω-cm; +as noted above, these thin device layers are not currently available in any other doping types or +concentrations. The nanopillars initially cover the entire surface of the substrate but were selectively +removed with photolithography for the electrical test structures [40]. Nanopillar height was varied +by adjusting the nanopillar growth period, with the longest growth period being 12 h. The ratio of +the N equivalent growth rate to the Ga equivalent growth rate during nanopillar growth was 6:1 +for the Set A and 3:1 for Set B. The nanopillars were grown at approximately 810 ◦C. More details +are provided in the Appendix. +Sample fabrication +After nanopillar growth, each 2 cm × 2 cm chip was fabricated into a testing platform to measure +its thermoelectric properties. Each completed chip yields 2 four-point electrical resistivity devices, +2 Seebeck coefficient devices, and 92 thermal conductivity test membranes ranging in nominal size +from 400 µm × 400 µm to 700 µm × 700 µm. Ohmic contact pads were formed using 20 nm Ti/200 +nm Al metal stacks annealed in argon at 500 ◦C for 1 minute. +Thermoelectric metrology +Raman thermometry: We used a 325-nm He-Cd laser as a heating source that was propagating +anti-parallel to a low intensity 532-nm laser used as a Raman probe. The nanopillared Si mem- +branes were positioned such that the side with nanopillars was exposed to the low intensity 532-nm +Raman probe, while the 325-nm beam was absorbed on the unpatterned side of the membrane. This +optical alignment allowed for more accurate estimation of absorbed laser power due to the ∼60-nm +absorption depth at the 325-nm wavelength, precluding transmission to the nanopillars on the op- +posite side of the membrane. The beam diameters at the 1/e2 points were 25 µm and 0.8 µm for +the 325-nm and 532-nm lasers, respectively. The nanopillars are transparent to the green probe +beam though some scattering occurred as the beam passed through them. The reflectance R of +the bottom side of the membranes was measured to be 0.57 for the UV beam, and the absorbed +beam power was calculated as the incident beam power multiplied by (1 − R). The beam power +was measured with an optical power meter close to where it impinged on the specimen and then +corrected for transmission of the intervening optics. The temperature dependence of the Si Raman +peak was calibrated by heating a Si chip with a strip heater and measuring its temperature with a +thermocouple while acquiring Raman data. The resulting data was fit with the quadratic equation +T(◦C) = 23.2 − 50.4(∆ν − 1.1(∆ν)2) where ∆ν is the temperature-induced shift in the Raman +peak position in wavenumbers (cm−1). The linear term of this equation agrees well with previous +∗Vendor is identified to adequately specify the source material. This identification does not imply recommendation +or endorsement by the National Institute of Standards and Technology, nor does it imply that the product identified +is necessarily the best available for the purpose. +9 + +evaluations that report ∆T/∆ν = −46 K/cm−1, initially by the work of Mendez and Cardona and +verified by others including Reparaz et al. [41, 42]. +Seebeck coefficient: The Seebeck coefficient measurement was performed via a steady-state method +with the geometry shown in the Appendix, Fig. A4. Two meandering Ti/Al wires were lithographi- +cally defined 10 µm from the Si device layer to serve as thermometers with a ∼100-Ω resistance. Prior +to measurement, both resistors R1 and R2 were calibrated to within 0.1 K. An additional pair of +Ti/Al wires was patterned in direct contact with either end of the Si device layer to measure the +Seebeck voltage. Two 1-kΩ chip resistors, serving as heaters, were glued to one end of the chip and +used to provide a thermal gradient along the length of Si device layer. The heaters provided up to +25 mW of power yielding a maximum ∆T of 3.5 K. The heater current, thermopower voltage Vth, +temperatures at R1 and R2, and temperature gradient across the Si device layer ∆T were recorded +as a function of time with initial sample temperature at 277 K. All calibrations and measurements +were performed in ice water to maintain a constant bath temperature. +Electrical resistivity: The electrical resistivity was measured using a standard four-point probe test +structure shown in the Appendix, Fig. A4. The quantity ∆V across the two inner contacts was +measured as a function of current across the two outer contacts over the range of 0 nA−100 nA. All +tested devices showed a linear, ohmic response, allowing for resistivity ρ to be calculated from the +membrane width w, thickness t, length L, and measured resistance R as ρ = Rwt/L. +Atomic models +The theoretical investigations are based on atomic models comprising a Si membrane with GaN +nanopillars standing on the surface. Both material portions were modeled as single crystals under +room-temperature equilibrium conditions. The Tersoff potential was used for the interatomic in- +teractions. The parameters of the Si-Si and Ga-N interactions were taken from Refs. [43] and [44], +respectively. For the Si-Ga and Si-N interactions, the potential parameters were mixed following +the Tersoff multicomponent combination rules [45]. Two sizes of NPMs were investigated: one that +is nearly 15 times smaller than a nominal experimental unit cell (shown in Fig. 2f, right), and a +smaller version for the coalescence investigation (shown in Fig. 4g). In the model of the coalesced +NPM, the top of the nanopillar was laterally extended to partially connect with adjacent nanopil- +lars. This was done by adding three primitive-cell layers of GaN around the tip of the nanopillar +forming a cross-like cross section when viewed from the top (cut view A-A in Fig. 4g). +Lattice dynamics calculations: The phonon band structures for the examined GaN-on-Si NPM +unit cells were obtained by solving the quasiharmonic lattice dynamics eigenvalue problem using +the GULP software [46]. Bloch periodic boundary conditions were applied along in-plane directions +and free boundary conditions were applied in the z direction and around the nanopillar. The phonon +frequencies were computed at a set of allowed wave vectors ranging from Γ to X in the Brillouin +zone with a resolution of 128 points. There are 3N phonon branches in the band structure, where +N is the total number of atoms in the unit cell. +The average group velocity ratio Gr is a quantity that characterizes the reduction in the +group velocities across the entire phonon spectrum [25]. It is defined as Gr = GNPM /GMem, +where Gi is the average group velocity of either an NPM or a membrane calculated by Gi = +(1/nκnm) �nκ +κ +�nm +m vg(κ, m). Here, κ is the wave number (scalar component of the wave vector +κ along the Γ−X direction), m is the branch number, nκ is the number of wave-number points +considered, and nm = 3N is the total number of phonon branches. The group velocity vg(κ, m) is +10 + +defined as the slope of the phonon frequency with respect to the wave number κ for branch m. +For characterization of nanopillar resonant mode localization, we examine the mode shape cor- +responding to each point in the phonon band structure. We then compute the mode participation +ratio pr, which is defined for a mode at wave vector κ and branch number m by Ref. [22, 25] +pr(κ, m) = +1 +N �N +i=1[�3 +j=1 φ∗ +ij(κ, m)φij(κ, m)]2 , +(2) +where φij(κ, m) is the displacement component corresponding to atom i and direction j of the +normalized mode shape. The formula comprises two summations. The first is over the total number +of atoms N in a unit cell, i.e., N = NBase + NPillar for an NPM, where NBase is the number of +atoms in the base membrane and NPillar is the number of atoms in the nanopillar. The second +summation is over the three directions of motion per atom. The inverse of this quantity pr indi- +cates the degree of modal localization over the entire unit cell considered without being specific +to a particular region, e.g., the nanopillar or base membrane portion of an NPM unit cell. This +calculation is performed for both an NPM and a bare membrane. In an NPM, a large number of the +modes exhibit high concentrations of vibrations in the nanopillar portion, yielding a low value of pr. +Molecular dynamics simulations: Equilibrium molecular dynamics (EMD) simulations were exe- +cuted to predict the in-plane lattice thermal conductivity of the GaN-on-Si NPMs sized at nearly +1/15 of the nominal experimental unit cell, with the height of the nanopillar being varied (see +atomic model dimensions in Fig. 2). A single unit cell was used as the simulation cell with periodic +boundary conditions applied along the x and y directions and free boundary condition applied in +z direction and around the nanopillar. The empirical interatomic potentials were identical to those +used in the LD calculations. The time integration step was set as ∆t = 0.5 fs. First, a canonical en- +semble MD with a Langevin heat reservoir was allowed to run for 0.3 ns to enable the whole system +to reach equilibrium at 300 K. Then, a microcanonical ensemble (NVE) was run for 3 ns; mean- +while, the heat current was recorded at each time step. At the end of the simulations, the thermal +conductivity was calculated by the Green-Kubo formula, [47] k = 1/(2V kBT 2) +� ∞ +0 ⟨J(τ) � J(τ)⟩dτ +where kB is the Boltzmann constant, V is the system material volume, and J is the heat flux along +the direction of transport. Finally, the thermal conductivity was averaged over the two in-plane +directions over six simulations with different initial velocities (i.e., a total of 12 cases), and the +statistical errors were obtained according to the method described in [47]. All EMD simulations +were performed in LAMMPS [48]. +Acknowledgments +This research was partially supported by the Advanced Research Projects Agency−Energy (ARPA- +E) under grant number DE-AR0001056. +* +APPENDIX +This supplemental information document contains tables of data from the figures used in the +main article, further details of the methods and results, including a full set of SEM images of +the specimens used in the study and examples of the Raman temperature vs. absorbed power +curves, discussion of possible additional heat transport mechanisms, and discussion of the Al and +11 + +Ga diffusion into the membrane. Commercial equipment and instruments are identified in order to +adequately specify certain procedures. In no case does such identification imply recommendation +or endorsement by the National Institute of Standards and Technology, nor does it imply that the +products identified are necessarily the best available for the purpose. +A +Tabular data +This section contains details of specimen synthesis and morphology and data from Fig. 3 in the +main article. +Table A1: Specimen characteristics for Sets A and B. +Run No. +Nanopillar +Growth +kNPM/kMem +height (µm) +time (h) +Bare (A) +NA +NA +0.99 ± 0.06 +D420 (A) +0.324 +1.7 +1.03 ± 0.05 +D421 (A) +0.785 +3.2 +0.98 ± 0.04 +D422 (A) +1.950 +6.2 +0.85 ± 0.02 +D423 (A) +3.300 +12.2 +0.79 ± 0.01 +D442 (B) +NA +0.083 +1.01 ± 0.01 +D469 (B) +0.77 +2.6 +0.93 ± 0.04 +D480 (B) +1.430 +2.6 +0.80 ± 0.02 +D443 (B) +2.140 +5.1 +0.87 ± 0.01 +D481 (B) +5.700 +10.1 +0.98 ± 0.01 +Table A2: Nanopillar dimensions for Sets A and B. +Run No. +Nanopillar +Root diam. +Fill +Avg. tip +height (µm) +(nm) +fraction +area (µm2) +D420 (A) +0.324 ± 0.035 +30 ± 4 +0.33 +0.0029 +D421 (A) +0.785 ± 0.130 +40 ± 11 +0.25 +0.0041 +D422 (A) +1.950 ± 0.200 +50 ± 11 +0.38 +0.010 +D423 (A) +3.300 ± 0.320 +75 ± 22 +0.35 +0.016 +D469 (B) +0.77 ± 0.050 +65 ± 14 +0.39 +0.0064 +D480 (B) +1.430 ± 0.080 +70 ± 25 +0.52 +0.019 +D443 (B) +2.140 ± 0.040 +60 ± 12 +0.54 +0.023 +D481 (B) +5.700 ± 0.140 +130 ± 36 +0.39 +0.060 +B +Nanopillar synthesis +SOI substrates (SEH America) were diced into 2 cm by 2 cm squares prior to growth and process- +ing. Immediately prior to loading for growth, the substrates were cleaned with solvents, oxygen +plasma in a reactive ion etching system, and approximately two-minute exposure to hydrogen +12 + +Table A3: Thermal conductivity, electrical conductivity, and Seebeck coefficient data from Fig. 3 of +the main article. The thermal conductivity values were obtained by multiplying the relative values +in Table A1 by 60 W/m · K, as discussed in the text. +Run No. +Nanopillar +Thermal cond. +Electrical cond. +Seebeck coef. +height (µm) +kNPM(W/m · K) +σ (S/m) +(µV/K) +Bare (A) +NA +59.6 ± 3.4 +3.6 ± 0.5 +1500 ± 12 +D420 (A) +0.324 +61.9 ± 2.8 +2.8 ± 0.2 +955 ± 47 +D421 (A) +0.785 +58.5 ± 2.6 +41.4 ± 5 +800 ± 8 +D422 (A) +1.950 +50.9 ± 1.0 +103 ± 30 +730 ± 5 +D423 (A) +3.300 +47.2 ± 0.5 +233 ± 30 +610 ± 1.6 +D442 (B) +NA +60.4 ± 0.6 +170 ± 20 +711 ± 1 +D469 (B) +0.77 +55.6 ± 2.2 +no data +no data +D480 (B) +1.430 +47.7 ± 1.0 +150 ± 30 +676 ± 2 +D443 (B) +2.140 +52.4 ± 0.7 +320 ± 50 +788 ± 1 +D481 (B) +5.700 +58.8 ± 0.8 +220 ± 50 +719 ± 3 +fluoride (HF) vapor. The substrates were outgassed three times at successively higher tempera- +tures in the MBE loadlock chamber, preparation chamber, and growth chamber, with the final +outgas reaching between 870 ◦C and 890 ◦C. The Si surface initially displayed a 1×1 reflection +high energy electron diffraction (RHEED) pattern that brightened and sharpened as surface oxides +desorbed. Around 720 ◦C, the 1×1 pattern changed into a 2×1 pattern. Substrate temperatures +throughout the growth were measured with an estimated uncertainty of 8 ◦C using a back-side py- +rometer described elsewhere [49]. Element fluxes were estimated from separate growth calibration +runs in which the growth rate of planar films was derived from optical interference fringes at a +variety of different Ga and Al beam equivalent pressures under group-III-limited conditions. The +nitrogen-limited growth rate for a variety of plasma conditions was estimated from the transition +from a spotty to streaky RHEED pattern as the gallium flux was increased [50]. For specimen Set +A (no buffer layer), the Ga and N equivalent planar growth rates were 110 ± 10 nm/h and 650 ± 30 +nm/h, respectively. For these runs, the Si surface was exposed to the N plasma for 60 s at 740 ◦C, +then nanopillar growth was initiated at a low temperature of 700 ◦C for 12 minutes. The remainder +of the nanopillar growth took place at 810 ◦C. For specimen Set B (AlN buffer layer), the Ga and +N equivalent planar growth rates were 250 ± 20 nm/h and 650 ± 30 nm/h, respectively. Buffer +layer deposition was preceded by a 15 s N plasma exposure and a 1 s Al exposure. The buffer layer +itself was deposited at an actual growth rate of 165 ± 20 nm/h with the N flux lower than the Al +flux to promote formation of a N-polar surface. The growth temperature for the buffer layer was +850 ◦C, where we estimate that approximately half of the Al reevaporated rather than be incorpo- +rated. GaN nanopillar growth was initiated on the AlN buffer layers at 770 ◦C for 5 minutes before +increasing to 810 ◦C for the main nanopillar growth stage. The GaN nanopillars grown without a +buffer layer were not intentionally doped, leading to an n-type carrier concentration below 1 × 1016 +cm−3. The GaN nanopillars on AlN were lightly doped with Si, and thus had an n-type carrier +concentration in the 1017 cm−3 range. +As shown in the main article, the nanopillars tended to get thicker along their length and coalesce +near the tips, and this coalescence increased with nanopillar height. The average tip area is an +approximate indicator of the degree of coalescence. For the specimen Set B (with AlN buffer layers), +13 + +a h = 0.32 �m +b h = 0.77 �m +c h = 0.79 �m +d h = 1.4 �m +e h = 1.95 �m +f h = 2.1 �m +g h = 3.3 �m +h h = 5.7 �m +200 nm +300 nm +300 nm +400 nm +500 nm +500 nm +1000 nm +2000 nm +Figure A1: SEM images of specimens in this study taken from a tilt angle of 45◦. Images in the +left column portray the specimen set grown without a buffer layer (Set A), and the right column +portrays the specimen set grown with an AlN buffer layer (Set B). Run numbers are (a) D420, (b) +D469, (c) D421, (d) D480, (e) D422, (f) D443, (g) D423, (h) D481. +the coalescence occurred faster as a function of height. This change is most likely due to the higher +Ga flux during nanopillar growth for this series. The flux was increased to increase the fill fraction +of the nanopillars, defined as the ratio of the total tip area in the top view images relative to the +corresponding image area, but also increased coalescence at the tips. As can be seen in Table A2 +and Fig. A1, in general the density of the nanopillars at their roots is more similar than the tip +area from run to run within a set. SEM images for the both specimen sets are given in Figs. A1 +and A2. +14 + +a h = 0.32 �m +b h = 0.77 �m +c h = 0.79 �m +d h = 1.4 �m +e h = 1.95 �m +f h = 2.1 �m +g h = 3.3 �m +h h = 5.7 �m +200 nm +320 nm +200 nm +200 nm +200 nm +200 nm +200 nm +200 nm +Figure A2: SEM images of specimens in this study viewed from the top. Images in the left column +portray the specimen Set A, and the right column portrays the specimen Set B. Run numbers are +(a) D420, (b) D469, (c) D421, (d) D480, (e) D422, (f) D443, (g) D423, (h) D481. +C +Specimen fabrication +The fabrication steps are shown in Fig. A3. Photolithography is used to protect the nanopillar +forest with a 7 µm thick resist coating while exposed nanopillars are removed via sonication in +de-ionized water. Next, the Si device layer is patterned and etched down to the buried oxide layer +with a 5-cycle deep reactive-ion etch (DRIE) using SF6. The patterns for the electrical conductivity +and Seebeck coefficient measurements are in Fig. A4a and A4b, respectively. A 30 s CF4 dry etch +is used to remove any native oxide on the Si before a 20 nm Ti/ 200 nm Al metal stack is deposited +to form probe contacts and thermometers. Deposition is followed by a 1 min 500 ◦C anneal in +argon to ensure ohmic contacts. The top-side of the chip is then wax bonded to a sapphire carrier +15 + +GaN nanopillar +Si membrane +Si handle +SiO2 +Ti/Al +a +b +c +d +e +Figure A3: Fabrication flow diagram. +a +b +Figure A4: Electrical patterns for measuring (a) electrical conductivity, and (b) Seebeck coefficient. +wafer to protect the nanopillars and Si device layer during back-side processing. The membranes for +thermal conductivity measurements are lithographically patterned and the Si handle layer etched +16 + +Heater +Heater +R1 +/+ +Si +3000 μm +V- +V+ +V +th +500 μm +R2 ++ +Sio +V- +V+Si +V+ +5000 μm +SiO2 +V- +500μmFigure A5: Raman temperature measurement schematic. +with a 900 cycle DRIE followed by a 15 min HF vapor phase etch to remove the buried oxide. The +carrier wafer is then suspended upside down in a beaker of acetone to dissolve the wax and release +the completed test devices. +D +Raman thermometry +A schematic of the Raman thermometry experiment is shown in Fig. A5. The setup consists of +a Coherent Verdi V6 532 nm CW laser that is attenuated with neutral density (ND) filters to +an average power of ∼ 300 µW at the sample. An infinity-corrected 40× Nikon objective with a +numerical aperture of 0.60 was employed to focus the 532 nm beam on the samples with a spot +diameter of ∼0.8 µm. The light is then collected in a backscattering geometry and sent through a +Raman filter (Semrock RazorEdge Filter) and then into a 0.5 m spectrometer (Acton Spectra Pro +2500i) that disperses the Stokes-shifted Raman light with an 1800 groove/mm grating onto a liquid +nitrogen cooled CCD array, providing sub cm−1 energy resolution. In order to heat the samples +for thermometry studies, we used a 325 nm He-Cd CW laser (Kimmon) and a variety of ND filter +combinations to control incident laser fluence of the pump beam. The pump beam is collinear and +17 + +Spectrometer and +CCD Array +Raman Filter +Beam Splitter +532 nm Laser +Imaging CCD +ND Filter +White Light Source +.Flip Beam Splitter +Beam Splitter +325 nm Laser +40x Objective +Sample +Lens +Variable +ND Filterpropagating antiparallel to the 532 nm Raman probe beam. The 325 nm laser beam is focused onto +the membranes with a 2.0-cm focal length lens mounted on a micrometer stage. Alignment of the +two beams was tested by observing the Raman peak shift while making minor adjustments to the +325-nm lens position. The spot diameter, defined at 1/e2 intensity locations, was measured to be +∼25 µm at the focal point by passing a knife edge through the beam and fitting the transmitted +intensity to an error function. The pump beam average incident power was varied between 0 to 8.5 +mW, resulting in fluences ranging from ∼0 to 3.5 kW/cm2. A flip-mounted pellicle beam splitter +was used in conjunction with white light illumination source to image the sample and verify the +Raman probe was proper centered on suspended membrane prior to each thermometry experiment. +In the main text, we utilize the following radial temperature profile (see Eq. 12 in Ref. [31]): +∆T = T(r) − Tamb = Pabsln(r/R)β(r)/(2πdk), +(A1) +where Pabs is the absorbed power from the 325-nm laser, Tamb is the ambient temperature, R is +the radius of the membrane, d is the effective conductive thickness, and k is the membrane thermal +conductivity. We evaluate some of the features of this profile to confirm our assumption that the +probe beam is sampling a region with approximately uniform temperature. We have made the +assumption that the edge of the membrane is at ambient temperature for large membranes like +those we use here, with R on the order of 375 µm. The function β is defined as +β(r) = 1 + Ei(−r2/r2 +0) − Ei(−R2/r2 +0) +2ln(R/r) +, +(A2) +where R is the radius of the membrane, and Ei is the exponential integral function [51]. The radius +r0 of the heating laser beam is defined for an intensity distribution I = (P/πr2 +0)exp(−r2/r2 +0). Note +that this definition of r0 omits the conventional factor of 2 in the argument of the exponential for a +Gaussian beam intensity distribution, and thus our measured beam diameter of 25 µm corresponds +to r0 = 8.8 µm. Ei(x) becomes negligible for large negative arguments, and hence for R >> r0, as +is the case here, the second term in the numerator can be omitted. The behavior of Ei near r = 0 +can be evaluated using a Taylor series expansion for negative real arguments [52]: +Ei(x) = γ + ln|x| + x + +x2 +2 · 2! + +x3 +3 · 3!... +(A3) +Evaluating the temperature model numerically, we find that the temperature within the radius of +the probe beam, 400 nm, is uniform within 0.02 K for 325-nm spot diameters ranging from 25 +to 71 µm and uniform within 0.1 K for a smaller spot diameter of 10 µm (see Fig. A6). We can +also evaluate the limiting behavior of the radial portion of the temperature profile near r = 0 by +substituting the first two terms of the Taylor expansion as follows: +ln(R/r)β(r) = ln(R/r) +� +1 + γ + ln| − r2/r2 +0| +2ln(R/r) +� += ln(R/r) +�2ln(R/r) + γ + 2ln(r/r0) +2ln(R/r) +� += γ/2 + ln(R/r) + ln(r/r0) += γ/2 + ln(R/r0) +(A4) +This temperature model can also be used to estimate the inverse slope of ∆T versus Pabs eval- +uated at the center of the 325-nm beam, which we measured to be 0.025 mW/K in bare mem- +branes. As shown in Table A4, the measured value is reproduced by a combination of a membrane +18 + +Distance from center (nm), r +�T(0)-�T(r) (K) +100 +200 +300 +400 +0 +-0.10 +-0.08 +-0.06 +-0.04 +-0.02 +0.00 +k = 80, 60 W/m K. +, +, +, +r0 = 25 �m +r0 = 8.8 �m +r0 = 3.5 �m +Figure A6: Radial variation of ∆T relative to its value at r = 0 as a function of radius for +representative values of membrane thermal conductivity k and 325 nm beam radii r0. +thermal conductivity of 60 W/m·K and a beam diameter of 71 µm, or a membrane thermal conduc- +tivity of 80 W/m·K and a beam diameter of 25 µm. The difficulty in measuring the beam diameter +accurately and aligning the two beams perfectly translates into difficulty in determining the ab- +solute thermal conductivity of the membrane; hence we primarily use the Raman thermometry +method for determining relative changes in the membrane thermal conductivity. +Table A4: Temperature model outputs for representative membrane and beam parameters. ∆T is +evaluated at r = 0 and Pabs = 1 mW. +r0 +Beam +k +k +(µm) +diam. +60 W/m·K +80 W/m·K +(µm) +Inv. Slope +∆T (K) +Inv. Slope +∆T (K) +3.54 +10 +0.015 +65.7 +0.020 +49.3 +8.8 +25 +0.019 +53.5 +0.025 +40.1 +25 +71 +0.025 +39.7 +0.034 +29.8 +Examples of the Raman thermometry data are shown in Fig. A7, where we plot Raman-derived +temperature at the center of the heated spot as a function of absorbed laser power for representative +specimens with nanopillars and their corresponding control membranes without nanopillars. The +Raman thermometry data show predominantly linear temperature increases for absorbed powers +up to ∼2.0 mW. In order to determine the ratio of thermal conductivities described in Eq. (1), +we have carried out linear fits to the Raman thermometry data. For highest accuracy, we fit data +points in the range of 40 ◦C to 120 ◦C. We exclude lower temperatures because the Raman peak +shifts are small and the corresponding uncertainty is high for these points. For higher temperatures, +the response starts to become nonlinear due to the decrease in the thermal conductivity of the Si +membrane with increasing temperature. Data for multiple membranes from the same run were fit +individually and then their slopes b were averaged using a weighting factor of σb/b, where σb is the +19 + +10 +20 +30 +40 +0 +Growth time, tg (ks) +Electrical conductivity, � (S/m) +0 +50 +100 +150 +200 +250 +� +Figure A7: Examples of nanopillared and bare membrane temperature in the center of the heated +spot as a function of absorbed UV laser power. (a) Set A and (b) Set B. +standard deviation of the set of measured slopes. The resulting ratios are shown in Table A1 with +corresponding uncertainties. As portrayed in the table, kNPM/kMem values are predominantly less +than unity with a minimum value of 0.79, i.e., a 21 % reduction in the thermal conductivity. We +note here that run number D442 was a control sample with only an AlN buffer layer on the Si +membrane. It exhibits a thermal conductivity that is nearly identical to the pristine membrane. As +discussed in the main article, partial coalescence of the nanopillars reduces the NPM effect and +leads to higher thermal conductivities. +E +Possible additional heat transport mechanisms +The presence of the nanopillar roots on the membranes may be viewed as a source of surface +roughness, and thus might be thought to increase surface/boundary scattering of phonons and +thereby reduce the thermal conductivity, similar to investigations of rough Si nanowires [8, 9, 14] +and rough Si membranes [10]. However, the average width of the nanopillar roots among the various +fabrication runs in our investigation is 65 nm (see Table A2). This is significantly larger than the +range of phonon wavelengths (∼5 to ∼60 angstroms for Si at room temperature [16]), which is +the size scale relevant for boundary scattering due to roughness [10, 14]. Furthermore, if boundary +scattering of any form is dominant†, the thermal conductivity would not increase with an increase +in nanopillar coalescence occurring predominantly at the tips; yet as noted in the main article we +observe a significant rise in k in the severely coalesced case of 5.7 µm tall nanopillars (see Fig. 3a). +Secondly, it is conceivable that a portion of the heat might take a path through the coalesced +nanopillars which would effectively serve as a parallel “branch” for the thermal transport for those +cases, resulting in a larger overall area through which the heat-carrying phonons travel. While this +is unlikely because the phonons will scatter at the coalesced nanopillar tips, the increased overall +†For example, in Ref. [53] the presence of nanopillars was interpreted as a source of increased boundary scattering. +20 + +cross-sectional area implies lower k for the same power transmitted−thus improving the reduction +rather than deteriorating it as we observe for the 5.7 µm tall nanopillars in Fig. 3. +Furthermore, we note that any transport of heat to the environment rather than being delivered +to the substrate at the edges of either type of membrane (i.e., base membrane in an NPM or bare +membrane) would result in an overestimation of the integrated thermal power P flowing through +the membrane, which we assumed to be equal to Pabs. One example of such transport would be +heat loss through the small but finite thermal conductivity of the surrounding air. We here show +that any such mechanism results in an overestimation of the true thermal conductivity ktrue of the +membrane. Postulating that this mechanism results in P = ηPabs, where η is a number less than +1, and defining D = 2πd/(γ/2 + ln(R/r0)), we can combine Eqs. A1 and A4 and solve for ktrue as +ktrue = 1 +D +� ∂∆T(0) +∂(ηPabs) +�−1 += 1 +D +�1 +η +∂∆T(0) +∂Pabs +�−1 += η 1 +D +�∂∆T(0) +∂Pabs +�−1 += ηkcalc, +(A5) +where kcalc is the value we would calculate for k using our original data analysis method. Because +η is less than 1, the true value ktrue would be less than our calculated value kcalc. +In taking the ratio of kNPM/kMem, any heat conduction through the surrounding air would +affect both the NPM membrane and the bare membrane, and therefore the heat loss would cancel +in taking this ratio, provided the nanopillars do not affect local interactions with the air. Although +the nanopillars bear a superficial resemblance to heat sink structures in which fins are used to +conduct heat away from an object and increase the surface area subject to air cooling, the spacing +between the nanopillars on our membranes is smaller than or comparable to the mean free path of air +molecules at atmospheric pressure, around 65 nm [54]. This tight spacing precludes the development +of natural convective or conductive regions in the gaps between nanopillars or any effective increase +in membrane surface area by the nanopillars. Instead, the high thermal conductivity along the +axis of the nanopillars (compared with air) effectively moves the membrane/air interface to the +plane defined by the tips of the nanopillars. However, even if the nanopillars were to increase air +cooling to some small degree, the η value for NPM membranes would then be smaller than that +for the bare membranes, and thus the true ratio of thermal conductivities would be smaller than +our measured ratio, i.e., we would be underestimating the NPM effect. This calculation can also +be understood qualitatively in that any loss of heat to the environment reduces the temperature +difference generated across the path of electrical current for a given thermal power input, and +therefore lowers the thermoelectric performance of a structure. +F +Dopant diffusion into the Si membrane +Al and Ga are both p-type dopants in silicon, and will diffuse into to the thin Si membrane at +high temperature, producing changes in membrane carrier concentration and electrical conductivity +that depend on time at high temperature and available dopant concentrations. For the specimen set +without buffer layers, Set A, Ga flux is continually arriving at the Si membrane surface between the +roots of the nanopillars. From the measured electrical conductivity values, the Ga concentration +in the membrane reaches a maximum value of 4×1016 cm−3 for the specimen with the longest +growth time, D423. We infer the carrier concentration from typical relationships between electrical +conductivity and carrier concentration for Si [55]. As shown in Fig. A8, the increase in σ is linearly +dependent on growth time. This dependence implies that the Ga concentration and hence the hole +concentration is a function of the total flux impinging on the top surface during NPM growth. The +21 + +a +b +0.5 +1.0 +1.5 +2.0 +2.5 +0 +Adsorbed power, P (mW) +0.5 +1.0 +1.5 +2.0 +2.5 +0 +Adsorbed power, P (mW) +Temperature, T (oC) +20 +40 +60 +80 +100 +200 +300 +Temperature, T (oC) +20 +40 +60 +80 +100 +200 +300 +D480: various membranes +D442: buffer only +Average slope for +each set +D423: various membranes +Various bare membranes +Average slope for +each set +Figure A8: Electrical conductivity of specimen Set A showing a linear increase as a function of +nanopillar growth time. The change is due to diffusion of Ga into the Si membrane. +maximum carrier concentration corresponds to a Ga concentration in the membrane of less than 1 +ppm. Figure A8 also illustrates that there was significant spatial variation in the measured electrical +conductivity. For most of the runs, two areas on the die were sampled for the electrical conductivity, +and although each measurement had low uncertainty, the spatial variations are on the order of 50 +% to 150 % for the specimens with the highest σ. The actual Ga flux to the wafer is likely much +more uniform that this, but the local environment of randomly growing nanopillars would lead to +large variations in shadowing of the surface by the nanopillars. +In order to avoid the complication of diffusion-induced electrical conductivity changes within +a specimen set, Set B employed an 8 nm thick AlN buffer layer as a diffusion blocking layer. AlN +has been used to block diffusion of Mg in GaN/AlGaN structures [56] and of Al in TiN contact +layers [57]. As shown in the main article, this very thin buffer layer achieved our goal of removing the +dependence of electrical conductivity on growth time. The electrical conductivity for the specimen +with the AlN buffer layer only (and no nanopillars) is still significantly higher than that measured for +a pristine Si membrane, around 170 S/m vs. 4 S/m, and we attribute this increase to Al diffusion +during the buffer layer growth. Al diffusion out of AlN buffer layers is a contributing factor to +conductive loss in RF devices grown on GaN-on-Si substrates [58]. Al diffuses more readily in Si +than Ga [59–61], but Al flux was only present during the AlN buffer layer growth, lasting 180 +s. The AlN buffer did appear to be effective as a barrier to subsequent Ga diffusion based on +the conductivity of the specimen with the longest growth time, D481. 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Phys., vol. 10, no. 4, p. 434, 1971. +26 + diff --git a/hNE3T4oBgHgl3EQf4AvA/content/tmp_files/load_file.txt b/hNE3T4oBgHgl3EQf4AvA/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..243ef9af75cde098917ab51c6bc5dae5a11cdd30 --- /dev/null +++ b/hNE3T4oBgHgl3EQf4AvA/content/tmp_files/load_file.txt @@ -0,0 +1,1434 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf,len=1433 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='04769v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='mes-hall] 12 Jan 2023 Semiconductor thermal and electrical properties decoupled by localized phonon resonances∗ Bryan T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Spann†‡a, Joel C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Weber‡a, Matt D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Brubakera, Todd E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Harveya, Lina Yangb, Hossein Honarvar§c, Chia-Nien Tsaic, Andrew C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Treglia¶d, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Leed, Mahmoud I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Hussein‖c,d, and Kris A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Bertness∗∗a aPhysical Measurement Laboratory, National Institute of Standards and Technology (NIST), Boulder, CO 80302 USA bSchool of Aerospace Engineering, Beijing Institute of Technology, Beijing 100081, China cAnn and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Smead Department of Aerospace Engineering Sciences, University of Colorado Boulder, Boulder, Colorado 80303, USA dDepartment of Physics, University of Colorado Boulder, Boulder, Colorado 80302, USA Abstract Thermoelectric materials convert heat into electricity through thermally driven charge trans- port in solids, or vice versa for cooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' To be competitive with conventional energy-generation technologies, a thermoelectric material must possess the properties of both an electrical conduc- tor and a thermal insulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' However, these properties are normally mutually exclusive because of the interconnection of the scattering mechanisms for charge carriers and phonons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Recent theoretical investigations on sub-device scales have revealed that silicon membranes covered by nanopillars exhibit a multitude of local phonon resonances, spanning the full spectrum, that couple with the heat-carrying phonons in the membrane and collectively cause a reduction in the in-plane thermal conductivity−while, in principle, not affecting the electrical properties be- cause the nanopillars are external to the pathway of voltage generation and charge transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Here this effect is demonstrated experimentally for the first time by investigating device-scale suspended silicon membranes with GaN nanopillars grown on the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' The nanopillars cause up to 21 % reduction in the thermal conductivity while the electrical conductivity and the Seebeck coefficient remain unaffected, thus demonstrating an unprecedented decoupling in the semiconductor’s thermoelectric properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' The measured thermal conductivity behavior for coalesced nanopillars and corresponding lattice-dynamics calculations provide further evidence that the reductions are mechanistically tied to the phonon resonances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' This finding breaks a longstanding trade-off between competing properties in thermoelectricity and paves the way for engineered high-efficiency solid-state energy recovery and cooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' ∗Contribution of an agency of the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' government;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' not subject to copyright.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' †Current affiliation: Lockheed Martin Space, Advanced Technology Center, Louisville, CO 80027 ‡Equal contributors §Current affiliation: ConcertAI, Cambridge, MA 02138, USA ¶Current affiliation: Department of Physics, Colorado State University, Fort Collins CO 80523 ‖Corresponding author: Mahmoud I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Hussein (mih@colorado.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='edu) ∗∗Corresponding author: Kris A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Bertness (kris.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='bertness@nist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='gov) 1 1 Introduction Thermoelectric (TE) materials enable electrical power generation, refrigeration, and heating, all in the solid state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Since no moving mechanical components, fluid systems, or chemical reactions are involved, TE devices provide good reliability, stability, and overall practicality [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' On the other hand, their low efficiency, around 3-6 % in commercial devices, is a significant obstacle that im- pedes competitive wide-scale use as a replacement to traditional electrical power generation and fluid-based refrigeration/heat pump technologies [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' The performance of a TE material under a temperature gradient is based on a well-defined figure of merit, ZT = [(S)2σ/k)]T, where S is the Seebeck coefficient, σ is the electrical conductivity, k is the thermal conductivity, and T is the average temperature between the hot and cold sides of the material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' The main challenge facing TE material performance is the tight coupling between these properties among nearly all classes of inorganic and organic materials;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' in particular, there is an inherent trade-off between exhibiting a low k while simultaneously possessing a high σ and a high S−a combination of attributes needed for a significant increase in ZT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' This trade-off has stood as a key limitation to TE technological development and proliferation since the early days of discovery of the Seebeck [3] and Peltier [4] effects close to two hundred years ago [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Increase of ZT by thermal conductivity reduction is a widely pursued strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Central to this path are phonon confinement [6] and the key scattering mechanisms available for impeding phonon transport;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' these include phonon-phonon scattering (which increases with temperature) [7], bound- ary scattering (such as rough boundaries) [8–10], and scattering by impurities and internal barri- ers [11] (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' 1a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' With the advent of nanotechnology, nanostructuring has enabled precise access and control of the internal microstructure of existing materials, especially semiconductors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' A pre- vailing approach is the introduction of obstacles, such as holes, inclusions, and interfaces, within the interior of the TE medium to enhance phonon scattering and reduce k [12, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' However, in addition to scattering the phonons, the motion of charge carriers is likely to be impeded by the same obstacles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' While it is possible to tune the separation distances between scattering centers to selectively scatter phonons with longer mean free paths (MFPs) and minimize electron/hole scat- tering at shorter MFPs, the problem remains constrained because the allowable range of selective scattering is limited by the inherent overlap in the intrinsic phonon and charge carrier MFP dis- tributions of the material [14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' This in turn negates the possibility of a true decoupling of the phononic and electronic properties and subsequent realization of a substantial increase in ZT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' 2 Results and Discussion Departing from this constraining trade-off strategy, here we demonstrate decoupling of the ther- mal conductivity reduction from the remaining TE properties by mechanistic means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' This is done by forming a thin, suspended membrane of Si with a random arrangement of closely-packed GaN nanopillars standing on its top surface, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=', exterior to the membrane nominal cross-section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' The membrane thickness and nanopillar spacing are selected to fall within the range of the phonon MFP distribution for Si, which is estimated to average at around 200-300 nm at room temper- ature [16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Similarly, the nanopillar feature sizes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=', the height and width, are selected to fall within the MFP distribution of GaN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' ab initio calculations predict that most of the thermal conductivity of GaN arises from phonons with MFPs greater than 200 nm [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' The dominant portion of the heat transported in this nanostructured material is carried by traveling phonons, because generally the electronic contribution to the thermal conductivity in silicon is negligible even with heavy doping [19, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' The atoms making up the nanopillars,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' on the other hand,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' gener- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='Transport ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='L >Λ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='k drop by phonon scattering ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='Base ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='membrane ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='L ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='Nanopillar ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='L ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='Λ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='k drop by phonon-vibron couplings ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='L ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='L ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='Nanop ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='L ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='Λ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='≤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='L ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='Λ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='k drop by boundary scattering ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='L ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='drop by boundary scattering ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='Vibron: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='Phonon: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='Average ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='phonon MFP: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='Λ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='≤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='Electron: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='L >Λ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='k drop by obstacle scattering ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='L ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='Λ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='k drop by obstacle scattering ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='drop by obstacle scattering ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='≤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='Average ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='electron MFP: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='Λ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='e ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='Phonon-vibron ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='coupling ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='Internal ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='scatterers: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='c ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='Membrane ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='Membrane ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='Bulk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='Bulk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='NPM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='No obstacles ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='Transport ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='Figure 1: Prime mechanisms of thermal conductivity reduction in a semiconductor for ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='TE conversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' (a) bulk and reduced-dimension configurations where the key factors for k reduc- tion are phonon-phonon scattering (top), and phonon confinement and scattering off rough surfaces (bottom), respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' (b) corresponding configurations where optimized internal scattering obsta- cles such as holes, inclusions, and interfaces dominate the scattering (contemporary approach);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' (c) NPM configuration in the form of a nanopillared membrane where the prime mechanism of k reduction is resonance hybridization and the resulting phonon group velocity reductions and mode localizations (current approach).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' ate vibrons, or wavenumber-independent phonon resonances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' These two types of waves, the trav- elling and the standing, couple (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' 1c) and cause a substantial portion of the energy of the heat-carrying phonons to modally localize in the nanopillars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' In addition, the coupling causes the base-membrane phonon group velocities to drop significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' These two effects lead to a reduction in the lattice thermal conductivity along the membrane portion and form the basis of the notion of a nanophononic metamaterial (NPM) [21–27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' This mechanism of phonon hybridizations and resonance localizations−which, in principle, takes place across the full phonon spectrum−is inde- pendent of the mechanisms of voltage generation and electrical charge transport and is therefore not expected to affect the Seebeck coefficient or the electrical conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Previous theoretical investigations using molecular dynamics (MD) simulations have shown the presence of phonon-vibron couplings [23] and predicted up to two orders of magnitude reduction in the thermal conductivity [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' However, these studies were done on model sizes on the order of 10-20 nm for the base membrane thickness due to computational limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Small nanostructures are more amenable to coherent wave effects;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' the key challenge is sustaining these effects at larger scales closer to the average MFP [27–29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Here, we demonstrate the first experimental evidence for both the thermal conductivity reduction by nanoresonators (designated as the NPM effect [21]) and the decoupling with the electrical properties, S and σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Importantly, this demonstration is accomplished with device-scale structures, with the smallest dimension being the membrane thickness of 200 3 SFigure 2: Nanofabricated samples of GaN-on-Si NPMs and corresponding lattice dy- namics properties (a) Schematic of the NPM unit cell, (b) SEM image of GaN nanopillars on a Si membrane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' (c) Optical microscope image of a suspended membrane, which appears lighter due to its partial transparency in the visible spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' The nanopillars produced a textured appearance, and (d) schematic of the Raman thermometry measurement geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' (e) Conventional unit cell of Si;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' primitive unit cell of GaN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' (f) atomic displacements for a bare membrane mode indicating intense motion (left) and for a corresponding NPM mode indicating localized motion in the nanopillars and minimal motion in the base membrane (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' (g) Phonon band structure, and (h) group velocity (left) and mode participation (right) distributions of Si membrane with (red) or without (blue) GaN nanopillars standing on the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' The resonance hybridization (phonon-vibron) coupling phenomenon is illustrated in the circular inset in (g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' The thermal conductivity along the base membranes decreases as the nanopillars increase in height, consistent with NPM theory [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Electrical conductivity and Seebeck coefficient measure- ments on the same structures show that the nanopillars do not degrade the electrical properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' We also show that the behavior of the thermal conductivity for coalesced nanopillars provides evidence 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='2 a Wave vector, K LD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='8 Frequency, w (THz) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='2 NPMmode 0 0 5 10 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='5 Group velocity, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Mode participatic nel g (nm/ps) ratio, P,= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='5431 nm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='5186 nm aA = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='58 nm a a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' =.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='3186 nm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='2759 nm a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='Al= 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='43 nm Membrane mode 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='3186 nm Modes at center of circled hybridization zone in patry sport Silicon handler s nc Membrane d Raman thermome Side view Top view Suspended Probe 米 membrane beam Trans UV Transport Oxide heatin layer beam 750mm g LD Lattice dynamic a A=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='50 nm aAby=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='59 nm Membrane 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='8 equency, w (THz) NPM 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='19-265.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='52 nm Hybridization 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='6 zone 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='4a Molecular b NPM 50-130 nm beam epitaxy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='77-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='7 um GaN nanopillar Transport ≥~200 nm Si base Transport 500 r membrane ~80-160 nm e f Atomic models Si Ga Nthat the reductions are primarily due to phonon resonances and not boundary scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' The thermal conductivity test structures are illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' The GaN nanopillars were grown on silicon-on-insulator (SOI) substrates via plasma-assisted molecular-beam epitaxy (MBE), see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' 2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' The GaN nanopillars formed spontaneously at high growth temperature and high N:Ga flux ratio [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Specimen sets with varying nanopillar height were grown with the expectation that taller, more massive nanopillars would produce more vibrons and therefore a greater reduction in the thermal conductivity [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' The samples are of two types, Set A in which GaN nanopillar growth was initiated directly on the Si after a brief nitridation step, and Set B in which a 8-nm AlN buffer was grown prior to nanopillar growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' As described in more detail in the Appendix, the sets differ in their electrical conductivity variation with nanopillar height because of different degrees of diffusion of Ga and Al into the membrane during high-temperature nanopillar growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Set A displays an increase in electrical conductivity as a function of nanopillar height, while Set B displays approximately constant electrical conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Suspended membranes were formed by etching from the backside of the substrate to the buried oxide layer, then removing the oxide layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' The as- purchased SOI device layer thickness of 200 nm thus becomes the final membrane thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' We note that SOI substrates with such thin device layers are only available with very light p-type doping, and therefore the electrical conductivity of these structures is not optimal for high ZT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' This limitation is not fundamental and does not interfere with the novelty of the mechanism for thermal conductivity reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' The membranes were heated with a strongly absorbed ultraviolet (UV) laser beam incident from the unpatterned lower side, and the specimen temperature was measured at the center of the hot spot using a green laser beam incident from the top side (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' 2d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' The temperature was determined by the shift in frequency of the Si Raman peak appearing near 520 cm−1 at room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Raman thermometry is a non-contact technique that has been widely used to measure the thermal conductivity of a variety of thin membranes [10, 31–34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Following the development given in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' [31] for bare (unpillared) membranes, the lateral thermal transport is governed by a radial heat equation with a source heating term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' We find ∆T(r) = T(r) − Tamb = Pabsln(r/R)β(r)/(2πdiki) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' where Pabs is the absorbed power from the heating laser with beam radius r0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' r is the radial distance from the center of the laser spot,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Tamb is the ambient temperature,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' R is the radius of the membrane (to the boundary where it attaches to the silicon wafer),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' and di and ki are the effective conductive thickness and thermal conductivity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' where i represents either a bare membrane “Mem” or a nanopillared-covered membrane “NPM”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' As described in the Appendix, the radial temperature variation is small within the probe beam diameter, and thus the measured temperature difference relative to Tamb can be equated to ∆T(0), for which ln(r/R)β(r) becomes ln(R/r0) + γ/2, where γ is the Euler constant = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='57721 to five significant digits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' In this study, we are primarily interested in the effects of the added nanopillars on the surface of the Si membranes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' In order to single out the nanopillar effects on the thermal conductivity, we differentiate the previous equation with respect to the absorbed laser power and take the ratio of this differential expression for the specimens with nanopillars and the specimens with bare membranes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=', ∂∆TMem/∂Pabs ∂∆TNPM/∂Pabs = kNPMdNPM kMemdMem .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' (1) In our measurements, the power of the 325 nm beam was varied and the slope of the tempera- ture versus absorbed power was used to derive the relative ratio of the thermal conductivities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' We convert the relative changes in k to estimates of absolute thermal conductivity by multiplying a typical thermal conductivity of 200-nm thick Si membranes, 60 W/m·K [34, 35], by the ratio of the inverse slope of each sample to the average value of the inverse slope for membranes without nanopillars, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='0251 mW/K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Although dNPM is greater on average than dMem because of the pres- 5 1 2 3 4 5 6 0 1 2 3 4 5 6 0 40 50 60 70 80 1600 800 400 1200 0 Thermal conductivity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' k (W/m K) Seebeck coefficient,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' S ( V/K) Normlized figure of merit,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' ZT* Electrical conductivity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' �� (S/m) 0 1 2 3 4 0 100 200 300 400 500 Without buffer layer (Set A) With AlN buffer layer (Set B) Without buffer layer (Set A) With AlN buffer layer (Set B) Without buffer layer (Set A) With AlN buffer layer (Set B) With AlN buffer layer a b c d 10 30 50 70 90 0 100 200 300 Nanopillar height,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' h (nm) MD Simulations Thermal conductivity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' k (W/m K) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Excluded due to coalescence Excluded due to coalescence 1 2 3 4 5 6 0 1 2 3 4 5 6 0 Nanopillar height, h ( m) � Nanopillar height, h ( m) � � � Nanopillar height, h ( m) � Nanopillar height, h ( m) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Figure 3: Measurements of TE properties of GaN-on-Si NPMs with varying nanopillar height (a) Thermal conductivity, (b) electrical conductivity, (c) Seebeck coefficient, and (d) ZT ∗ figure of merit normalized with respect to bare membrane value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' In (a), thermal conductivity predictions by MD simulations for smaller (by a factor of ∼15) but proportionally-sized models are shown in green;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' arrows point to relevant axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' The AlN buffer layer (Set B) minimized diffusion of GaN into the Si membrane that dominated electrical properties in Set A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Data points circled in blue represent samples with coalesced nanopillars and were excluded from the curve fittings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Solid (dashed) curves represent phenomenon influenced (uninfluenced) by the nanopillar vibrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' ence of the nanopillars, we make the assumption that these two thicknesses are equal and cancel in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' This assumption tends to underestimate the thermal conductivity reduction by the NPM effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' In the SI, we discuss how surface roughness variation and heat loss to the environment are not consequential in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' As can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' 3a, the thermal conductivity for the specimens displays a significant re- duction as the height of the nanopillars increases, with a maximum reduction of 21 % ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='4 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' The source of this reduction is explained by examining the phonon band structure of the NPM unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' For our models, we consider a representative unit cell with a Si base width of 85 nm and thickness of 200 nm, supporting a GaN nanopillar with a square cross-section, a width of 55 nm, and a height targeted to vary from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='5 to 4 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' A corresponding atomic model was created with all dimensions ∼15 times smaller for feasible computation (see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' 2e,f and Methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' 2g, the nanopillars fundamentally transform the membrane band structure by adding a population of localized modes that appear as horizontal lines spanning the Brillouin zone;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' these represent the resonance/vibron modes that couple with the underlying membrane phonon disper- sion modes throughout the spectrum (the NPM effect).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' The localizations manifest physically as illustrated in the atomic motion close-up inserts in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' 2f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' The outcome is strong reductions in the phonon group velocities vg and their mode participation ratios pr which quantify the extent 6 Figure 4: Nanopillar coalescence: Evidence of NPM effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' SEM images of specimen with (a) least coalescence and (b) greatest coalescence, both in tilt view 45◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Top views of (a) and (b) are shown in (c) and (d), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' (e) Plot of average tip area versus nanopillar height showing that coalescence increased with nanopillar height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Atomic model of unit cell (f) without coalescence and (g) with coalescence (base membrane brown, nanopillar purple), and corresponding (h) phonon band structure and group-velocity distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' The average group velocity for NPM normalized with respect to corresponding bare membrane is shown to increase by 53 % with coalescence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' of mode localization in the NPM unit cell;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' see definitions in Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' These two factors directly contribute to reducing the in-plane thermal conductivity [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Equilibrium MD simulations were also conducted on the same atomic-scale NPM model, followed by application of the Green-Kubo method, producing a trend similar to the experimental trend of a reduction in k with nanopillar height (see green curve in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' 3a and Methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' The MD results indicate a reduction of nearly 92 %, which is higher than the experimental reduction because of the smaller features sizes compared to the phonon MFP distributions of Si and GaN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' This similarity in trends shows that the NPM effect describes the data we observe experimentally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Unlike strategies of introducing defects that also slow electronic carrier transport, we see no negative impact on the electrical conductivity of the specimens (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' 3b), while both sets display similar reductions in k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' As Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' 3b and 3c show, the σ and S values for Set B are unaffected by the presence of increasingly taller nanopillars, while k is reduced for all but the severely coalesced specimen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='The low value of σ, around 200 S/m, is due to the low doping in the samples (see Meth- ods and Appendix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' This data rules out the possibility of scattering-induced reductions in carrier 7 Gr= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='15 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' =023 Gr X 0 2 4 6 ctor, K Group velocity, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' (nm/ps)Average tip area, Atip (μm* E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='05 B- Without buffer layer (Set A) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='3 m O- With AIN buffer layer (Set B 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='04 requency, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='05 0田 0 0 1 2 3 4 5 6 Nanopillar height, h (μum) Wave veTheory: LD Not coalesced g Coalesced .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='5558 nm tm Cross section A-A 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='5952 nm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='2586 nm um 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='5046 nm h 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='431 nm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='5 NPM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='45 (not coalesced NPM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='4 (coalesced) N 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='35 MernbraneExperiment: SEM Not coalesced h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='77 um Coalesced h = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='7 a 200 nm nm Not coalesced h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='77 μm Coalesced h = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='7 d 200 nm 200 nm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='06 Biiffemobility or density from the presence of the nanopillar forest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' As explained previously, Set A shows an increase in electrical conductivity that we attribute to coincidental Ga diffusion and not to improvement in mobility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' The Seebeck coefficients for Set A show the typical decrease as carrier concentrations increase [19, 36, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Thus we have clearly shown that the thermal properties and electrical properties of the nanopillared membranes have been decoupled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Under ideal circumstances, theory predicts that having larger nanopillars attached to the Si membranes should reduce the thermal conductivity by increasing the number of vibrons available for coupling with the base-membrane phonons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' As can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' 3a, the initial decreases in k with increasing nanopillar height reverse themselves for Set B, with the NPM effect extin- guished at a nanopillar height of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='7 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' This reversal is explained by an unavoidable coalescence of neighboring nanopillars as the nanopillar height increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' We observe that the coalescence occurs predominantly near the tips rather than at the roots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' A comparison of two extreme cases is given in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' 4a-d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' We quantify the coalescence by calculating an average tip area using standard image analysis techniques;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' the complete image set is available in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' The tip areas plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' 4e show that most of the specimens in this study display some degree of coalescence, and the effect is significantly (∼ 3×) stronger for the tallest nanopillars in Set B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' The nullification of the observed NPM thermal conductivity reduction by coalescence is also seen in quasiharmonic lattice dynamics calculations, as shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' 4f-i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' The phonon band structure shows that vibron states (horizontal black lines) move to higher frequencies when the nanopillars touch at the tips and thus reduce the NPM effect at the lower frequency regime which is dominant in the thermal transport [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Furthermore, an increase in the average group velocities across the spectrum is ob- served due to having less isolated nanoresonators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' These changes cause an increase in k relative to nanopillars with unconnected tips, which provides further proof that the thermal conductivity reduction is due to the NPM effect and not scattering of phonons from the nanopillar roots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' More broadly, the results offer an experimental demonstration of the role of wave effects in thermal transport in nanostructures with feature sizes on the order of a few hundred nanometers, at room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' This finding establishes a unique analogy with acoustics, given that the introduction of substructures to induce intrinsic local resonances has been widely utilized in the form of acoustic metamaterials [38];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' here the concept is experimentally realized−for the first time−at the nanoscale for influencing the thermal conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' 3 Conclusions The ultimate target of decoupling TE properties is to enable a route for increasing ZT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' 3d, we see that the NPM effect has increased the relative ZT by a factor of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='7, raising the absolute value from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='42×10−3 for the bare membrane to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='12×10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' The theory predicts that significantly larger enhancements are possible in more ideal specimens with larger ratio of nanopillar-to-membrane volume [25, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Our results demonstrate that these gains are obtained by the NPM effect without degradation in the electrical properties of membranes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' By increasing doping in the base membrane, the numerator in the ZT expression will also increase to provide significant additional gains in the ZT absolute value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Furthermore, these results have been demonstrated in base membranes with robust dimensions and in a material that is technologically advanced and inexpensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' The enhancement through nanostructure-induced resonances would apply to other semiconductors as well, including common TE materials [39], provided the phonon MFP distribution has significant overlap with the nanostructure features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Together these results point to a long-sought solution to the problem of maximizing TE material performance by breaking the coupling between the thermal 8 and electrical properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' 4 Methods Section Nanopillar synthesis MBE growth: GaN nanopillars were grown by catalyst-free MBE with a plasma-assisted nitrogen source onto the Si(100) device layer prior to membrane etching and release.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' The SOI substrates (SEH America∗) had device, buried oxide, and carrier layer thicknesses of 200 nm, 380 nm, and 675 µm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' The device layer was lightly boron doped with a resistivity of 28 Ω-cm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' as noted above, these thin device layers are not currently available in any other doping types or concentrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' The nanopillars initially cover the entire surface of the substrate but were selectively removed with photolithography for the electrical test structures [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Nanopillar height was varied by adjusting the nanopillar growth period, with the longest growth period being 12 h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' The ratio of the N equivalent growth rate to the Ga equivalent growth rate during nanopillar growth was 6:1 for the Set A and 3:1 for Set B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' The nanopillars were grown at approximately 810 ◦C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' More details are provided in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Sample fabrication After nanopillar growth, each 2 cm × 2 cm chip was fabricated into a testing platform to measure its thermoelectric properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Each completed chip yields 2 four-point electrical resistivity devices, 2 Seebeck coefficient devices, and 92 thermal conductivity test membranes ranging in nominal size from 400 µm × 400 µm to 700 µm × 700 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Ohmic contact pads were formed using 20 nm Ti/200 nm Al metal stacks annealed in argon at 500 ◦C for 1 minute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Thermoelectric metrology Raman thermometry: We used a 325-nm He-Cd laser as a heating source that was propagating anti-parallel to a low intensity 532-nm laser used as a Raman probe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' The nanopillared Si mem- branes were positioned such that the side with nanopillars was exposed to the low intensity 532-nm Raman probe, while the 325-nm beam was absorbed on the unpatterned side of the membrane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' This optical alignment allowed for more accurate estimation of absorbed laser power due to the ∼60-nm absorption depth at the 325-nm wavelength, precluding transmission to the nanopillars on the op- posite side of the membrane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' The beam diameters at the 1/e2 points were 25 µm and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='8 µm for the 325-nm and 532-nm lasers, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' The nanopillars are transparent to the green probe beam though some scattering occurred as the beam passed through them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' The reflectance R of the bottom side of the membranes was measured to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='57 for the UV beam, and the absorbed beam power was calculated as the incident beam power multiplied by (1 − R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' The beam power was measured with an optical power meter close to where it impinged on the specimen and then corrected for transmission of the intervening optics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' The temperature dependence of the Si Raman peak was calibrated by heating a Si chip with a strip heater and measuring its temperature with a thermocouple while acquiring Raman data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' The resulting data was fit with the quadratic equation T(◦C) = 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='2 − 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='4(∆ν − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='1(∆ν)2) where ∆ν is the temperature-induced shift in the Raman peak position in wavenumbers (cm−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' The linear term of this equation agrees well with previous ∗Vendor is identified to adequately specify the source material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' This identification does not imply recommendation or endorsement by the National Institute of Standards and Technology, nor does it imply that the product identified is necessarily the best available for the purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' 9 evaluations that report ∆T/∆ν = −46 K/cm−1, initially by the work of Mendez and Cardona and verified by others including Reparaz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' [41, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Seebeck coefficient: The Seebeck coefficient measurement was performed via a steady-state method with the geometry shown in the Appendix, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' A4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Two meandering Ti/Al wires were lithographi- cally defined 10 µm from the Si device layer to serve as thermometers with a ∼100-Ω resistance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Prior to measurement, both resistors R1 and R2 were calibrated to within 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='1 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' An additional pair of Ti/Al wires was patterned in direct contact with either end of the Si device layer to measure the Seebeck voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Two 1-kΩ chip resistors, serving as heaters, were glued to one end of the chip and used to provide a thermal gradient along the length of Si device layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' The heaters provided up to 25 mW of power yielding a maximum ∆T of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='5 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' The heater current, thermopower voltage Vth, temperatures at R1 and R2, and temperature gradient across the Si device layer ∆T were recorded as a function of time with initial sample temperature at 277 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' All calibrations and measurements were performed in ice water to maintain a constant bath temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Electrical resistivity: The electrical resistivity was measured using a standard four-point probe test structure shown in the Appendix, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' A4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' The quantity ∆V across the two inner contacts was measured as a function of current across the two outer contacts over the range of 0 nA−100 nA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' All tested devices showed a linear, ohmic response, allowing for resistivity ρ to be calculated from the membrane width w, thickness t, length L, and measured resistance R as ρ = Rwt/L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Atomic models The theoretical investigations are based on atomic models comprising a Si membrane with GaN nanopillars standing on the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Both material portions were modeled as single crystals under room-temperature equilibrium conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' The Tersoff potential was used for the interatomic in- teractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' The parameters of the Si-Si and Ga-N interactions were taken from Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' [43] and [44], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' For the Si-Ga and Si-N interactions, the potential parameters were mixed following the Tersoff multicomponent combination rules [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Two sizes of NPMs were investigated: one that is nearly 15 times smaller than a nominal experimental unit cell (shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' 2f, right), and a smaller version for the coalescence investigation (shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' 4g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' In the model of the coalesced NPM, the top of the nanopillar was laterally extended to partially connect with adjacent nanopil- lars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' This was done by adding three primitive-cell layers of GaN around the tip of the nanopillar forming a cross-like cross section when viewed from the top (cut view A-A in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' 4g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Lattice dynamics calculations: The phonon band structures for the examined GaN-on-Si NPM unit cells were obtained by solving the quasiharmonic lattice dynamics eigenvalue problem using the GULP software [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Bloch periodic boundary conditions were applied along in-plane directions and free boundary conditions were applied in the z direction and around the nanopillar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' The phonon frequencies were computed at a set of allowed wave vectors ranging from Γ to X in the Brillouin zone with a resolution of 128 points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' There are 3N phonon branches in the band structure, where N is the total number of atoms in the unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' The average group velocity ratio Gr is a quantity that characterizes the reduction in the group velocities across the entire phonon spectrum [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' It is defined as Gr = GNPM /GMem, where Gi is the average group velocity of either an NPM or a membrane calculated by Gi = (1/nκnm) �nκ κ �nm m vg(κ, m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Here, κ is the wave number (scalar component of the wave vector κ along the Γ−X direction), m is the branch number, nκ is the number of wave-number points considered, and nm = 3N is the total number of phonon branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' The group velocity vg(κ, m) is 10 defined as the slope of the phonon frequency with respect to the wave number κ for branch m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' For characterization of nanopillar resonant mode localization, we examine the mode shape cor- responding to each point in the phonon band structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' We then compute the mode participation ratio pr, which is defined for a mode at wave vector κ and branch number m by Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' [22, 25] pr(κ, m) = 1 N �N i=1[�3 j=1 φ∗ ij(κ, m)φij(κ, m)]2 , (2) where φij(κ, m) is the displacement component corresponding to atom i and direction j of the normalized mode shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' The formula comprises two summations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' The first is over the total number of atoms N in a unit cell, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=', N = NBase + NPillar for an NPM, where NBase is the number of atoms in the base membrane and NPillar is the number of atoms in the nanopillar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' The second summation is over the three directions of motion per atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' The inverse of this quantity pr indi- cates the degree of modal localization over the entire unit cell considered without being specific to a particular region, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=', the nanopillar or base membrane portion of an NPM unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' This calculation is performed for both an NPM and a bare membrane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' In an NPM, a large number of the modes exhibit high concentrations of vibrations in the nanopillar portion, yielding a low value of pr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Molecular dynamics simulations: Equilibrium molecular dynamics (EMD) simulations were exe- cuted to predict the in-plane lattice thermal conductivity of the GaN-on-Si NPMs sized at nearly 1/15 of the nominal experimental unit cell, with the height of the nanopillar being varied (see atomic model dimensions in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' A single unit cell was used as the simulation cell with periodic boundary conditions applied along the x and y directions and free boundary condition applied in z direction and around the nanopillar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' The empirical interatomic potentials were identical to those used in the LD calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' The time integration step was set as ∆t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='5 fs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' First, a canonical en- semble MD with a Langevin heat reservoir was allowed to run for 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='3 ns to enable the whole system to reach equilibrium at 300 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Then, a microcanonical ensemble (NVE) was run for 3 ns;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' mean- while, the heat current was recorded at each time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' At the end of the simulations, the thermal conductivity was calculated by the Green-Kubo formula, [47] k = 1/(2V kBT 2) � ∞ 0 ⟨J(τ) � J(τ)⟩dτ where kB is the Boltzmann constant, V is the system material volume, and J is the heat flux along the direction of transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Finally, the thermal conductivity was averaged over the two in-plane directions over six simulations with different initial velocities (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=', a total of 12 cases), and the statistical errors were obtained according to the method described in [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' All EMD simulations were performed in LAMMPS [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Acknowledgments This research was partially supported by the Advanced Research Projects Agency−Energy (ARPA- E) under grant number DE-AR0001056.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' APPENDIX This supplemental information document contains tables of data from the figures used in the main article, further details of the methods and results, including a full set of SEM images of the specimens used in the study and examples of the Raman temperature vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' absorbed power curves, discussion of possible additional heat transport mechanisms, and discussion of the Al and 11 Ga diffusion into the membrane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Commercial equipment and instruments are identified in order to adequately specify certain procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' In no case does such identification imply recommendation or endorsement by the National Institute of Standards and Technology, nor does it imply that the products identified are necessarily the best available for the purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' A Tabular data This section contains details of specimen synthesis and morphology and data from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' 3 in the main article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Table A1: Specimen characteristics for Sets A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Run No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Nanopillar Growth kNPM/kMem height (µm) time (h) Bare (A) NA NA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='99 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='06 D420 (A) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='324 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='03 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='05 D421 (A) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='785 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='98 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='04 D422 (A) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='950 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='85 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='02 D423 (A) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='300 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='79 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='01 D442 (B) NA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='083 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='01 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='01 D469 (B) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='77 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='93 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='04 D480 (B) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='430 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='80 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='02 D443 (B) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='140 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='87 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='01 D481 (B) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='700 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='98 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='01 Table A2: Nanopillar dimensions for Sets A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Run No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Nanopillar Root diam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Fill Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' tip height (µm) (nm) fraction area (µm2) D420 (A) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='324 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='035 30 ± 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='0029 D421 (A) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='785 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='130 40 ± 11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='0041 D422 (A) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='950 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='200 50 ± 11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='010 D423 (A) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='300 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='320 75 ± 22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='016 D469 (B) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='77 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='050 65 ± 14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='0064 D480 (B) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='430 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='080 70 ± 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='52 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='019 D443 (B) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='140 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='040 60 ± 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='023 D481 (B) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='700 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='140 130 ± 36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='060 B Nanopillar synthesis SOI substrates (SEH America) were diced into 2 cm by 2 cm squares prior to growth and process- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Immediately prior to loading for growth, the substrates were cleaned with solvents, oxygen plasma in a reactive ion etching system, and approximately two-minute exposure to hydrogen 12 Table A3: Thermal conductivity, electrical conductivity, and Seebeck coefficient data from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' 3 of the main article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' The thermal conductivity values were obtained by multiplying the relative values in Table A1 by 60 W/m · K, as discussed in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Run No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Nanopillar Thermal cond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Electrical cond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Seebeck coef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' height (µm) kNPM(W/m · K) σ (S/m) (µV/K) Bare (A) NA 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='6 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='5 1500 ± 12 D420 (A) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='324 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='9 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='2 955 ± 47 D421 (A) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='785 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='5 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='6 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='4 ± 5 800 ± 8 D422 (A) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='950 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='9 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='0 103 ± 30 730 ± 5 D423 (A) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='300 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='5 233 ± 30 610 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='6 D442 (B) NA 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='6 170 ± 20 711 ± 1 D469 (B) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='77 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='6 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='2 no data no data D480 (B) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='430 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='7 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='0 150 ± 30 676 ± 2 D443 (B) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='140 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='7 320 ± 50 788 ± 1 D481 (B) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='700 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='8 220 ± 50 719 ± 3 fluoride (HF) vapor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' The substrates were outgassed three times at successively higher tempera- tures in the MBE loadlock chamber, preparation chamber, and growth chamber, with the final outgas reaching between 870 ◦C and 890 ◦C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' The Si surface initially displayed a 1×1 reflection high energy electron diffraction (RHEED) pattern that brightened and sharpened as surface oxides desorbed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Around 720 ◦C, the 1×1 pattern changed into a 2×1 pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Substrate temperatures throughout the growth were measured with an estimated uncertainty of 8 ◦C using a back-side py- rometer described elsewhere [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Element fluxes were estimated from separate growth calibration runs in which the growth rate of planar films was derived from optical interference fringes at a variety of different Ga and Al beam equivalent pressures under group-III-limited conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' The nitrogen-limited growth rate for a variety of plasma conditions was estimated from the transition from a spotty to streaky RHEED pattern as the gallium flux was increased [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' For specimen Set A (no buffer layer), the Ga and N equivalent planar growth rates were 110 ± 10 nm/h and 650 ± 30 nm/h, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' For these runs, the Si surface was exposed to the N plasma for 60 s at 740 ◦C, then nanopillar growth was initiated at a low temperature of 700 ◦C for 12 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' The remainder of the nanopillar growth took place at 810 ◦C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' For specimen Set B (AlN buffer layer), the Ga and N equivalent planar growth rates were 250 ± 20 nm/h and 650 ± 30 nm/h, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Buffer layer deposition was preceded by a 15 s N plasma exposure and a 1 s Al exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' The buffer layer itself was deposited at an actual growth rate of 165 ± 20 nm/h with the N flux lower than the Al flux to promote formation of a N-polar surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' The growth temperature for the buffer layer was 850 ◦C, where we estimate that approximately half of the Al reevaporated rather than be incorpo- rated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' GaN nanopillar growth was initiated on the AlN buffer layers at 770 ◦C for 5 minutes before increasing to 810 ◦C for the main nanopillar growth stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' The GaN nanopillars grown without a buffer layer were not intentionally doped, leading to an n-type carrier concentration below 1 × 1016 cm−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' The GaN nanopillars on AlN were lightly doped with Si, and thus had an n-type carrier concentration in the 1017 cm−3 range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' As shown in the main article, the nanopillars tended to get thicker along their length and coalesce near the tips, and this coalescence increased with nanopillar height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' The average tip area is an approximate indicator of the degree of coalescence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' For the specimen Set B (with AlN buffer layers), 13 a h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='32 �m b h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='77 �m c h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='79 �m d h = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='4 �m e h = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='95 �m f h = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='1 �m g h = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='3 �m h h = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='7 �m 200 nm 300 nm 300 nm 400 nm 500 nm 500 nm 1000 nm 2000 nm Figure A1: SEM images of specimens in this study taken from a tilt angle of 45◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Images in the left column portray the specimen set grown without a buffer layer (Set A), and the right column portrays the specimen set grown with an AlN buffer layer (Set B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Run numbers are (a) D420, (b) D469, (c) D421, (d) D480, (e) D422, (f) D443, (g) D423, (h) D481.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' the coalescence occurred faster as a function of height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' This change is most likely due to the higher Ga flux during nanopillar growth for this series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' The flux was increased to increase the fill fraction of the nanopillars, defined as the ratio of the total tip area in the top view images relative to the corresponding image area, but also increased coalescence at the tips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' As can be seen in Table A2 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' A1, in general the density of the nanopillars at their roots is more similar than the tip area from run to run within a set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' SEM images for the both specimen sets are given in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' A1 and A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' 14 a h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='32 �m b h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='77 �m c h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='79 �m d h = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='4 �m e h = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='95 �m f h = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='1 �m g h = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='3 �m h h = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='7 �m 200 nm 320 nm 200 nm 200 nm 200 nm 200 nm 200 nm 200 nm Figure A2: SEM images of specimens in this study viewed from the top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Images in the left column portray the specimen Set A, and the right column portrays the specimen Set B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Run numbers are (a) D420, (b) D469, (c) D421, (d) D480, (e) D422, (f) D443, (g) D423, (h) D481.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' C Specimen fabrication The fabrication steps are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' A3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Photolithography is used to protect the nanopillar forest with a 7 µm thick resist coating while exposed nanopillars are removed via sonication in de-ionized water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Next, the Si device layer is patterned and etched down to the buried oxide layer with a 5-cycle deep reactive-ion etch (DRIE) using SF6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' The patterns for the electrical conductivity and Seebeck coefficient measurements are in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' A4a and A4b, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' A 30 s CF4 dry etch is used to remove any native oxide on the Si before a 20 nm Ti/ 200 nm Al metal stack is deposited to form probe contacts and thermometers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Deposition is followed by a 1 min 500 ◦C anneal in argon to ensure ohmic contacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' The top-side of the chip is then wax bonded to a sapphire carrier 15 GaN nanopillar Si membrane Si handle SiO2 Ti/Al a b c d e Figure A3: Fabrication flow diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' a b Figure A4: Electrical patterns for measuring (a) electrical conductivity, and (b) Seebeck coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' wafer to protect the nanopillars and Si device layer during back-side processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' The membranes for thermal conductivity measurements are lithographically patterned and the Si handle layer etched 16 Heater Heater R1 /+ Si 3000 μm V- V+ V th 500 μm R2 + Sio V- V+Si V+ 5000 μm SiO2 V- 500μmFigure A5: Raman temperature measurement schematic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' with a 900 cycle DRIE followed by a 15 min HF vapor phase etch to remove the buried oxide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' The carrier wafer is then suspended upside down in a beaker of acetone to dissolve the wax and release the completed test devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' D Raman thermometry A schematic of the Raman thermometry experiment is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' A5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' The setup consists of a Coherent Verdi V6 532 nm CW laser that is attenuated with neutral density (ND) filters to an average power of ∼ 300 µW at the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' An infinity-corrected 40× Nikon objective with a numerical aperture of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='60 was employed to focus the 532 nm beam on the samples with a spot diameter of ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='8 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' The light is then collected in a backscattering geometry and sent through a Raman filter (Semrock RazorEdge Filter) and then into a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='5 m spectrometer (Acton Spectra Pro 2500i) that disperses the Stokes-shifted Raman light with an 1800 groove/mm grating onto a liquid nitrogen cooled CCD array, providing sub cm−1 energy resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' In order to heat the samples for thermometry studies, we used a 325 nm He-Cd CW laser (Kimmon) and a variety of ND filter combinations to control incident laser fluence of the pump beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' The pump beam is collinear and 17 Spectrometer and CCD Array Raman Filter Beam Splitter 532 nm Laser Imaging CCD ND Filter White Light Source .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='Flip Beam Splitter Beam Splitter 325 nm Laser 40x Objective Sample Lens Variable ND Filterpropagating antiparallel to the 532 nm Raman probe beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' The 325 nm laser beam is focused onto the membranes with a 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='0-cm focal length lens mounted on a micrometer stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Alignment of the two beams was tested by observing the Raman peak shift while making minor adjustments to the 325-nm lens position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' The spot diameter, defined at 1/e2 intensity locations, was measured to be ∼25 µm at the focal point by passing a knife edge through the beam and fitting the transmitted intensity to an error function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' The pump beam average incident power was varied between 0 to 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='5 mW, resulting in fluences ranging from ∼0 to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='5 kW/cm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' A flip-mounted pellicle beam splitter was used in conjunction with white light illumination source to image the sample and verify the Raman probe was proper centered on suspended membrane prior to each thermometry experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' In the main text, we utilize the following radial temperature profile (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' 12 in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' [31]): ∆T = T(r) − Tamb = Pabsln(r/R)β(r)/(2πdk), (A1) where Pabs is the absorbed power from the 325-nm laser, Tamb is the ambient temperature, R is the radius of the membrane, d is the effective conductive thickness, and k is the membrane thermal conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' We evaluate some of the features of this profile to confirm our assumption that the probe beam is sampling a region with approximately uniform temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' We have made the assumption that the edge of the membrane is at ambient temperature for large membranes like those we use here, with R on the order of 375 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' The function β is defined as β(r) = 1 + Ei(−r2/r2 0) − Ei(−R2/r2 0) 2ln(R/r) , (A2) where R is the radius of the membrane, and Ei is the exponential integral function [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' The radius r0 of the heating laser beam is defined for an intensity distribution I = (P/πr2 0)exp(−r2/r2 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Note that this definition of r0 omits the conventional factor of 2 in the argument of the exponential for a Gaussian beam intensity distribution, and thus our measured beam diameter of 25 µm corresponds to r0 = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='8 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Ei(x) becomes negligible for large negative arguments, and hence for R >> r0, as is the case here, the second term in the numerator can be omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' The behavior of Ei near r = 0 can be evaluated using a Taylor series expansion for negative real arguments [52]: Ei(x) = γ + ln|x| + x + x2 2 · 2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' + x3 3 · 3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='. (A3) Evaluating the temperature model numerically, we find that the temperature within the radius of the probe beam, 400 nm, is uniform within 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='02 K for 325-nm spot diameters ranging from 25 to 71 µm and uniform within 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='1 K for a smaller spot diameter of 10 µm (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' A6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' We can also evaluate the limiting behavior of the radial portion of the temperature profile near r = 0 by substituting the first two terms of the Taylor expansion as follows: ln(R/r)β(r) = ln(R/r) � 1 + γ + ln| − r2/r2 0| 2ln(R/r) � = ln(R/r) �2ln(R/r) + γ + 2ln(r/r0) 2ln(R/r) � = γ/2 + ln(R/r) + ln(r/r0) = γ/2 + ln(R/r0) (A4) This temperature model can also be used to estimate the inverse slope of ∆T versus Pabs eval- uated at the center of the 325-nm beam,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' which we measured to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='025 mW/K in bare mem- branes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' As shown in Table A4, the measured value is reproduced by a combination of a membrane 18 Distance from center (nm), r �T(0)-�T(r) (K) 100 200 300 400 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='00 k = 80, 60 W/m K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' , , , r0 = 25 �m r0 = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='8 �m r0 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='5 �m Figure A6: Radial variation of ∆T relative to its value at r = 0 as a function of radius for representative values of membrane thermal conductivity k and 325 nm beam radii r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' thermal conductivity of 60 W/m·K and a beam diameter of 71 µm, or a membrane thermal conduc- tivity of 80 W/m·K and a beam diameter of 25 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' The difficulty in measuring the beam diameter accurately and aligning the two beams perfectly translates into difficulty in determining the ab- solute thermal conductivity of the membrane;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' hence we primarily use the Raman thermometry method for determining relative changes in the membrane thermal conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Table A4: Temperature model outputs for representative membrane and beam parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' ∆T is evaluated at r = 0 and Pabs = 1 mW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' r0 Beam k k (µm) diam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' 60 W/m·K 80 W/m·K (µm) Inv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Slope ∆T (K) Inv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Slope ∆T (K) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='54 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='015 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='020 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='3 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='8 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='019 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='025 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='1 25 71 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='025 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='034 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='8 Examples of the Raman thermometry data are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' A7, where we plot Raman-derived temperature at the center of the heated spot as a function of absorbed laser power for representative specimens with nanopillars and their corresponding control membranes without nanopillars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' The Raman thermometry data show predominantly linear temperature increases for absorbed powers up to ∼2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='0 mW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' In order to determine the ratio of thermal conductivities described in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' (1), we have carried out linear fits to the Raman thermometry data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' For highest accuracy, we fit data points in the range of 40 ◦C to 120 ◦C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' We exclude lower temperatures because the Raman peak shifts are small and the corresponding uncertainty is high for these points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' For higher temperatures, the response starts to become nonlinear due to the decrease in the thermal conductivity of the Si membrane with increasing temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Data for multiple membranes from the same run were fit individually and then their slopes b were averaged using a weighting factor of σb/b, where σb is the 19 10 20 30 40 0 Growth time, tg (ks) Electrical conductivity, � (S/m) 0 50 100 150 200 250 � Figure A7: Examples of nanopillared and bare membrane temperature in the center of the heated spot as a function of absorbed UV laser power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' (a) Set A and (b) Set B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' standard deviation of the set of measured slopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' The resulting ratios are shown in Table A1 with corresponding uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' As portrayed in the table, kNPM/kMem values are predominantly less than unity with a minimum value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='79, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=', a 21 % reduction in the thermal conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' We note here that run number D442 was a control sample with only an AlN buffer layer on the Si membrane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' It exhibits a thermal conductivity that is nearly identical to the pristine membrane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' As discussed in the main article, partial coalescence of the nanopillars reduces the NPM effect and leads to higher thermal conductivities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' E Possible additional heat transport mechanisms The presence of the nanopillar roots on the membranes may be viewed as a source of surface roughness, and thus might be thought to increase surface/boundary scattering of phonons and thereby reduce the thermal conductivity, similar to investigations of rough Si nanowires [8, 9, 14] and rough Si membranes [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' However, the average width of the nanopillar roots among the various fabrication runs in our investigation is 65 nm (see Table A2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' This is significantly larger than the range of phonon wavelengths (∼5 to ∼60 angstroms for Si at room temperature [16]), which is the size scale relevant for boundary scattering due to roughness [10, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Furthermore, if boundary scattering of any form is dominant†, the thermal conductivity would not increase with an increase in nanopillar coalescence occurring predominantly at the tips;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' yet as noted in the main article we observe a significant rise in k in the severely coalesced case of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='7 µm tall nanopillars (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' 3a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Secondly, it is conceivable that a portion of the heat might take a path through the coalesced nanopillars which would effectively serve as a parallel “branch” for the thermal transport for those cases, resulting in a larger overall area through which the heat-carrying phonons travel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' While this is unlikely because the phonons will scatter at the coalesced nanopillar tips, the increased overall †For example, in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' [53] the presence of nanopillars was interpreted as a source of increased boundary scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' 20 cross-sectional area implies lower k for the same power transmitted−thus improving the reduction rather than deteriorating it as we observe for the 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='7 µm tall nanopillars in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Furthermore, we note that any transport of heat to the environment rather than being delivered to the substrate at the edges of either type of membrane (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=', base membrane in an NPM or bare membrane) would result in an overestimation of the integrated thermal power P flowing through the membrane, which we assumed to be equal to Pabs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' One example of such transport would be heat loss through the small but finite thermal conductivity of the surrounding air.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' We here show that any such mechanism results in an overestimation of the true thermal conductivity ktrue of the membrane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Postulating that this mechanism results in P = ηPabs, where η is a number less than 1, and defining D = 2πd/(γ/2 + ln(R/r0)), we can combine Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' A1 and A4 and solve for ktrue as ktrue = 1 D � ∂∆T(0) ∂(ηPabs) �−1 = 1 D �1 η ∂∆T(0) ∂Pabs �−1 = η 1 D �∂∆T(0) ∂Pabs �−1 = ηkcalc, (A5) where kcalc is the value we would calculate for k using our original data analysis method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Because η is less than 1, the true value ktrue would be less than our calculated value kcalc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' In taking the ratio of kNPM/kMem, any heat conduction through the surrounding air would affect both the NPM membrane and the bare membrane, and therefore the heat loss would cancel in taking this ratio, provided the nanopillars do not affect local interactions with the air.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Although the nanopillars bear a superficial resemblance to heat sink structures in which fins are used to conduct heat away from an object and increase the surface area subject to air cooling, the spacing between the nanopillars on our membranes is smaller than or comparable to the mean free path of air molecules at atmospheric pressure, around 65 nm [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' This tight spacing precludes the development of natural convective or conductive regions in the gaps between nanopillars or any effective increase in membrane surface area by the nanopillars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Instead, the high thermal conductivity along the axis of the nanopillars (compared with air) effectively moves the membrane/air interface to the plane defined by the tips of the nanopillars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' However, even if the nanopillars were to increase air cooling to some small degree, the η value for NPM membranes would then be smaller than that for the bare membranes, and thus the true ratio of thermal conductivities would be smaller than our measured ratio, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=', we would be underestimating the NPM effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' This calculation can also be understood qualitatively in that any loss of heat to the environment reduces the temperature difference generated across the path of electrical current for a given thermal power input, and therefore lowers the thermoelectric performance of a structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' F Dopant diffusion into the Si membrane Al and Ga are both p-type dopants in silicon, and will diffuse into to the thin Si membrane at high temperature, producing changes in membrane carrier concentration and electrical conductivity that depend on time at high temperature and available dopant concentrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' For the specimen set without buffer layers, Set A, Ga flux is continually arriving at the Si membrane surface between the roots of the nanopillars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' From the measured electrical conductivity values, the Ga concentration in the membrane reaches a maximum value of 4×1016 cm−3 for the specimen with the longest growth time, D423.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' We infer the carrier concentration from typical relationships between electrical conductivity and carrier concentration for Si [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' A8, the increase in σ is linearly dependent on growth time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' This dependence implies that the Ga concentration and hence the hole concentration is a function of the total flux impinging on the top surface during NPM growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' The 21 a b 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='5 0 Adsorbed power, P (mW) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content='5 0 Adsorbed power, P (mW) Temperature, T (oC) 20 40 60 80 100 200 300 Temperature, T (oC) 20 40 60 80 100 200 300 D480: various membranes D442: buffer only Average slope for each set D423: various membranes Various bare membranes Average slope for each set Figure A8: Electrical conductivity of specimen Set A showing a linear increase as a function of nanopillar growth time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' The change is due to diffusion of Ga into the Si membrane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' maximum carrier concentration corresponds to a Ga concentration in the membrane of less than 1 ppm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Figure A8 also illustrates that there was significant spatial variation in the measured electrical conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' For most of the runs, two areas on the die were sampled for the electrical conductivity, and although each measurement had low uncertainty, the spatial variations are on the order of 50 % to 150 % for the specimens with the highest σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' The actual Ga flux to the wafer is likely much more uniform that this, but the local environment of randomly growing nanopillars would lead to large variations in shadowing of the surface by the nanopillars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' In order to avoid the complication of diffusion-induced electrical conductivity changes within a specimen set, Set B employed an 8 nm thick AlN buffer layer as a diffusion blocking layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' AlN has been used to block diffusion of Mg in GaN/AlGaN structures [56] and of Al in TiN contact layers [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' As shown in the main article, this very thin buffer layer achieved our goal of removing the dependence of electrical conductivity on growth time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' The electrical conductivity for the specimen with the AlN buffer layer only (and no nanopillars) is still significantly higher than that measured for a pristine Si membrane, around 170 S/m vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' 4 S/m, and we attribute this increase to Al diffusion during the buffer layer growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Al diffusion out of AlN buffer layers is a contributing factor to conductive loss in RF devices grown on GaN-on-Si substrates [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' Al diffuses more readily in Si than Ga [59–61], but Al flux was only present during the AlN buffer layer growth, lasting 180 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' The AlN buffer did appear to be effective as a barrier to subsequent Ga diffusion based on the conductivity of the specimen with the longest growth time, D481.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' For this run, the electrical conductivity would have more than doubled if Ga diffusion occurred at the same rate as in Set A, but in fact it remained unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' This diffusion is not a fundamental limitation for practical applications because it is only signif- icant when the starting materials have very low background carrier concentrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' SOI with 200 nm device layers are only available with low B doping concentrations (<1×1015 cm−3) due to their method of fabrication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' For practical thermoelectric applications, much higher electrical conductiv- ity would be needed, and doping methods would be developed for the membranes to increase the electrical conductivity and therefore increase ZT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' 22 References [1] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE3T4oBgHgl3EQf4AvA/content/2301.04769v1.pdf'} +page_content=' M.' metadata={'source': 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index 0000000000000000000000000000000000000000..2ff99504e9fbc14713253221a95c64f174fc8559 --- /dev/null +++ b/ktAyT4oBgHgl3EQfyPnx/content/tmp_files/2301.00682v1.pdf.txt @@ -0,0 +1,635 @@ +arXiv:2301.00682v1 [gr-qc] 27 Dec 2022 +On the stability of electrostatics stars with modified non-gauge +invariance, Einstein-Maxwell gravity +H. Ghaffarnejad1, +T. Ghorbani 2 and F. Eidizadeh 3 +Faculty of Physics, Semnan University, P.C. 35131-19111, Semnan, Iran +Abstract +We use modified Einstein-Maxwell gravity where correction parts +are nonminimally directional coupling between the Maxwell vector +potential field and Ricci tensor and Ricci scalar fields to produce La- +grangian density of the fields for a spherically symmetric static curved +metric field. Euler-Lagrange equations are nonlinear second order dif- +ferential equations and have not an analytic closed form of the solu- +tions and so we must be solve these via dynamical systems approach. +By regarding this method we obtained some stable solutions near the +critical points. To determine stable (unstable) nature for the obtained +solutions we must solve secular equation of the Jacobi matrix of the +field equations and determine sign of the eigenvalues. +1 +Introduction +To describe the stability of a stellar compact object, in usual way, it is nec- +essary to consider the Tolman-Oppenheimer-Volkoff equations [1] and the +equation of state of the star. Stability criteria of relativistic spherical sym- +metric compact objects with isotropic pressure in the framework of general +relativity include boundary conditions, non-singularity, electric charge, sur- +face redshift, energy conditions, the speed of sound in causal conditions and +relativistic adiabatic index. In a stable model, the energy and pressure densi- +ties are finite at the center of compact object and decrease uniformly toward +the boundary. The metric potentials are regular and the electric field inten- +sity is zero at the center and increases towards the surface. In addition, the +gravitational redshift follows Zs < 2 and four energy conditions are satisfied, +the speed of sound is less than the speed of light and decreases uniformly +1E-mail address: hghafarnejad@semnan.ac.ir +2E-mail address: tohidghobani@semnan.ac.ir +3E-mail address: f.eidizadeh@gmail.com +1 + +toward the surface. In addition, the adiabatic index is strongly higher than 4 +3 +[2]. Relativistic Compact object with gravity and strong internal density have +two different pressures, radial and tangential [3]. The stability of a stellar +model can be increased by an anisotropic repulsive force that ∆ = pt−pr > 0. +This property leads to more compact stable configurations compared to the +states of isotropic [4]. +Hydrostatic equilibrium of solutions of anisotropic +relativistic stars in scale-dependent gravity, where Newton’s constant is al- +lowed to vary with radial coordinates across the star, shows that a decrease +in Newton’s constant across objects leads to slightly more massive and com- +pact stars [5]. +A stability analysis for Einstein-Kline-Gordon model with +static real scalar field interaction express that the initial value of the field +at the origin is a function of the energy density of the matter at the origin +and in the far regions the field behaves Yukawa-like potential. Such a model +for compact stellar object is stable if the gradient of the total mass versus +energy density is positive and the weak energy condition is satisfied (positive +total density) [6]. The stability of the star can be investigated in the pres- +ence of both electric and magnetic fields. Solving the Einstein-Maxwell field +equations for compact objects with the charged anisotropic fluid model gives +more stable solutions than for neutral stars. The presence of charges creates +a repulsive force against the gravitational force, and this factor causes denser +stable stars, higher maximum mass and larger redshift [7]. Charged quarks +can create more stable quark stars than neutron nuclei. Also, for a white +dwarf with a charged perfect fluid, there is a direct correlation between the +increase in electric charge and its size. Near the surface of the star, the radial +pressure is close to zero and the electric charge density is non-zero, leading +to a stable star with more mass [8]. The mass-radius relation of some kinds +of Neutron stars, which can contain a core of quark matter, have a large +frequency range of radial fluctuations near the transition point in their core +versus mass. These induce nonlinear general relativistic effects which cause +to be the stars unstable dynamically. The core of the Neutron stars becomes +several times larger, making the Neutron stars highly unstable [9]. While for +the charged boson-fermion stars with a charged fluid related to fermion and +a complex scalar field related to boson, the charge increase can reduce the +stellar radius and create a denser and more massive star. In the whole pa- +rameter space, the critical curve can show stable and unstable regions [10]. +If the number of baryons in compact pulsar-like stars exceeds the critical +value 109, the strangeon star model is proposed. In fact the strangeon star +atmosphere model describes the radiation from interstellar medium accreted +2 + +plasma atmosphere on a strangeon star surface and its spectrum. This ob- +ject could simply be regarded as the upper layer of a normal neutron star +because the radiation from strangeon matter can be neglected [11]. +The +atmosphere is in radiative, thermal equilibrium and two-temperature. The +strangeon star spectrum is based on bremsstrahlung from an extremely thin +hydrogen plasma. More details of this model are described in [12]. Since the +extra strange flavor provides more degrees of freedom to lower the Fermi en- +ergy in the free quark approximation, macroscopic bulk strong matter with +3-flavor symmetry (up, down, and strange quarks) is more stable than up +quark matter. The difference in the strangeness level between a strange star +and a typical neutron star can have a profound effect on the magnetospheres +activity associated with the coherent radio emission of the Compact stars. +Content of this paper is as follows: +In section 2 we describe shortly generalized Einstein Maxwell gravity. Then +we obtain Lagrangian form of the model for a general spherically symmetric +static metric and corresponding Euler-Lagrange equations. These are non- +linear second order differential equations and so we use dynamical systems +approach to solve them in the section 3. In the latter section we obtain alter- +native first order differential equations in phase space and calculate Jacobi +matrix of the equations. We determine eigenvalues of the Jacobi matrix by +solving the secular equation. At last we determine sign of the eigenvalues +for which the system has stable natures for some suitable numeric values of +the parameter of the coupling constant of the model. They are collected in +a table. Section 4 dedicated to concluding remarks and outlook of the work. +2 +The gravity model +As a generalization of the Einstein-Maxwell gravity theory we consider non- +minimally coupling between the Maxwell vector potential Aµ and the Recci +scalar and the Ricci tensor such that [13] +I = − +� +dx4√g +�1 +4FµνF µν + α +2 A2R + β +2 RµνAµAν +� +, +(2.1) +where g is absolute value of determinant of the metric field and anti symmet- +ric electromagnetic tensor field Fµν is defined versus the partial derivatives +of the four vector electromagnetic potential Aµ as follows. +Fµν = ∇µAν − ∇νAµ = ∂µAν − ∂νAµ +(2.2) +3 + +with A2 = gµνAµAν and Rµν is Ricci tensor. It is easy to check that this +model has not gauge invariance symmetry same as [14] in which the ac- +tion functional remain unchanged by transforming Aµ → Aµ + ∂µξ. In this +transformation ξ is gauge field for which Fµν → Fµν. In this work we like +to investigate effects of the electromagnetic fields on metric field solutions +of a spherically symmetric stellar object. For spherically symmetric time- +independent static metric +ds2 = −X(r)dt2 + Y (r)dr2 + r2(dθ2 + sin2 θdϕ2) +(2.3) +we know that non-vanishing components of the vector potential is just At(r) = +φ(r) means electric potential for a electrostatics spherical stellar object which +by substituting into the action function (2.1) we obtain exact form for the +Lagrangian density of the fields such that +L = +2π +√ +XY +� +˙φ2 + βφ2 +� ˙Y +Y + Y − 1 +�� +(2.4) +where we defined ˙ as logarithmic derivative of the radial coordinate as +˙ = d +dτ = r d +dr = +d +d ln(r/D) +(2.5) +in which D is a suitable length parameter. Also we use the ansatz +α = −β +2 +(2.6) +to remove second order derivatives of the X field coming from the Ricci scalar +and Ricci tensor. By defining an new field ψ(r) such that +˙φ = ψφ +(2.7) +one can show that the Euler-Lagrange equations of the fields φ(r), X(r), Y (r) +are obtained after some simple mathematical calculations respectively as fol- +lows. +˙ψ = −ψ2 +β (4βY + βψ + 3ψ2 − 3β) +(2.8) +˙X = −X +β (7βY − 4βψ + 5ψ2 − 5β) +(2.9) +4 + +and +˙Y = −Y +β (βY + ψ2 − β). +(2.10) +The above equations are non-linear first order differential equations not hav- +ing exact analytic solutions and so we can solve via dynamical systems ap- +proach. This is done in the next section. +3 +Metric solutions +General strategy in the dynamical system approach to obtain analytic solu- +tions for the fields equations ˙Xi = F(Xi, t) have 4 different steps as follows: +(a) By solving the equations ˙Xi = 0; i = 1, 2, 3, · · ·n for n dimensional phase +space of the system one determine the critical points. (b) For each of these +critical points he/she should calculate Jacobi matrix Jij = ∂ ˙Xi +∂Xj and (c) then +solve the corresponding secular equation det(Jij − Eδij) = 0 to obtain eigen- +values for each of critical points. (d) By choosing some eigenvalues which +have negative numeric values (if are real) or real part of them are nega- +tive (if they are complex numbers) he/she should solve alternative equations +˙Xi = Σn +j=1JijXj instead of the original equations ˙Xi = F(Xi, t). In fact the +obtained solutions are valid near the stable critical points (see [15] or [16] for +more discussion). +We now continue to use these 4 steps for the equations given in the previous +section. In this case ˙X = ˙Y = ˙ψ = 0 give us two different critical points as +(1) : +{ψc = 0, +Xc = 0, +Yc = 1} +(3.1) +(2) : +{ψc = 1 +2, +Xc = 0, +Yc = 1 − 1 +4β}. +At these critical points one can calculate the Jacobi matrix as follows. +J(1) +ij = ∂ ˙Xi +∂Xj += + + +0 +0 +0 +0 +−2 +0 +0 +0 +−1 + + +(3.2) +and +J(2) +ij = + + + +−(7β+2) +4β +0 +−1 +0 +1 +2β +0 +(1−4β) +4β2 +0 +(1−4β) +4β + + + . +(3.3) +5 + +To obtain eigenvalues E of the above Jacobi matrixes we should solve the +secular equation det(J(1,2) +ij +− Eδij) = 0 which reads +E(1) +1 += 0, +E(1) +2 += −2, +E(3) +3 += −1 +(3.4) +for the first critical point (1) and +E(2) +1 += 1 +4β, +E(2) +2 += −(1 + 11β) + 3 +� +(β − β+)(β − β−) +8β +(3.5) +E(2) +3 += −(1 + 11β) − 3 +� +(β − β+)(β − β−) +8β +where we defined +β± = −41 ± 4 +√ +109 +9 +. +(3.6) +By looking at these critical points one can infer that the first critical point (1) +is quasi stable because all three eigenvalues for it are not negative numbers. +The eigenvalues for the second critical point (2) are parametric. For β < 0 +the first eigenvalue is negative number E(2) +1 +< 0 and for β > 0 the first part +in the second and third eigenvalues are negative umbers E(2) +2,3 < 0 and for +β− < β < β+ in which β− = −9.1956 and β+ = 0.0844 the square roots +in these eigenvalues become imaginary and so nature of the system is spiral +stable. We give out numeric values for the eigenvalues (3.5) with describing +whose stability nature in the table 1. To obtain metric solutions near the +critical point (2) one should solve alternative equations instead of the original +equations (2.8),(2.9) and (2.10) such that +d +dτ + + +ψ +X +Y + + = + + + +−(7β+2) +4β +0 +−1 +0 +1 +2β +0 +(1−4β) +4β2 +0 +(1−4β) +4β + + + + + +ψ +X +Y + + +(3.7) +which have solutions with the following forms. +ψ(τ) = ψ+eE(2) +2 +τ + ψ−eE(2) +3 +τ +(3.8) +X(τ) = X0e(τ/2β) +(3.9) +and +Y (τ) = Y+eE(2) +2 +τ + Y−eE(2) +3 +τ +(3.10) +6 + +in which ψ±, Y± and X(0) = Xc are constants and should be fix by initial +conditions of the system. We collected numeric values for the eigenvalues +together with stability nature in the table 1. This metric solution is not for +a black hole because the horizon equations X(τ) = 0 and Y −1(τ) = 0 give +not a finite radius for position of horizon. In other words this metric solution +describes line element of internal region of an electrostatic star. +4 +Concluding remarks +In this work we added a nonminimal directionally interaction Lagrangian +between geometry and the electromagnetic vector potential for Einstein- +Maxwell gravity and investigate this additional contribution on a spherically +symmetric static space time of a stellar compact object. After to solve the +Euler-Lagrange equations of the fields via dynamical systems approach, we +were determine stabilization conditions of the obtained solutions near para- +metric critical points in phase space. We obtained that for negative values +of the coupling constant of the interaction part of the Lagrangian density +the solutions take on stable nature but not for positive coupling constant. +As extension of this work we like investigate magnetic fields effects of the +stellar matter on stabilization of the stellar compact object. In our previ- +ous work [14] we investigated effects of magnetic monopoles on stability of a +spherically symmetric stellar compact object for a modified gauge invariance +Einstein-Maxwell gravity. Same as the one, we checked this effect again for +the present work but we obtained a problem regretfully where the interaction +parameter must be complex (imaginary) which is not physical. This comes +from integral of the action functional for the polar coordinate θ which must +be used the Cauchy‘s integral residue. In fact the integrand is singular on +the poles θ = 0, π. Hence we like to investigate effects of magnetic multipoles +on stabilization of cylindrically symmetric stellar compact objects for the +present gravity model in our future work. Also we will seek that did stable +the stellar systems with these magnetic multipole sources? +7 + +Table 1: Numeric values of Jacobi matrix eigenvalues vs β. Here, ‘S‘ means +‘stable‘, ‘S.S‘, means ‘spiral stable‘, ‘Indet.‘ means ‘indeterminate‘ and +‘Q.S‘, means ‘quasi stable‘ +β +E1 +E2 +E3 +nature +-15. +-0.0166667 +-360.134 +-254.866 +S +-14. +-0.0178571 +-310.936 +-224.564 +S +-13. +-0.0192308 +-265.145 +-196.355 +S +-12. +-0.0208333 +-222.696 +-170.304 +S +-11. +-0.0227273 +-183.448 +-146.552 +S +-10. +-0.025 +-146.93 +-125.57 +S +-9. +-0.0277778 +-110.25 -4.5 I +-110.25 + 4.5 I +S.S +-8. +-0.03125 +-87. - 9.32738 I +-87. + 9.32738 I +S.S +-7. +-0.0357143 +-66.5 - 10.3531 I +-66.5 + 10.3531 I +S.S +-6. +-0.0416667 +-48.75 - 9.92157 I +-48.75 + 9.92157 I +S.S +-5. +-0.05 +-33.75 - 8.66025 I +-33.75 + 8.66025 I +S.S +-4. +-0.0625 +-21.5 - 6.91014 I +-21.5 + 6.91014 I +S.S +-3. +-0.0833333 +-12. - 4.91808 I +-12. + 4.91808 I +S.S +-2. +-0.125 +-5.25 - 2.90474 I +-5.25 + 2.90474 I +S.S +-1. +-0.25 +-1.25 - 1.11803 I +-1.25 + 1.11803 I +S.S +0. +Complex Infinity +0. +0. +Indet. +1. +0.25 +-0.354356 +-2.64564 +Q.S +2. +0.125 +-2.27689 +-9.22311 +Q.S +3. +0.0833333 +-6.0418 +-19.4582 +Q.S +4. +0.0625 +-11.7181 +-33.2819 +Q.S +5. +0.05 +-19.3375 +-50.6625 +Q.S +6. +0.0416667 +-28.9178 +-71.5822 +Q.S +7. +0.0357143 +-40.4696 +-96.0304 +Q.S +8. +0.03125 +-54. +-124. +Q.S +9. +0.0277778 +-69.5138 +-155.486 +Q.S +10. +0.025 +-87.0145 +-190.486 +Q.S +11. +0.0227273 +-106.505 +-228.995 +Q.S +12. +0.0208333 +-127.986 +-271.014 +Q.S +13. +0.0192308 +-151.46 +-316.54 +Q.S +14. +0.0178571 +-176.928 +-365.572 +Q.S +15. +0.0166667 +-204.391 +-418.109 +Q.S +8 + +References +[1] F. 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Ge, Z. Li, Y. Men, R. Xu, ‘The opti- +cal/UV excess of X-ray dim isolated neutron star: I. bremsstrahlung +emission from a strangeon star plasma atmosphere‘,ApJ 837 81 (2017), +arXiv:1603.08288 [astro-ph.HE] +[13] M. S. Turner and L. M. Widrow, ‘Inflation Produced, Large Scale Mag- +netic Fields‘, Phys.Rev. D 37, 2743 (1988). +[14] H. Ghaffarnejad and L. Naderi, ‘Modified Gauge Invariance Ein- +stein Maxwell Gravity and Stability of Spherical Stars with Magnetic +Monopoles ‘, arXiv:2212.09485 [gr-qc] +[15] H. +Ghaffarnejad, +E. +Yaraie,‘Dynamical +system +approach +to +scalar-vector-tensor cosmology‘,Gen Relativ Gravit 49, +49 (2017); +arXiv:1604.06269 [physics.gen-ph] +[16] H. Ghaffarnejad and H. Gholipour, ‘Bianchi I metric solutions with +nonminimally coupled Einstein-Maxwell gravity theory‘, Gen. Relativ. +Gravit 53, 1 (2021); arXiv:2003.14216 [gr-qc] +10 + +(a) +(b) +(c) +(d) +(e) +(f) +Figure 1: Arrow diagrams for the critical point (3.1) with β < 0 describing stable (sink) +nature +11 + diff --git a/ktAyT4oBgHgl3EQfyPnx/content/tmp_files/load_file.txt b/ktAyT4oBgHgl3EQfyPnx/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1e9123ecc48b7319d09326061edcec67313bbcc3 --- /dev/null +++ b/ktAyT4oBgHgl3EQfyPnx/content/tmp_files/load_file.txt @@ -0,0 +1,400 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf,len=399 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content='00682v1 [gr-qc] 27 Dec 2022 On the stability of electrostatics stars with modified non-gauge invariance, Einstein-Maxwell gravity H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' Ghaffarnejad1, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' Ghorbani 2 and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' Eidizadeh 3 Faculty of Physics, Semnan University, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' 35131-19111, Semnan, Iran Abstract We use modified Einstein-Maxwell gravity where correction parts are nonminimally directional coupling between the Maxwell vector potential field and Ricci tensor and Ricci scalar fields to produce La- grangian density of the fields for a spherically symmetric static curved metric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' Euler-Lagrange equations are nonlinear second order dif- ferential equations and have not an analytic closed form of the solu- tions and so we must be solve these via dynamical systems approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' By regarding this method we obtained some stable solutions near the critical points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' To determine stable (unstable) nature for the obtained solutions we must solve secular equation of the Jacobi matrix of the field equations and determine sign of the eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' 1 Introduction To describe the stability of a stellar compact object, in usual way, it is nec- essary to consider the Tolman-Oppenheimer-Volkoff equations [1] and the equation of state of the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' Stability criteria of relativistic spherical sym- metric compact objects with isotropic pressure in the framework of general relativity include boundary conditions, non-singularity, electric charge, sur- face redshift, energy conditions, the speed of sound in causal conditions and relativistic adiabatic index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' In a stable model, the energy and pressure densi- ties are finite at the center of compact object and decrease uniformly toward the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' The metric potentials are regular and the electric field inten- sity is zero at the center and increases towards the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' In addition, the gravitational redshift follows Zs < 2 and four energy conditions are satisfied, the speed of sound is less than the speed of light and decreases uniformly 1E-mail address: hghafarnejad@semnan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content='ir 2E-mail address: tohidghobani@semnan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content='ir 3E-mail address: f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content='eidizadeh@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content='com 1 toward the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' In addition, the adiabatic index is strongly higher than 4 3 [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' Relativistic Compact object with gravity and strong internal density have two different pressures, radial and tangential [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' The stability of a stellar model can be increased by an anisotropic repulsive force that ∆ = pt−pr > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' This property leads to more compact stable configurations compared to the states of isotropic [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' Hydrostatic equilibrium of solutions of anisotropic relativistic stars in scale-dependent gravity, where Newton’s constant is al- lowed to vary with radial coordinates across the star, shows that a decrease in Newton’s constant across objects leads to slightly more massive and com- pact stars [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' A stability analysis for Einstein-Kline-Gordon model with static real scalar field interaction express that the initial value of the field at the origin is a function of the energy density of the matter at the origin and in the far regions the field behaves Yukawa-like potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' Such a model for compact stellar object is stable if the gradient of the total mass versus energy density is positive and the weak energy condition is satisfied (positive total density) [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' The stability of the star can be investigated in the pres- ence of both electric and magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' Solving the Einstein-Maxwell field equations for compact objects with the charged anisotropic fluid model gives more stable solutions than for neutral stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' The presence of charges creates a repulsive force against the gravitational force, and this factor causes denser stable stars, higher maximum mass and larger redshift [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' Charged quarks can create more stable quark stars than neutron nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' Also, for a white dwarf with a charged perfect fluid, there is a direct correlation between the increase in electric charge and its size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' Near the surface of the star, the radial pressure is close to zero and the electric charge density is non-zero, leading to a stable star with more mass [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' The mass-radius relation of some kinds of Neutron stars, which can contain a core of quark matter, have a large frequency range of radial fluctuations near the transition point in their core versus mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' These induce nonlinear general relativistic effects which cause to be the stars unstable dynamically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' The core of the Neutron stars becomes several times larger, making the Neutron stars highly unstable [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' While for the charged boson-fermion stars with a charged fluid related to fermion and a complex scalar field related to boson, the charge increase can reduce the stellar radius and create a denser and more massive star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' In the whole pa- rameter space, the critical curve can show stable and unstable regions [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' If the number of baryons in compact pulsar-like stars exceeds the critical value 109, the strangeon star model is proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' In fact the strangeon star atmosphere model describes the radiation from interstellar medium accreted 2 plasma atmosphere on a strangeon star surface and its spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' This ob- ject could simply be regarded as the upper layer of a normal neutron star because the radiation from strangeon matter can be neglected [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' The atmosphere is in radiative, thermal equilibrium and two-temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' The strangeon star spectrum is based on bremsstrahlung from an extremely thin hydrogen plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' More details of this model are described in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' Since the extra strange flavor provides more degrees of freedom to lower the Fermi en- ergy in the free quark approximation, macroscopic bulk strong matter with 3-flavor symmetry (up, down, and strange quarks) is more stable than up quark matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' The difference in the strangeness level between a strange star and a typical neutron star can have a profound effect on the magnetospheres activity associated with the coherent radio emission of the Compact stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' Content of this paper is as follows: In section 2 we describe shortly generalized Einstein Maxwell gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' Then we obtain Lagrangian form of the model for a general spherically symmetric static metric and corresponding Euler-Lagrange equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' These are non- linear second order differential equations and so we use dynamical systems approach to solve them in the section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' In the latter section we obtain alter- native first order differential equations in phase space and calculate Jacobi matrix of the equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' We determine eigenvalues of the Jacobi matrix by solving the secular equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' At last we determine sign of the eigenvalues for which the system has stable natures for some suitable numeric values of the parameter of the coupling constant of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' They are collected in a table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' Section 4 dedicated to concluding remarks and outlook of the work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' 2 The gravity model As a generalization of the Einstein-Maxwell gravity theory we consider non- minimally coupling between the Maxwell vector potential Aµ and the Recci scalar and the Ricci tensor such that [13] I = − � dx4√g �1 4FµνF µν + α 2 A2R + β 2 RµνAµAν � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content='1) where g is absolute value of determinant of the metric field and anti symmet- ric electromagnetic tensor field Fµν is defined versus the partial derivatives of the four vector electromagnetic potential Aµ as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' Fµν = ∇µAν − ∇νAµ = ∂µAν − ∂νAµ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content='2) 3 with A2 = gµνAµAν and Rµν is Ricci tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' It is easy to check that this model has not gauge invariance symmetry same as [14] in which the ac- tion functional remain unchanged by transforming Aµ → Aµ + ∂µξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' In this transformation ξ is gauge field for which Fµν → Fµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' In this work we like to investigate effects of the electromagnetic fields on metric field solutions of a spherically symmetric stellar object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' For spherically symmetric time- independent static metric ds2 = −X(r)dt2 + Y (r)dr2 + r2(dθ2 + sin2 θdϕ2) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content='3) we know that non-vanishing components of the vector potential is just At(r) = φ(r) means electric potential for a electrostatics spherical stellar object which by substituting into the action function (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content='1) we obtain exact form for the Lagrangian density of the fields such that L = 2π √ XY � ˙φ2 + βφ2 � ˙Y Y + Y − 1 �� (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content='4) where we defined ˙ as logarithmic derivative of the radial coordinate as ˙ = d dτ = r d dr = d d ln(r/D) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content='5) in which D is a suitable length parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' Also we use the ansatz α = −β 2 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content='6) to remove second order derivatives of the X field coming from the Ricci scalar and Ricci tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' By defining an new field ψ(r) such that ˙φ = ψφ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content='7) one can show that the Euler-Lagrange equations of the fields φ(r), X(r), Y (r) are obtained after some simple mathematical calculations respectively as fol- lows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' ˙ψ = −ψ2 β (4βY + βψ + 3ψ2 − 3β) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content='8) ˙X = −X β (7βY − 4βψ + 5ψ2 − 5β) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content='9) 4 and ˙Y = −Y β (βY + ψ2 − β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content='10) The above equations are non-linear first order differential equations not hav- ing exact analytic solutions and so we can solve via dynamical systems ap- proach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' This is done in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' 3 Metric solutions General strategy in the dynamical system approach to obtain analytic solu- tions for the fields equations ˙Xi = F(Xi, t) have 4 different steps as follows: (a) By solving the equations ˙Xi = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' i = 1, 2, 3, · · ·n for n dimensional phase space of the system one determine the critical points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' (b) For each of these critical points he/she should calculate Jacobi matrix Jij = ∂ ˙Xi ∂Xj and (c) then solve the corresponding secular equation det(Jij − Eδij) = 0 to obtain eigen- values for each of critical points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' (d) By choosing some eigenvalues which have negative numeric values (if are real) or real part of them are nega- tive (if they are complex numbers) he/she should solve alternative equations ˙Xi = Σn j=1JijXj instead of the original equations ˙Xi = F(Xi, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' In fact the obtained solutions are valid near the stable critical points (see [15] or [16] for more discussion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' We now continue to use these 4 steps for the equations given in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' In this case ˙X = ˙Y = ˙ψ = 0 give us two different critical points as (1) : {ψc = 0, Xc = 0, Yc = 1} (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content='1) (2) : {ψc = 1 2, Xc = 0, Yc = 1 − 1 4β}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' At these critical points one can calculate the Jacobi matrix as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' J(1) ij = ∂ ˙Xi ∂Xj = \uf8eb \uf8ed 0 0 0 0 −2 0 0 0 −1 \uf8f6 \uf8f8 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content='2) and J(2) ij = \uf8eb \uf8ec \uf8ed −(7β+2) 4β 0 −1 0 1 2β 0 (1−4β) 4β2 0 (1−4β) 4β \uf8f6 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content='3) 5 To obtain eigenvalues E of the above Jacobi matrixes we should solve the secular equation det(J(1,2) ij − Eδij) = 0 which reads E(1) 1 = 0, E(1) 2 = −2, E(3) 3 = −1 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content='4) for the first critical point (1) and E(2) 1 = 1 4β, E(2) 2 = −(1 + 11β) + 3 � (β − β+)(β − β−) 8β (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content='5) E(2) 3 = −(1 + 11β) − 3 � (β − β+)(β − β−) 8β where we defined β± = −41 ± 4 √ 109 9 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content='6) By looking at these critical points one can infer that the first critical point (1) is quasi stable because all three eigenvalues for it are not negative numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' The eigenvalues for the second critical point (2) are parametric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' For β < 0 the first eigenvalue is negative number E(2) 1 < 0 and for β > 0 the first part in the second and third eigenvalues are negative umbers E(2) 2,3 < 0 and for β− < β < β+ in which β− = −9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content='1956 and β+ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content='0844 the square roots in these eigenvalues become imaginary and so nature of the system is spiral stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' We give out numeric values for the eigenvalues (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content='5) with describing whose stability nature in the table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' To obtain metric solutions near the critical point (2) one should solve alternative equations instead of the original equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content='8),(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content='9) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content='10) such that d dτ \uf8eb \uf8ed ψ X Y \uf8f6 \uf8f8 = \uf8eb \uf8ec \uf8ed −(7β+2) 4β 0 −1 0 1 2β 0 (1−4β) 4β2 0 (1−4β) 4β \uf8f6 \uf8f7 \uf8f8 \uf8eb \uf8ed ψ X Y \uf8f6 \uf8f8 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content='7) which have solutions with the following forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' ψ(τ) = ψ+eE(2) 2 τ + ψ−eE(2) 3 τ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content='8) X(τ) = X0e(τ/2β) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content='9) and Y (τ) = Y+eE(2) 2 τ + Y−eE(2) 3 τ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content='10) 6 in which ψ±, Y± and X(0) = Xc are constants and should be fix by initial conditions of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' We collected numeric values for the eigenvalues together with stability nature in the table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' This metric solution is not for a black hole because the horizon equations X(τ) = 0 and Y −1(τ) = 0 give not a finite radius for position of horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' In other words this metric solution describes line element of internal region of an electrostatic star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' 4 Concluding remarks In this work we added a nonminimal directionally interaction Lagrangian between geometry and the electromagnetic vector potential for Einstein- Maxwell gravity and investigate this additional contribution on a spherically symmetric static space time of a stellar compact object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' After to solve the Euler-Lagrange equations of the fields via dynamical systems approach, we were determine stabilization conditions of the obtained solutions near para- metric critical points in phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' We obtained that for negative values of the coupling constant of the interaction part of the Lagrangian density the solutions take on stable nature but not for positive coupling constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' As extension of this work we like investigate magnetic fields effects of the stellar matter on stabilization of the stellar compact object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' In our previ- ous work [14] we investigated effects of magnetic monopoles on stability of a spherically symmetric stellar compact object for a modified gauge invariance Einstein-Maxwell gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' Same as the one, we checked this effect again for the present work but we obtained a problem regretfully where the interaction parameter must be complex (imaginary) which is not physical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' This comes from integral of the action functional for the polar coordinate θ which must be used the Cauchy‘s integral residue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' In fact the integrand is singular on the poles θ = 0, π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' Hence we like to investigate effects of magnetic multipoles on stabilization of cylindrically symmetric stellar compact objects for the present gravity model in our future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' Also we will seek that did stable the stellar systems with these magnetic multipole sources?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' 7 Table 1: Numeric values of Jacobi matrix eigenvalues vs β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' Here, ‘S‘ means ‘stable‘, ‘S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content='S‘, means ‘spiral stable‘, ‘Indet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content='‘ means ‘indeterminate‘ and ‘Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content='S‘, means ‘quasi stable‘ β E1 E2 E3 nature 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content='0166667 360.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content='134 254.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content='866 S 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' Naderi, ‘Modified Gauge Invariance Ein- stein Maxwell Gravity and Stability of Spherical Stars with Magnetic Monopoles ‘, arXiv:2212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content='09485 [gr-qc] [15] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' Ghaffarnejad, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' Yaraie,‘Dynamical system approach to scalar-vector-tensor cosmology‘,Gen Relativ Gravit 49, 49 (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' arXiv:1604.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content='06269 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content='gen-ph] [16] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' Ghaffarnejad and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' Gholipour, ‘Bianchi I metric solutions with nonminimally coupled Einstein-Maxwell gravity theory‘, Gen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' Relativ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' Gravit 53, 1 (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content=' arXiv:2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content='14216 [gr-qc] 10 (a) (b) (c) (d) (e) (f) Figure 1: Arrow diagrams for the critical point (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} +page_content='1) with β < 0 describing stable (sink) nature 11' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktAyT4oBgHgl3EQfyPnx/content/2301.00682v1.pdf'} diff --git a/ldE3T4oBgHgl3EQfiAoP/content/tmp_files/2301.04575v1.pdf.txt b/ldE3T4oBgHgl3EQfiAoP/content/tmp_files/2301.04575v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..642e725d704bf3889f293db2fc90bca1baa5d609 --- /dev/null +++ b/ldE3T4oBgHgl3EQfiAoP/content/tmp_files/2301.04575v1.pdf.txt @@ -0,0 +1,1698 @@ +Wilson lines construction of sl3 toroidal conformal +blocks +Vladimir Belavin, Pietro Oreglia, J. Ramos Cabezas +Physics Department, Ariel University, Ariel 40700, Israel. +E-mail: vladimirbe@ariel.ac.il, pietro.oreglia@msmail.ariel.ac.il, +juanra@ariel.ac.il +Abstract: We study W3 toroidal conformal blocks for degenerate primary fields in AdS/CFT +context. In the large central charge limit W3 algebra reduces to sl3 algebra and sl3 blocks +are defined as contributions to W3 blocks coming from the generators of sl3 subalgebra. We +consider the construction of sl3 toroidal blocks in terms of Wilson lines operators of 3d Chern- +Simons gravity in the thermal AdS3 space-time. According to the correspondence, degenerate +primary fields are associated with Wilson lines operators acting in the corresponding finite- +dimensional sl3 representations. We verify this dual construction for one-point toroidal block +using sl3 tensor technique in the bulk theory and an algorithm based on AGT correspondence +in the boundary CFT. +arXiv:2301.04575v1 [hep-th] 11 Jan 2023 + +Contents +1 +Introduction +1 +2 +Preliminaries: AdS3 and W3 CFT +4 +2.1 +W3 Conformal Field Theory +4 +2.2 +Brief review of 3d Chern-Simons gravity theory +5 +3 +One-point conformal block from CFT +7 +4 +One-point conformal blocks from AGT +8 +5 +Conformal Blocks through Wilson line operators +9 +5.1 +Computation of the Wilson line operators +11 +5.1.1 +Zero-point conformal block +13 +5.1.2 +One-point conformal block +14 +6 +Discussion +16 +A Matrix Elements +16 +B Weyl character formula +19 +C AGT relation +20 +D Examples +21 +1 +Introduction +Conformal blocks (CB’s) determine the holomorphic contributions to the correlation functions +that appear after fixing the OPE channels [1]. CB’s are generally fixed by the symmetry +algebra and depend on the topology of the CFT Riemann surface. +In this work, we are +interested in W3 one-point conformal block on the torus in the large central charge limit. +The one-point correlation function of a primary field Φα1 with conformal dimension h1 +on the torus is defined as1 +⟨Φα1(z1, ¯z1)⟩ = Trα +� +qL0 ¯q +¯L0Φα1(z1, ¯z1) +� += +� +α +Cαα1α|F(α, α1, q)|2 , +(1.1) +1Throughout this paper we omit factor (q¯q)− c +24 which can be easily restored. +– 1 – + +where Trα is the trace taken over a module of the symmetry algebra associated with the +primary field Φα in the intermediate OPE channel, q is the elliptic parameter of the torus +q = e2πiτ and L0 is the generator of the algebra satisfying L0 |h1⟩ = h1 |h1⟩. Here F(α, α1, q) +is the one-point holomorphic toroidal conformal block (for more details, see, e.g., [2]). +In recent years, the AdS3/CFT2 provided a new formulation of conformal blocks in +terms of geodesic networks in AdS space-time, see e.g., [3–16], or Wilson network operators +of Chern-Simons gravity, which will be the subject of the present consideration.2 +The latter approach is based on the higher-spin version of AdS/CFT correspondence [36], +which, in general, identifies the minimal model cosets +SU(N)k ⊕ SU(N)1 +SU(N)k+1 +(1.2) +in ’t Hooft limit, as the holographic dual of the higher spin theory [37]. +Here we study +a different limit, the so-called semiclassical limit, where we have, from the bulk side, an +SL(N) × SL(N) Chern-Simons theory, while from the CFT side, a non-unitary WN CFT +model considered in the large central charge limit, with N fixed. We recall that primary +fields in the WN model are labelled by a pair of SL(N) highest-weights (Λ+, Λ−), which are +both highest-weights of a finite-dimensional representation of SL(N) [38]. One can identify +two different kinds of primaries fields: +• heavy operators, identified by (0, Λ−), which have scale dimensions ∆ ∼ c, and corre- +spond to flat SL(N) × SL(N) connections in the bulk; +• light operators, labelled by (Λ+, 0), whose scale dimensions go as ∆ ∼ o(1), and are +related to perturbative matter in the bulk. +In this paper we focus on the light operators. This allows us to consider the large c limit by +restricting the set of generators, which in general can be written as {Ln, W 3 +n, . . . , W N +n }, to +those of the types +Ln for |n| < 2 , +W s +n for |n| < s , +(1.3) +with s = 3, . . . , N. By looking at the commutation relations (for explicit form in N = 3 +case, see sec. 2), it can be seen that these operators can be identified as the generators of +the slN algebra. In the spherical topology (for N = 2, 3) this program has been implemented +in [39]. Regarding the toroidal topology, it was shown in [40] that sl2 one-point toroidal block +F(−j, −j1, q)sl2, where −j and −j1 are respectively the intermediate and external conformal +dimensions, is given by +F(−j, −j1, q)sl2 = Trj +� +Wj[zb, zb + 2πτ]Ij;j,j1 +� +⊗ Wj1[zb, z1] |lw⟩1 . +(1.4) +In the dual description j and j1 are the spins of sl2 Chern-Simons gauge group. The trace +Trj(· · · ) is taken over the representation with spin j, z1 is the point on the boundary of the +2For recent development on CB’s in the holography context, see, e.g., [17–35]. +– 2 – + +solid torus, which is a geometric representation of the thermal AdS3, zb is an arbitrary point +in the bulk of AdS3, |lw⟩1 is the lowest-weight state of the representation with spin j1 and +Ij;j,j1 is the intertwining operator associated with the representations j and j1. The factors +Wi[x, y] (for i = j, j1; x = zb and y = zb + 2πτ, z1) denote the Wilson line operators +Wi[x, y] = P exp +� +− +� y +x +Ω +� += exp +� +(x − y)(L1 + 1 +4L−1) +� +. +(1.5) +Here L1 and L−1 are the lowering and raising operators of sl2 algebra in the representation +with spin i. +In this work we consider the W3 algebra; we are interested in fully degenerate primary +fields Φα(z, ¯z), characterized by a pair of quantum numbers (hα, qα) (conformal dimension +and W3 charge, respectively).3 +In the semiclassical limit c → ∞, or b → 0 in Toda-like +parameterization [41], c = 2+24(b+1/b)2, the parameter α corresponding to the intermediate +primary field Φα in (1.1) is given by +α → −bj , +j = m1w1 + m2w2 , +(1.6) +where w1, w2 are sl3 fundamental weights and m1, m2 are non-negative integers. For the +external primary Φα1 characterized by (hα1, qα1) +α1 → −bj1 , +j1 = aw1 , +(1.7) +where a is a non-negative integer. The external field Φα1 is restricted to the so-called semi- +degenerate form, to avoid the multiplicities problem of conformal blocks in W3 CFT, see, +e.g., [41]. +In what follows, we focus on the one-point blocks with external fields satisfying (1.7), +which can be equivalently written as +W−1Φα1(0, 0) |0⟩ = 3q1 +2h1 +L−1Φα1(0, 0) |0⟩ . +(1.8) +The outline of the paper is the following. In section 2, we recall the necessary facts about +W3 CFT and AdS3 Chern-Simons gravity. In sections 3 and 4, we compute F(α, α1, q)sl3 +and check it by comparing with the large central charge limit of W3 block of light operators, +using the algorithm based on AGT relation. +To this end we propose a relation between +the light W3 and sl3 one-point blocks, which is similar to the one existing between sl2 and +Virasoro light blocks in the large central charge limit [42]. In section 5, we describe the dual +construction for sl3 one-point block in terms of the Wilson lines operators. We find that the +expression obtained in section 3 can be represented by the lhs of (1.4). Our conclusions are +collected in section 6. Appendices A, B, C, D contain some technical details related to sl3 +matrix elements, sl3 Weyl character formula, AGT relation, and Wilson lines description of +conformal blocks respectively. +3Explicit form of (hα, qα) in terms of α will be given below. +– 3 – + +2 +Preliminaries: AdS3 and W3 CFT +2.1 +W3 Conformal Field Theory +The symmetry of the W3 CFT is generated by the energy-momentum tensor T(z) (a spin-2 +current) and the additional spin-3 current W(z). Their expansions as Laurent series read +T(z) = +∞ +� +n=−∞ +Ln +zn+2 , +W(z) = +∞ +� +n=−∞ +Wn +zn+3 . +(2.1) +The modes Ln and Wm generate the W3 algebra, that is, they satisfy the commutation +relations +[Ln, Lm] = (n − m)Ln+m + c +12(n3 − n)δn+m,0 , +[Ln, Wm] = (2n − m)Wn+m , +[Wn, Wm] = +c +3 · 5!(n2 − 1)(n2 − 4)nδn+m,0 + +16 +22 + 5c(n − m)Λn+m+ ++ (n − m) +30 +� +2m2 + 2n2 − mn − 8 +� +Ln+m , +(2.2) +where +Λm = +� +p ⩽ −2 +LpLm−p + +� +p ⩾ −1 +Lm−pLp − 3(m + 2)(m + 3) +10 +Lm . +(2.3) +In the limit c → ∞ these commutation relations reduce to the ones of the sl3 algebra, +generated by +{L−1, L0, L1, W−1, W1, W0, W−2, W2} , +(2.4) +that satisfy +[Ln, Lm] = (n − m)Ln+m , +[Ln, Wm] = (2n − m)Wn+m , +[Wn, Wm] = (n − m) +� 1 +15(n + m + 2)(n + m + 3) − 1 +6(n + 2)(m + 2) +� +Ln+m . +(2.5) +W3 primary fields. +The conformal dimension hα and charge qα of a primary field Φα in a +W3 CFT are expressed as +hα = 1 +2(α, 2Q − α) , +qα = i +� +48 +22 + 5c +3 +� +i=1 +(ei, α − Q) , +(2.6) +Here Q = (b + 1 +b)(w1 + w2), ei are the weights of the fundamental representation +e1 = w1 , +e2 = w2 − w1 , +e3 = −w2 , +(2.7) +and α is a vector on the root space given by +αr1r2s1s2 = b +� +(1 − r1)w1 + (1 − r2)w2 +� ++ 1 +b +� +(1 − s1)w1 + (1 − s2)w2 +� +, +(2.8) +– 4 – + +with r1, r2, s1 and s2 positive integers. In the large central charge limit, we have that (2.8) +becomes +α → −bj, +j = m1w1 + m2w2 . +(2.9) +As explained before, we are mostly interested in semi-degenerate fields (1.7) in the large +central charge limit, for which the conformal dimension and the charge assume the values +hα = −m1 − m2, +qα = i +3 +� +2 +5(m2 − m1) . +(2.10) +W3 module. +A W3 highest-weight vector |hα, qα⟩ given by +|hα, qα⟩ = lim +z→0 Φα(z) |0⟩ , +(2.11) +satisfies the conditions +L0 |hα, qα⟩ = hα |hα, qα⟩ , +W0 |hα, qα⟩ = qα |hα, qα⟩ , +(2.12) +Ln |hα, qα⟩ = Wn |hα, qα⟩ = 0 , +n > 0 . +(2.13) +The W3 module associated with this highest-weight vector is spanned by the basis of descen- +dant states +L−I |hα, qα⟩ = L−i1 . . . L−imW−j1 . . . W−jn |hα, qα⟩ , +I = {i1, . . . , im; j1, . . . , jn} , +(2.14) +with +1 ≤ i1 ≤ · · · ≤ im , +1 ≤ j1 ≤ · · · ≤ jn . +(2.15) +The sum +N = +� +ia,jb∈I +ia + jb +(2.16) +is called level of the state. Similarly, the sl3 module is defined when in the set I, the indices +are restricted to +i1, . . . , im = 1 , +j1, . . . , jn = 1, 2 , +(2.17) +that is, when descendants are generated only by L−1, W−1 and W−2. For the basis states +of W3 and sl3 modules, we introduce the following convenient notation +L−I |hα, qα⟩ = |hα, qα, N⟩ . +(2.18) +2.2 +Brief review of 3d Chern-Simons gravity theory +It is known [43] that the 2 + 1-dimensional Einstein-Hilbert action S with a negative cosmo- +logical constant can be written in terms of the Chern-Simons action +S = SCS[A] − SCS[ ¯A] , +(2.19) +– 5 – + +where +SCS[A] = k +4π +� +M3 Tr +� +A ∧ dA + 2 +3A ∧ A ∧ A +� +. +(2.20) +The constant k is related to the 3-dimensional Newton constant G3 as +k = +r +4G3 +, +(2.21) +with r the AdS3 radius, Tr stands for the invariant Killing form, M3 is the 3-dimensional +space-time and A ( ¯A) is the (anti-) chiral gauge connection, that is, a one-form valued in +the gauge group, given by a composition of vielbein and spin-connection. In the case we are +interested in, that is the SL(N) × SL(N) Chern-Simons theory, A is an SL(N) connection. +The equations of motion that follow from (2.20) are given by (the same equations are valid +for the anti-chiral connection) +dA + A ∧ A = 0 , +(2.22) +which are the flatness conditions on the connection. It can be shown [44] that, by choosing +the proper boundary conditions, in the local coordinates xµ = (ρ, z, ¯z) with ρ ≥ 0 the radial +one, the solution of (2.22) can be written as the gauge-transformed Ω: +A = U −1ΩU + U −1dU , +(2.23) +with +Ω = +� +L1 − 2π6T(z) +c +L−1 +� +dz , +U = eρL0 . +(2.24) +Here T(z) is the holomorphic boundary energy-momentum tensor, while c is the central +charge, defined through the Brown-Henneaux relation c = +3r +2G3 [45]. L1, L−1 are two of the +generators of the sl2 subalgebra of the gauge algebra slN. We are interested in the space-time +with periodic time conditions, that is to say, the thermal AdS3, in which case the stress-energy +tensor is +T(z) = − c +48π , +(2.25) +and the connection Ω becomes +Ω = +� +L1 + 1 +4L−1 +� +dz , +(2.26) +together with the identifications z ∼ z + 2π and z ∼ z + 2πτ, with iτ ∈ R<0. +Finally, we define the Wilson line operators. Given a chiral connection A defined in a +representation R of slN and a path l that connects two points x1, x2 ∈ M3, we define the +Wilson line operator WR[l] as +WR[l] = P exp +� � +l +A +� +, +(2.27) +where P denotes the path-ordering operator. +– 6 – + +3 +One-point conformal block from CFT +In this section, we define the holomorphic W3 and sl3 one-point conformal blocks. The result +of the calculation of the W3 block will be given in the next section. The holomorphic W3 +one-point conformal block F(α, α1, q) is given by +F(α, α1, q) = +1 +⟨h0, q0| Φα1(z) |h0, q0⟩ +� +M,N=0 +M=N +⟨h0, q0, M| qL0Φα1(z) |h0, q0, N⟩ G−1 +MN , +(3.1) +where hα := h0, qα := q0, the states |h0, q0, N⟩ belong to the W3 module (2.14), and G−1 +MN is +the inverse of the Shapavalov matrix +GMN = ⟨h0, q0, M|h0, q0, N⟩ . +(3.2) +For general Φα1, the matrix elements +⟨h0, q0, M| qL0Φα1(z) |h0, q0, N⟩ +(3.3) +are not defined uniquely. For this reason, we focus on the case where the external field Φα1(z) +satisfies (1.8), from which it follows that +− h2 +1 +5 = 9q2 +1 +2 +. +(3.4) +This constraint is satisfied when the vector j1 corresponding to the field Φα1 is given by +j1 = aw1 or j1 = aw2. Without loss of generality, we choose j1 = aw1, which implies +h1 = −a , +q1 = − i +3 +� +2 +5a . +(3.5) +Similarly, the holomorphic sl3 one-point conformal block F(α, α1, q)sl3 is defined by (3.1), +but now the states |h0, q0, N⟩ belong to the sl3 module, obtained by restricting the set I (2.14) +to the conditions (2.17). For the fields Φα, Φα1 with h’s and q’s given by (2.10) and (3.5) +respectively, we compute the contribution to the sl3 one-point conformal block up to the +second level. The matrix elements involved in this computation are listed in appendix A. +Here we present the result +F(α, α1, q)sl3 = 1 + +�18m1m2 − a2(m1 + m2) − 3a(m1 + m2) +9m1m2 +� +q + ++ +1 +162 +�(a + 6)(a + 3)a(a − 3) +m2 − 1 ++ ++ +� +(a + 3)a − 18m1 +�� +(a + 3)a − 36(m1 − 1) +� +m1(m1 − 1) ++ +− (a + 3)a +� +(a + 3)a(m1 − 1) + 36(m1 + 1) +� +(m1 + 1)m2 ++ ++ 2(a + 3)a(a − 3m1)(a + 3m1 + 3) +(m1 + 1)m1(m1 + m2 + 1) +� +q2 + . . . . +(3.6) +– 7 – + +As we see from the matrix elements in appendix A, the computation becomes quite tedious, +even at the second level. For this reason, it is clear that another method is needed to compute +a general expression of the one-point conformal block. Nevertheless, these first terms of (3.6) +will help us to compare with the computation from the Wilson lines approach. +Zero-point Conformal Block. In the case when Φα1 is given by the identity operator +and Φα is fully degenerate, the sl3 one-point conformal block F(α, α1, q)sl3 reduces to the +sl3 character χsl3 +j +of the finite-dimensional representation with highest-weight vector j = +m1w1 +m2w2. This character can be computed using the Weyl character formula (the details +of this computation are given in appendix B), +χsl3 +j += q−m1−m2(−1 + q1+m1)(−1 + q1+m2)(−1 + q2+m1+m2) +(−1 + q)3(1 + q) +. +(3.7) +4 +One-point conformal blocks from AGT +In this section, we move to compute the W3 one-point conformal block (3.1) in the limit +c → ∞ up to the second level of the module (2.14). +To this end we use the algorithm +based on the AGT correspondence (notations and some details of the algorithm are given in +appendix C), which leads to the following expression +F(α, α1, q) = 1 + +�18m1m2 − a2(m1 + m2) − 3a(m1 + m2) +9m1m2 +� +q + ++ +�(a + 3)2(a − 3m1 − 3)(a − 3m2) +81(m1 + 1)m2 ++ ++ (a + 1) +� +a2(m1 + m2) + 3a(m1 + m2) − 9(a + 3)m1m2 +� +9m1m2 ++ +− (a + 3)2(a − 3m1) +27m1 +− (a + 3)2(a − 3m2 − 3) +27(m2 + 1) ++ 1 +2(a2 − a − 2) + ++ (a + 6)(a + 3)(a − 3m1 + 3)(a − 3m1) +162m1(m1 − 1) +− (a + 3)(a + 3m2 + 3)(a − 3m2) +27m2(m2 + 1) ++ ++ (a + 6)(a + 3)(a − 3m2 + 3)(a − 3m2) +162m2(m2 − 1) ++ ++ (a + 3)(a − 3m1)(a + 3m1 + 3)(a − 3m1 − 3m2 − 3) +81m1(m1 + 1)(m1 + m2 + 1) ++ +1 +18(a + 6)(a + 3) + a +3 + 1 +� +q2 + . . . . +(4.1) +Now, we propose the following formula that relates the W3 block to the sl3 one in the following +way +F(α, α1, q) +F(α, α1, q)sl3 += χW3 +χsl3 , +(4.2) +where χW3 and χsl3 are W3 and sl3 (for infinite-dimensional representations) characters re- +spectively, given by +– 8 – + +χW3 = +1 +�∞ +i=1(1 − qi)2 , +χsl3 = +1 +(1 − q)3(1 + q) , +(4.3) +where the second line is obtained by fixing the proper normalization of (3.7) consistent with +the standard normalization of CB’s and then taking the limit (m1, m2 → ∞). By substitut- +ing (4.1) in (4.2) and solving for F(α, α1, q)sl3, we recover the expression (3.6). This confirms, +up to the second level, our proposal. The motivation of this conjectured formula comes from +the fact that a similar relation [42] holds for the Virasoro block FVir in the large central +charge limit and sl2 block, namely +FVir +Fsl2 += χVir +χsl2 = +1 − q +�∞ +i=1(1 − qi) . +(4.4) +The relation (4.2) allows effectively to compute higher-level contributions to the sl3 block. +Note that the characters and blocks involved in (4.2) refer to infinite-dimensional represen- +tations. +Nevertheless, we can still use this formula for finite-dimensional representations, +truncating the series up to an appropriate level. +5 +Conformal Blocks through Wilson line operators +In [40], it was proved that the expectation value of the Wilson network operator computes +the sl2 toroidal one-point conformal block. This section aims to show that a similar result is +obtained in the case of sl3 algebra. +In the case of the spherical topology and the sl3 algebra, in [39], it was given the Wilson +lines description of the sl3 four-point block. Here we will combine the ideas of [40] and [39] +to proceed with our task. Before introducing the object we start with some definitions and +notations. In the subsequent, we will work with finite-dimensional representations R of sl3 +that, in general, are characterized by a highest-weight vector j given by (2.9). We will label +the finite-dimensional representations R of sl3 by the vectors α, α1, having in mind that +they correspond to highest-weight vectors j, j1 according to (2.9) and (1.7). The Wilson line +operators in thermal AdS3 are given by +Wα[x, y] = P exp +� +− +� y +x +Ω +� += exp +� +(x − y)(L1 + 1 +4L−1) +� +, +(5.1) +where L1, L−1 are defined in the representation Rα of the sl3 algebra. +The Wilson network operator allows to compute the conformal block and contains a +number of elements (see, e.g., [40]) that are listed below for completeness. +1. Φ(hα,qα)(zα, ¯zα) on the boundary is attached to a bulk-to-boundary Wilson line operator +Wα[zb, zα] acting in the representation Rα, which connects the boundary point zα to +the bulk point zb. +– 9 – + +2. Bulk-to-bulk Wilson line operator Wβ[z1, z2] in the representation Rβ, which connects +two bulk points and acts trivially for a contractible loop. In the toroidal topology, there +exists a non-trivial bulk-to-bulk Wilson loop operator Wβ[zb, zb + 2πτ], associated with +a non-contractible cycle. +3. A vertex in the bulk, obtained when three Wilson line operators, associated with the +representations Rα, Rβ and Rγ, meet each other in the same point. This vertex is +described by the trivalent intertwining operator +Iα;β,γ : +Rβ ⊗ Rγ −→ Rα , +(5.2) +defined by the invariance property +Iα;β,γUβUγ = UαIα;β,γ , +(5.3) +where Ua, (a = α, β, γ), are elements of the gauge group. +With a proper choice of the gauge, we can simplify the expression of the Wilson loop +operator [40]. Due to the gauge covariance of the Wilson line operator, we can perform a +gauge transformation of the connection +Ω = U �ΩU −1 , +(5.4) +where, if we choose as gauge element +U = e− i +2 L−1eiL1e−i π +2 L0 , +(5.5) +we obtain +�Ω = −iL0 dz . +(5.6) +With this particular choice, known as diagonal gauge, we have that the Wilson loop reads +Wi[0, 2πτ] = e2πiτL0 = qL0 +with q = e2πiτ . +(5.7) +The only element that is affected by this choice of the gauge is the external field, which has +to transform accordingly. We consider as the boundary state the lowest-weight state of the +representation that has to be transformed as +|lw⟩α −→ |� +lw⟩α ≡ U −1 +α +|lw⟩α , +(5.8) +where Uα is the gauge group element (5.5) in the representation Rα. +– 10 – + +5.1 +Computation of the Wilson line operators +We are interested in performing the computation of the Wilson network operator exploiting +the tensor products of representations of the sl3 algebra. This is based on the fact that every +state in a given representation of the algebra can be represented as a symmetric traceless +tensor. Let us show how this procedure works. +The fundamental representation of sl3 is characterized by the highest-weight vector j = +w1 = (1, 0). This representation contains three states given by (2.7), which for convenience, +we denote as +|e1⟩ = w1, +|e2⟩ = w2 − w1, +|e3⟩ = −w2 . +(5.9) +Similarly, we denote the states of the anti-fundamental representation (0, 1) (the conjugated +of the fundamental) +|¯e1⟩ = −w1, +|¯e2⟩ = w1 − w2, +|¯e3⟩ = w2 . +(5.10) +For a generic sl3 representation with a highest-weight vector +j = m1w1 + m2w2 ≡ (m1, m2) , +(5.11) +the whole vector space of the representation is obtained by applying the lowering generators +L1, W1, W2 to the highest-weight vector. Each of these states can be written as the tensor +product of the states of the fundamental representation (5.9) and its conjugate (5.10) in the +following way +T +k1....km2 +i1...im1 += +1 +m1!m2! |e(i1⟩ ⊗ |ei2⟩ ⊗ · · · ⊗ |eim1)⟩ |¯e(k1⟩ ⊗ |¯ek2⟩ ⊗ · · · ⊗ |¯ekm2)⟩ = += +1 +m1!m2! |e(i1...eim1)¯e(k1...¯ekm2)⟩ , +(5.12) +where the parentheses of lower and upper indices denote the symmetrization, and the tensor +T +k1....km2 +i1...im1 +has to be traceless. The highest-weight vector corresponds to the case when all +indices i are equal to 1 for the fundamental representation, while for the anti-fundamental +one, we require all the indices k to be equal to 3. Below, we will need to compute the tensor +product of the kind Rα ⊗ Rα1. In particular, here we need to extract the representation Rα, +i.e., we need the projection +P : Rα ⊗ Rα1 → Rα . +(5.13) +The role of this projector is played by the intertwining operator Iα;α,α1 (5.2). In the way +this intertwining operator is defined, its matrix elements are given by the Clebsch-Gordan +coefficients, i.e. +⟨a| Iα;α,α1 |b⟩ ⊗ |c⟩ = Cabc , +(5.14) +where a, b ∈ Rα, c ∈ Rα1, and Cabc is the Clebsch-Gordan coefficient. However, in what +follows, we are going to define this projector explicitly, since for sl3, the Clebsch-Gordan +coefficients are not known for generic representations. +– 11 – + +We want to show that the following expression holds, where F(α, α1, q)sl3 is the sl3 one- +point conformal block introduced in (3.6) +Vα|α1(τ) := +� +|α⟩∈Rα +⟨α| Wα[zb, zb + 2πτ]Iα;α,α1 |α⟩ ⊗ Wα1[zb, z1] |lw⟩α1 = += �C(α, α1)F(α, α1, q)sl3 , +(5.15) +where z1 is the point on the boundary of the solid torus that is a geometric representation +of the thermal AdS3, zb is an arbitrary point in the bulk of AdS3, |lw⟩α1 is the lowest-weight +state of the representation Rα1 with j1 = (a, 0) and �C(α, α1) is an overall constant irrelevant +to our discussion. By using the diagonal gauge (5.7) it can be shown that Vα|α1(τ) does not +depend on zb and z1, and it can be expressed as +Vα|α1(τ) = Trα ⟨α| qL0Iα;α,α1 |α⟩ ⊗ U −1 +α1 |lw⟩α1 , +(5.16) +where Trα = � +|α⟩∈Rα and U −1 +α1 is given by (5.5) in the representation Rα1. Now, properly +inserting resolutions of identities +1 = +� +Rβ +|β⟩ ⟨β| , +(5.17) +where � +Rβ stands for the sum over all the states in the representation Rβ, we can write, in +a symbolic way, +Vα|α1(τ) = +� +Rα +� +Rγ +⟨α| qL0 |α⟩ +� +⟨α| Iα;α,γ |α⟩ ⊗ |γ⟩ +� +⟨γ| U −1 +α1 |lw⟩α1 . +(5.18) +Let us see how to represent each of the factors in (5.18). +1. The first term can be represented in terms of the so-called Wigner D-matrix. We will +be interested in particular in the fundamental and anti-fundamental representations, +where the D-matrices read respectively +Dij := ⟨ei| qL0 |ej⟩ = +� +� +� +q 0 +0 +0 1 +0 +0 0 q−1 +� +� +� , +¯Dij := ⟨¯ei| qL0 |¯ej⟩ = +� +� +� +q−1 0 0 +0 +1 0 +0 +0 q +� +� +� . +(5.19) +2. The second factor in (5.18), the one involving the intertwining operator, acts as a +projector. Its explicit action will be explained later. +3. The third term consists in the coordinates of the transformed lowest-weight state of the +external representation. Again, focusing on the fundamental representation, we have +that the transformed lowest-weight state is4 +U −1 +fund |e3⟩ ∝ |e1⟩ + +√ +2 |e2⟩ + |e3⟩ . +(5.21) +4In the fundamental representation, we use the following forms of the generators of sl2 subalgebra of sl3 +L1 = +√ +2 +� +� +� +0 0 0 +1 0 0 +0 1 0 +� +� +� , +L0 = +� +� +� +1 0 +0 +0 0 +0 +0 0 −1 +� +� +� , +L−1 = +√ +2 +� +� +� +0 −1 +0 +0 +0 +−1 +0 +0 +0 +� +� +� . +(5.20) +– 12 – + +This means that, in general, defining +pk := ⟨ek| U −1 +fund |e3⟩ = δk,1 + +√ +2δk,2 + δk,3 +k = 1, 2, 3 , +(5.22) +we have that +⟨ek1ek2 · · · eka| U −1 +α1 |lw⟩α1 ∼ pk1pk2 · · · pka . +(5.23) +At this stage, we state the main result of this section. Even though we do not have, as of +this writing, a general proof of (5.15) for generic representations Rα, Rα1, we have tested it +for some non-trivial representations given by the following values +(m1, m2, a) = {(2, 2, 3), (2, 2, 6), (2, 3, 3), (2, 3, 6), (3, 3, 3), (3, 3, 6), (4, 4, 3), (4, 4, 6)} , +(5.24) +obtaining that in all these cases, (5.15) holds precisely. +Let us anticipate here that, to obtain a non-vanishing one-point toroidal conformal block, +we have to require a ≡ 0 mod 3. The details of the case (2, 2, 3) are given in appendix D as an +example. Furthermore, as a complementary motivation of (5.15) for the case when a = 0 (i.e., +when Rα1 is the trivial representation), we show explicitly that for representations Rα with +highest-weight vector j = (m1, 0) or j = (0, m2), Vα|α1 in (5.15) is equal to the character (3.7). +5.1.1 +Zero-point conformal block +The simplest case of (5.15) is when Rα1 is the trivial representation (one-dimensional), in +which case we expect to find the character (3.7) (or zero-point conformal block) of the sl3 +representation. Let us check this statement. +We show this by choosing as highest-weight vector of Rα the vector j = (m1, 0)5. Since +Rα1 is the trivial representation, the intertwining operator Iα;α,0 (or the projector (5.13)) +reduces to the identity operator, thus +Vα|0(τ) ≡ Vα = +� +i1,...,im1=1,2,3 +⟨ei1 . . . eim1| qL0 |e(i1 . . . eim1)⟩ = += +� +i1,...,im1=1,2,3 +Di1(i1 . . . Dim1im1) . +(5.25) +Due to the symmetrization of the indices, the above expression can be written as +Vα = +m1 +� +m,n,k=0 +m+n+k=m1 +qmq−n(1)k , +(5.26) +and after some algebra, we can show that (5.26) gives +Vα = +m1 +� +r=1 +� +q−r + qr� � +−M(m1 + r) + m1 + r +2 +− r + 1 +� +− M(m1) + m1 +2 + 1 , +(5.27) +5The same procedure holds in the case j = (0, m2). +– 13 – + +where M(r) = 1−(−1)r +4 +. By expanding χsl3 +j=(m1,0) of (3.7) in q, we can see that this character +is equal to (5.27). +We would like to obtain the same result in the general case when m2 ̸= 0, i.e., we want +to show the following equation +� +i1,...,im1=1,2,3 +k1,...,km2=1,2,3 +⟨¯ek1 . . . ¯ekm2 ei1 . . . eim1| qL0T +k1...km2 +i1...im1 += χsl3 +j +, +(5.28) +where χsl3 +j +is the character (3.7), and the tensor T +k1....km2 +i1...im1 +is the symmetric (in lower and +upper indices) and traceless tensor from (5.12). We do not have a general proof of (5.28). +However, the numerical results we have obtained for many values of j confirm this equation. +5.1.2 +One-point conformal block +In this section, we elaborate on the expression (5.16) (which we claimed that reproduces the +sl3 one-point conformal block F(α, α1, q)sl3) for general representations Rα and Rα1 labelled +by the highest-weight vectors j = (m1, m2) and j1 = (a, 0), respectively. The purpose is to +simplify this expression in order to compare it with F(α, α1, q)sl3. In what follows, we will +not give too much attention to overall constants since they are generally ineffective. For this +reason, some of the following equalities have to be considered to hold up to a constant. +The idea is to represent (5.16) as a symmetric traceless tensor. The first step will be +constructing a tensor with the right structure to represent the one-point block but without +the requirements of symmetry and tracelessness, which will be imposed at the end. So, let us +start defining the following tensor +b1···bm2 +a1···am1T +r1···rm2 +s1···sm1;k1···ka := += ⟨¯er1 · · · ¯erm2 es1 · · · esm1| qL0 |ea1 · · · eam1 ¯eb1 · · · ¯ebm2⟩ ⟨ek1 · · · eka| U −1 +α1 |lw⟩α1 , +(5.29) +which, thanks to the definitions (5.19), (5.22) and (5.23), can be rewritten as +b1···bm2 +a1···am1T +r1···rm2 +s1···sm1;k1···ka = Da1s1 · · · Dam1sm1 ¯Db1r1 · · · ¯Dbm2rm2pk1 · · · pka . +(5.30) +The tensor defined in (5.29) resembles the quantity (5.16) we are interested in. However, we +need to insert the intertwining operator. In order to do that, let us recall that it acts as a +projector (5.13), i.e., it translates in the insertion of a tensor P, which allows us to define +b1···bm2 +a1···am1M +c1···cm2 +d1···dm1 = +� +P +c1···cm2 +d1···dm1 +�s1···sm1;k1···ka +r1···rm2 +b1···bm2 +a1···am1T +r1···rm2 +s1···sm1;k1···ka . +(5.31) +In order to construct the tensor P, we follow the method described in [39]. It has to be a +product of sl3 invariant tensors, which are +δi +j , +ϵijk , +ϵijk . +(5.32) +– 14 – + +However, since the final result has to be traceless and symmetric, the only possible contrac- +tions we are interested in are with +δk +r , +ϵcsk , +(5.33) +where we have labelled the indices coherently to (5.31). If we suppose that P is made out +of x contractions with δk +r and y contractions with ϵcsk, since the remaining indices have to +be the same in number as the expected ones, i.e., m1 lower and m2 upper indices, a simple +counting shows us that a has to satisfy the condition +a = 3l , +l ∈ Z≥0 . +(5.34) +In this way, we obtain that the tensor that represents the intertwining operator is +� +P +c1···cm2 +d1···dm1 +�s1···sm1;k1···ka +r1···rm2 += δs1 +d1 · · · δ +sm1−l +dm1−lδk1 +dm1−l+1 · · · δkl +dm1δc1 +r1 · · · δ +cm2−l +rm2−l× +× δkl+1 +rm2−l+1 · · · δk2l +rm2ϵcm2−l+1sm1−l+1k2l+1 · · · ϵcm2sm1k3l . +(5.35) +Inserting this projector into (5.31), we end up with +b1···bm2 +a1···am1M +c1···cm2 +d1···dm2 = Da1d1 · · · Dam1−ldm1−l ¯Db1c1 · · · ¯Dbm2−lcm2−lpdm1−l+1 · · · pdm1× +× ¯pbm2−l+1 · · · ¯pbm2 A +cm2−l+1 +am1−l+1 · · · A +cm2 +am1, +(5.36) +where we have introduced the new quantities +¯pb := ¯Dbrpr , +Ac +a := ϵcskDaspk . +(5.37) +We want to make this tensor symmetric, and we do this procedure just formally, remembering +in what follows that we have to consider all the possible cyclic permutations of the indices. +Hence we write +b1···bm2 +a1···am1M +c1···cm2 +d1···dm1 ≡ +b1···bm2 +a1···am1M +(c1···cm2) +(d1···dm1) . +(5.38) +The last step consists in making this tensor traceless. We do this by exploiting the procedure +described in [39], where it is shown that a symmetric tensor can be made traceless as +b1···bm2 +a1···am1 � +M +c1···cm2 +d1···dm1 = +min +� +n=0 +Cnδ(c1 +(d1 · · · δcn +dn +b1···bm2 +a1···am1M +cn+1···cm2)f1···fn +dn+1···dm1)f1···fn , +(5.39) +where +Cn = (−1)n +(m1 + m2 − n + 1)! +(m1 + m2 + 1)!(m1 − n)!(m2 − n)!n! , +min = min(m1, m2) , +(5.40) +and in each summand, we take the sums over repeated indices fi. +Finally, (5.15) can be written as +b1···bm2 +a1···am1 � +M +(c1···cm2) +(d1···dm1)δb1c1 · · · δbm2cm2δa1d1 · · · δam1dm1 = �C(α, α1)F(α, α1, q)sl3 . +(5.41) +As of the date of this work, we have not found a general proof of (5.41), but many numerical +computations (which are partially described in the appendix D) confirm this equality. +– 15 – + +6 +Discussion +In this work we studied the semiclassical limit of W3 CFT on the torus, with a focus on the +sl3 one-point conformal blocks of fully degenerate primary fields for an external field satisfy- +ing (1.8). We compute the W3 block in the light c → ∞ limit using AGT correspondence. +In order to obtain the sl3 block we propose the relation (4.2). The utility of this relation +is twofold since it serves as a check of the dual representation for sl3 conformal block and +provides a computational method to calculate its higher-level contributions. +The dual description of sl3 one-point toroidal conformal block is given in terms of the +Wilson lines in 3d Chern-Simons gravity. +The proposal summarizes in (5.41). +The dual +approach seems to give more manageable computations compared to those based on the +original definition, at least in its construction. The simplest test of this proposal is the case +when the one-point block reduces to the zero-point block, which we confirmed analytically +and numerically in section 5.1.1. Further, we have performed more general tests for specific +cases of representations (5.24), obtaining that all these examples confirm (5.15). The absence +of a general expression of sl3 Clebsch-Gordan coefficients forces us to use the tensor technique +in section 5.1, the general proof of (5.41) remains an open problem. +From the Wilson line construction a natural generalization follows, that is the study +of the dual description in the cases of higher-point conformal blocks, which show additional +features, like the presence of different intermediate channels (usually called s and t), as shown +for the sl2 case in [40]. This generalization is straightforward in the context of the Wilson +lines formulation of the conformal blocks, according to the construction of section 5. Also, it +would be worth to extend this approach to the whole W3 CFT, which would require to focus +also on quantum corrections (for related consideration see [26]). +Another interesting question is how the Wilson line approach works if multiplicities in +the representations under consideration are present. From the dual construction it follows +that in such cases we can obtain multiple nonequivalent expressions for the projector (5.13). +For instance, if the external field is in the adjoint representation, there is a multiplicity two +for the conformal blocks in the W3 CFT [46], which corresponds to the fact that Clebsch- +Gordan coefficients are not uniquely defined [47]. In this case, the multiplicity is present also +in the definition of the projector. It would be interesting to establish a precise correspondence +between uncertainty related to the OPE multiplicities for conformal blocks and multiplicities +arising in the definition of the projector in the dual construction. +A +Matrix Elements +Here, we give the matrix elements that enter in the computation of the sl3 conformal block (3.6) +at the first and second levels of sl3 module. We use the algorithm described in [48] to compute +these matrix elements. +– 16 – + +First level. From the sl3 commutation relations, one can find the following products +⟨h0, q0|L1L−1|h0, q0⟩ = 2h0 , +⟨h0, q0|W1L−1|h0, q0⟩ = 3q0 , +⟨h0, q0|W1W−1|h0, q0⟩ = −h0/5 , +(A.1) +thus the Shapavalov matrix G at the first level is +G = +� +2h0 3q0 +3q0 − h0 +5 +� +. +(A.2) +The matrix elements at the first level are6 +⟨h0, q0|L1Φα1L−1|h0, q0⟩ = 2h0 + (h1 − 1) h1, +⟨h0, q0|W1Φα1L−1|h0, q0⟩ = 3q0 + 1 +2 (h1 − 1) q1, +⟨h0, q0|L1Φα1W−1|h0, q0⟩ = 3q0 − 1 +2 (h1 − 1) q1, +⟨h0, q0|W1Φα1W−1|h0, q0⟩ = 1 +5 (h1 − h0) − (h1 − 3) q2 +1 +4h1 +. +(A.3) +Second level. At this level, we consider the basis +L2 +−1 , +W2 +−1 , +L−1W−1 , +W−2 . +(A.4) +We obtain the following products +⟨h0, q0|L2 +1L2 +−1|h0, q0⟩ = 4h0 (2h0 + 1) , +⟨h0, q0|W2 +1L2 +−1|h0, q0⟩ = 18q2 +0 − 6h0 +5 , +⟨h0, q0|W1L1L2 +−1|h0, q0⟩ = 6 (2h0 + 1) q0, +⟨h0, q0|W2L2 +−1|h0, q0⟩ = 12q0, +⟨h0, q0|W2 +1W2 +−1|h0, q0⟩ = 1 +25h0 (2h0 + 1) , +⟨h0, q0|W1L1W2 +−1|h0, q0⟩ = 1 +5(−3) (2h0 + 3) q0, +⟨h0, q0|W2W2 +−1|h0, q0⟩ = 6q0 +5 , +⟨h0, q0|W1L1L−1W−1|h0, q0⟩ = 9q2 +0 − 2 +5h0 (h0 + 1) , +⟨h0, q0|W2L−1W−1|h0, q0⟩ = −1 +5 (4h0) , +⟨h0, q0|W2W−2|h0, q0⟩ = 8h0 +5 . +(A.5) +6All the matrix elements are proportional to ⟨h0, q0|Φα1|h0, q0⟩. But, for the sake of brevity, we omit this +factor. +– 17 – + +Hence the Shapavalov matrix G at the second level is +G = +� +� +� +� +� +4h0 (2h0 + 1) +18q2 +0 − 6h0 +5 +6 (2h0 + 1) q0 +12q0 +18q2 +0 − 6h0 +5 +1 +25h0 (2h0 + 1) +1 +5(−3) (2h0 + 3) q0 +6q0 +5 +6 (2h0 + 1) q0 1 +5(−3) (2h0 + 3) q0 9q2 +0 − 2 +5h0 (h0 + 1) − 1 +5 (4h0) +12q0 +6q0 +5 +− 1 +5 (4h0) +8h0 +5 +� +� +� +� +� . +(A.6) +The matrix elements at the second level are +⟨h0, q0|L2 +1Φα1L2 +−1|h0, q0⟩ = 8h2 +0 + (8 (h1 − 1) h1 + 4) h0 + (h1 − 1) h1 ((h1 − 1) h1 + 2) , +⟨h0, q0|L2 +1Φα1W2 +−1|h0, q0⟩ = 18q2 +0 − 6 (h1 − 1) q1q0 + +� +h3 +1 − 7h1 + 6 +� +q2 +1 +4h1 +− 1 +5 (h1 − 1) h1 (h1 + 4) − 6h0 +5 , +⟨h0, q0|L2 +1Φα1L−1W−1|h0, q0⟩ = 6 (2h0 + (h1 − 1) h1 + 1) q0 +− 1 +2 (h1 − 1) (4h0 + (h1 − 1) h1 + 2) q1, +⟨h0, q0|L2 +1Φα1W−2|h0, q0⟩ = 12q0 − 2 (h1 − 1) h1q1. +(A.7) +⟨h0, q0|W2 +1Φα1L2 +−1|h0, q0⟩ = 18q2 +0 + 6 (h1 − 1) q1q0 + +� +h3 +1 − 7h1 − 30 +� +q2 +1 +4h1 +− 1 +5h1 (h1 (h1 + 3) − 2) − 6h0 +5 , +⟨h0, q0|W2 +1Φα1W2 +−1|h0, q0⟩ = (h1 − 6) (h1 − 3) (h1 + 3) q4 +1 +16h3 +1 ++ (2h0 (h1 − 3) − 3 (h1 − 2) h1 + 12) q2 +1 +10h1 ++ 1 +25 +� +2h2 +0 + (1 − 4h1) h0 + h1 (3h1 + 2) +� +, +⟨h0, q0|W2 +1Φα1L−1W−1|h0, q0⟩ = 12 +40 +�15q2 +1 +h1 +− 5q2 +1 − 4h0 + 4h1 − 6 +� +q0+ ++ 1 +40 +� +12h2 +1 − 8 (h0 + 1) h1 + 8h0 − 5 ((h1 − 3) (h1 − 1) h1 − 36) q2 +1 +h2 +1 ++ 4 +� +q1, +⟨h0, q0|W2 +1Φα1W−2|h0, q0⟩ = −(h1 − 6) (h1 − 3) q3 +1 +2h2 +1 ++ 2 +5 (2h1 − 3) q1 + 6q0 +5 . +(A.8) +– 18 – + +⟨h0, q0|W1L1Φα1L2 +−1|h0, q0⟩ = 6 (2h0 + (h1 − 1) h1 + 1) q0+ ++ 1 +2 (h1 − 1) (4h0 + (h1 − 1) h1 + 2) q1, +⟨h0, q0|W1L1Φα1W2 +−1|h0, q0⟩ = 12 +40 +�15q2 +1 +h1 +− 5q2 +1 − 4h0 + 4h1 − 6 +� +q0+ ++ 1 +40 +(h1 − 1) q1 +� +5 (h1 − 6) (h1 + 3) q2 +1 − 4h2 +1 (−2h0 + 3h1 + 2) +� +h2 +1 +, +⟨h0, q0|W1L1Φα1L−1W−1|h0, q0⟩ = 1 +5 +� +−2h2 +0 − (h1 − 2) (h1 − 1) h0 + 45q2 +0 + h1 ((h1 − 1) h1 + 2) +� +− (h1 − 3) (2h0 + (h1 − 1) h1 + 2) q2 +1 +4h1 +, +⟨h0, q0|W1L1Φα1W−2|h0, q0⟩ = 2 +5 +� +h2 +1 + h1 − 2h0 +� +− ((h1 − 6) h1 + 3) q2 +1 +h1 +. +(A.9) +⟨h0, q0|W2Φα1L2 +−1|h0, q0⟩ = 2 (6q0 + (h1 − 1) h1q1) , +⟨h0, q0|W2Φα1W2 +−1|h0, q0⟩ = +� +h2 +1 − 9 +� +q3 +1 +2h2 +1 +− 3 +5 (h1 − 1) q1 + 6q0 +5 , +⟨h0, q0|W2Φα1L−1W−1|h0, q0⟩ = 1 +10 +�5 (h1 (3 − 2h1) + 3) q2 +1 +h1 +− 8h0 + 2h1 (h1 + 3) +� +, +⟨h0, q0|W2Φα1W−2|h0, q0⟩ = 8h0 +5 +− 4 (h1 − 3) q2 +1 +h1 +. +(A.10) +B +Weyl character formula +Here, we want to compute the character χsl3 +j +(3.7) of a finite-dimensional representation of +sl3 with highest-weight vector given by j = m1w1 + m2w2, where w1,2 are the fundamental +weights and m1, m2 are positive integers. Let us define some notations. The fundamental +weights satisfy the following products +(w1, w1) = (w2, w2) = 2 +3, +(w1, w2) = 1 +3 , +(B.1) +The Weyl vector is defined as +ρ = w1 + w2 , +(B.2) +and the Weyl group W of sl3 is formed by 6 elements, which are denoted as +W = [1, s1, s2, s1s2, s2s1, s1s2s1] . +(B.3) +– 19 – + +and they act on the vector j as follows +1(j) = j , +s1(j) = −m1w1 + (m1 + m2)w2 , +s2(j) = (m1 + m2)w1 − m2w2 , +s1s2(j) = −(m1 + m2)w1 + m1w2 , +s2s1(j) = m2w1 − (m1 + m2)w2 , +s1s2s1(j) = −m1w1 − m2w2 . +(B.4) +The signature ϵ(w) of a element w ∈ W is defined by +ϵ(w) = (−1)l(w) , +(B.5) +where l(w) is the length of w, that is the number of si that w contains. For example, the +signatures of the elements (B.3) are respectively +ϵ(W) = [1, −1, −1, 1, 1, −1] . +(B.6) +We compute the character χsl3 +j +from the Weyl character formula +χsl3 +j += +� +w∈W ϵ(w)e(w(j+ρ),2πiτρ) +� +w∈W ϵ(w)e(w(ρ),2πiτρ) +, +(B.7) +where we use q = e2πiτ, obtaining exactly (3.7). +C +AGT relation +In this section, we comment on some important details of the AGT relation we used to +compute the expression (4.1). +According to the AGT relation, the W3 conformal block +F(α, α1, q) (4.1) can be computed by +F(α, α1, q) = Zinst +SU(3) +∞ +� +i=1 +(1 − qi)1−2h1 , +(C.1) +where the infinite product is the so-called Heisenberg factor, and Zinst +SU(3) is the SU(3) instanton +partition function. +For the algorithm which we used to compute Zinst +SU(3) we refer to [49], +specifically the section 3.2.1. Here, we want to clarify the relations between the parameters +of F(α, α1, q) and Zinst +SU(3). +The partition function Zinst +SU(3) is given by equation (28) of [49] and depends on the pa- +rameters xi = (Q − α, ei), µ, ϵ1, ϵ2 which are related to our parameters as follows +µ = a/3, +ϵ1 = b, +ϵ2 = 1/b, +(C.2) +while α is given by (2.9) and Q, ei are according to our notations above. Finally, the expres- +sion (4.1) is given in the limit c → ∞ (or equivalently b → 0) of C.1. +– 20 – + +D +Examples +The simplest test7 of (5.41) we can perform is the case j = 2w1 + 2w2 = (2, 2), j1 = (3, 0). In +this case the tensor (5.38) becomes +b1b2 +a1a2Mc1c2 +d1d2 = 1 +4 ¯pb2(Da1d1pd2 + Da1d2pd1)( ¯Db1c1Ac2 +a2 + ¯Db1c2Ac1 +a2) . +(D.1) +The next step is to make this tensor traceless according to (5.39) and then take the contraction +according to (5.41). We can show that there will be 16 terms proportional to C1, and all the +terms proportional to C2 vanish. To write the result, we use the notation of the matrix B +Ba +b = ¯papb . +(D.2) +By using this notation, and denoting the trace of a generic matrix C by Tr[C] = [C], one can +show that +b1b2 +a1a2 � +M(c1c2) +(d1d2)δb1c1δb2c2δa1d1δa2d2 = += 1 +4 +� +C0 +� +[D]2[AB] + [D][ ¯DBA] + [ ¯D][DAB] + [DA ¯DB] +� ++ C1 +4 +� +[D ¯DBAT ] + [DAB ¯D] ++ [D ¯DBA] + [ ¯DD][AB] + [D][ ¯DBAT ] + [D][B ¯DA] + [ ¯DBDAT ] + [D ¯DBAB] ++ [ ¯D][DABT ] + [ ¯D][DAT B] + [DA ¯DBT ] + [D ¯DBAT BT ] + [ ¯D][D][AT B] ++ [D][ ¯DAT B] + [ ¯D][BDAT ] + [D ¯DAT B] +� ++ (0)C2 +� += += �C(2, 2, 3)(1 − 8 +5q2 − 4 +5q3 + 4 +5q5 + 8 +5q6 − q8) . +(D.3) +The firs three terms in q, (i.e. up to q2) of the above expression are exactly the terms obtained +by (3.6) in this case (m1, m2, a → (2, 2, 3)). This validates (5.41). +It is clear that the difficulty of this “simple” problem increases substantially when m1, +m2, a increase. We were able to work out the cases (5.24). Here we give the results and let +us emphasize that all these cases confirm (5.41). +Cases : +(m1, m2, a) = (2, 2, 6) +b1b2 +a1a2 � +M(b1b2) +(a1a2) = �C(2, 2, 6)(1 − 4q + 4q2 + . . . ) , +(D.4) +(m1, m2, a) = (2, 3, 3) +b1b2b3 +a1a2 � +M(b1b2b3) +(a1a2) += �C(2, 3, 3)(1 + q +3 − 7q2 +9 + . . . ) , +(D.5) +7Since the expression (3.6) contains contributions of two states to the first level and four states to the +second level, then m1, m2 ≥ 2. +– 21 – + +(m1, m2, a) = (2, 3, 6) +b1b2b3 +a1a2 � +M(b1b2b3) +(a1a2) += �C(2, 3, 6)(1 − 3q + q2 + . . . ) , +(D.6) +(m1, m2, a) = (3, 3, 3) +b1b2b3 +a1a2a3 � +M(b1b2b3) +(a1a2a3) = �C(3, 3, 3)(1 + 2q +3 + 2q2 +21 + . . . ) , +(D.7) +(m1, m2, a) = (3, 3, 6) +b1b2b3 +a1a2a3 � +M(b1b2b3) +(a1a2a3) = �C(3, 3, 6)(1 − 2q − 10q2 +7 ++ . . . ) , +(D.8) +(m1, m2, a) = (4, 4, 3) +b1b2b3b4 +a1a2a3a4 � +M(b1b2b3b4) +(a1a2a3d4) = �C(4, 4, 3)(1 + q + q2 + . . . ) , +(D.9) +(m1, m2, a) = (4, 4, 6) +b1b2b3b4 +a1a2a3a4 � +M(b1b2b3b4) +(a1a2a3a4) = �C(4, 4, 6)(1 − q − 5q2 +3 + . . . ) . +(D.10) +References +[1] A. 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Santachiara, AGT, N-Burge partitions and WN minimal models, +JHEP 10 (2015) 073, [1507.03540]. +– 24 – + diff --git a/ldE3T4oBgHgl3EQfiAoP/content/tmp_files/load_file.txt b/ldE3T4oBgHgl3EQfiAoP/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9d68e3a08335ed8f3049dc3df4ce170b73dd6129 --- /dev/null +++ b/ldE3T4oBgHgl3EQfiAoP/content/tmp_files/load_file.txt @@ -0,0 +1,976 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf,len=975 +page_content='Wilson lines construction of sl3 toroidal conformal blocks Vladimir Belavin, Pietro Oreglia, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' Ramos Cabezas Physics Department, Ariel University, Ariel 40700, Israel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' E-mail: vladimirbe@ariel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='il, pietro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='oreglia@msmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='ariel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='il, juanra@ariel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='il Abstract: We study W3 toroidal conformal blocks for degenerate primary fields in AdS/CFT context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' In the large central charge limit W3 algebra reduces to sl3 algebra and sl3 blocks are defined as contributions to W3 blocks coming from the generators of sl3 subalgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' We consider the construction of sl3 toroidal blocks in terms of Wilson lines operators of 3d Chern- Simons gravity in the thermal AdS3 space-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' According to the correspondence, degenerate primary fields are associated with Wilson lines operators acting in the corresponding finite- dimensional sl3 representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' We verify this dual construction for one-point toroidal block using sl3 tensor technique in the bulk theory and an algorithm based on AGT correspondence in the boundary CFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='04575v1 [hep-th] 11 Jan 2023 Contents 1 Introduction 1 2 Preliminaries: AdS3 and W3 CFT 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='1 W3 Conformal Field Theory 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='2 Brief review of 3d Chern-Simons gravity theory 5 3 One-point conformal block from CFT 7 4 One-point conformal blocks from AGT 8 5 Conformal Blocks through Wilson line operators 9 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='1 Computation of the Wilson line operators 11 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='1 Zero-point conformal block 13 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='2 One-point conformal block 14 6 Discussion 16 A Matrix Elements 16 B Weyl character formula 19 C AGT relation 20 D Examples 21 1 Introduction Conformal blocks (CB’s) determine the holomorphic contributions to the correlation functions that appear after fixing the OPE channels [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' CB’s are generally fixed by the symmetry algebra and depend on the topology of the CFT Riemann surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' In this work, we are interested in W3 one-point conformal block on the torus in the large central charge limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' The one-point correlation function of a primary field Φα1 with conformal dimension h1 on the torus is defined as1 ⟨Φα1(z1, ¯z1)⟩ = Trα � qL0 ¯q ¯L0Φα1(z1, ¯z1) � = � α Cαα1α|F(α, α1, q)|2 , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='1) 1Throughout this paper we omit factor (q¯q)− c 24 which can be easily restored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' – 1 – where Trα is the trace taken over a module of the symmetry algebra associated with the primary field Φα in the intermediate OPE channel, q is the elliptic parameter of the torus q = e2πiτ and L0 is the generator of the algebra satisfying L0 |h1⟩ = h1 |h1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' Here F(α, α1, q) is the one-point holomorphic toroidal conformal block (for more details, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=', [2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' In recent years, the AdS3/CFT2 provided a new formulation of conformal blocks in terms of geodesic networks in AdS space-time, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=', [3–16], or Wilson network operators of Chern-Simons gravity, which will be the subject of the present consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='2 The latter approach is based on the higher-spin version of AdS/CFT correspondence [36], which, in general, identifies the minimal model cosets SU(N)k ⊕ SU(N)1 SU(N)k+1 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='2) in ’t Hooft limit, as the holographic dual of the higher spin theory [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' Here we study a different limit, the so-called semiclassical limit, where we have, from the bulk side, an SL(N) × SL(N) Chern-Simons theory, while from the CFT side, a non-unitary WN CFT model considered in the large central charge limit, with N fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' We recall that primary fields in the WN model are labelled by a pair of SL(N) highest-weights (Λ+, Λ−), which are both highest-weights of a finite-dimensional representation of SL(N) [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' One can identify two different kinds of primaries fields: heavy operators, identified by (0, Λ−), which have scale dimensions ∆ ∼ c, and corre- spond to flat SL(N) × SL(N) connections in the bulk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' light operators, labelled by (Λ+, 0), whose scale dimensions go as ∆ ∼ o(1), and are related to perturbative matter in the bulk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' In this paper we focus on the light operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' This allows us to consider the large c limit by restricting the set of generators, which in general can be written as {Ln, W 3 n, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' , W N n }, to those of the types Ln for |n| < 2 , W s n for |n| < s , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='3) with s = 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' , N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' By looking at the commutation relations (for explicit form in N = 3 case, see sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' 2), it can be seen that these operators can be identified as the generators of the slN algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' In the spherical topology (for N = 2, 3) this program has been implemented in [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' Regarding the toroidal topology, it was shown in [40] that sl2 one-point toroidal block F(−j, −j1, q)sl2, where −j and −j1 are respectively the intermediate and external conformal dimensions, is given by F(−j, −j1, q)sl2 = Trj � Wj[zb, zb + 2πτ]Ij;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='j,j1 � ⊗ Wj1[zb, z1] |lw⟩1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='4) In the dual description j and j1 are the spins of sl2 Chern-Simons gauge group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' The trace Trj(· · · ) is taken over the representation with spin j, z1 is the point on the boundary of the 2For recent development on CB’s in the holography context, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=', [17–35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' – 2 – solid torus, which is a geometric representation of the thermal AdS3, zb is an arbitrary point in the bulk of AdS3, |lw⟩1 is the lowest-weight state of the representation with spin j1 and Ij;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='j,j1 is the intertwining operator associated with the representations j and j1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' The factors Wi[x, y] (for i = j, j1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' x = zb and y = zb + 2πτ, z1) denote the Wilson line operators Wi[x, y] = P exp � − � y x Ω � = exp � (x − y)(L1 + 1 4L−1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='5) Here L1 and L−1 are the lowering and raising operators of sl2 algebra in the representation with spin i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' In this work we consider the W3 algebra;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' we are interested in fully degenerate primary fields Φα(z, ¯z), characterized by a pair of quantum numbers (hα, qα) (conformal dimension and W3 charge, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='3 In the semiclassical limit c → ∞, or b → 0 in Toda-like parameterization [41], c = 2+24(b+1/b)2, the parameter α corresponding to the intermediate primary field Φα in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='1) is given by α → −bj , j = m1w1 + m2w2 , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='6) where w1, w2 are sl3 fundamental weights and m1, m2 are non-negative integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' For the external primary Φα1 characterized by (hα1, qα1) α1 → −bj1 , j1 = aw1 , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='7) where a is a non-negative integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' The external field Φα1 is restricted to the so-called semi- degenerate form, to avoid the multiplicities problem of conformal blocks in W3 CFT, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=', [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' In what follows, we focus on the one-point blocks with external fields satisfying (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='7), which can be equivalently written as W−1Φα1(0, 0) |0⟩ = 3q1 2h1 L−1Φα1(0, 0) |0⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='8) The outline of the paper is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' In section 2, we recall the necessary facts about W3 CFT and AdS3 Chern-Simons gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' In sections 3 and 4, we compute F(α, α1, q)sl3 and check it by comparing with the large central charge limit of W3 block of light operators, using the algorithm based on AGT relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' To this end we propose a relation between the light W3 and sl3 one-point blocks, which is similar to the one existing between sl2 and Virasoro light blocks in the large central charge limit [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' In section 5, we describe the dual construction for sl3 one-point block in terms of the Wilson lines operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' We find that the expression obtained in section 3 can be represented by the lhs of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' Our conclusions are collected in section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' Appendices A, B, C, D contain some technical details related to sl3 matrix elements, sl3 Weyl character formula, AGT relation, and Wilson lines description of conformal blocks respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' 3Explicit form of (hα, qα) in terms of α will be given below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' – 3 – 2 Preliminaries: AdS3 and W3 CFT 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='1 W3 Conformal Field Theory The symmetry of the W3 CFT is generated by the energy-momentum tensor T(z) (a spin-2 current) and the additional spin-3 current W(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' Their expansions as Laurent series read T(z) = ∞ � n=−∞ Ln zn+2 , W(z) = ∞ � n=−∞ Wn zn+3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='1) The modes Ln and Wm generate the W3 algebra, that is, they satisfy the commutation relations [Ln, Lm] = (n − m)Ln+m + c 12(n3 − n)δn+m,0 , [Ln, Wm] = (2n − m)Wn+m , [Wn, Wm] = c 3 · 5!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' (n2 − 1)(n2 − 4)nδn+m,0 + 16 22 + 5c(n − m)Λn+m+ + (n − m) 30 � 2m2 + 2n2 − mn − 8 � Ln+m , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='2) where Λm = � p ⩽ −2 LpLm−p + � p ⩾ −1 Lm−pLp − 3(m + 2)(m + 3) 10 Lm .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='3) In the limit c → ∞ these commutation relations reduce to the ones of the sl3 algebra, generated by {L−1, L0, L1, W−1, W1, W0, W−2, W2} , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='4) that satisfy [Ln, Lm] = (n − m)Ln+m , [Ln, Wm] = (2n − m)Wn+m , [Wn, Wm] = (n − m) � 1 15(n + m + 2)(n + m + 3) − 1 6(n + 2)(m + 2) � Ln+m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='5) W3 primary fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' The conformal dimension hα and charge qα of a primary field Φα in a W3 CFT are expressed as hα = 1 2(α, 2Q − α) , qα = i � 48 22 + 5c 3 � i=1 (ei, α − Q) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='6) Here Q = (b + 1 b)(w1 + w2), ei are the weights of the fundamental representation e1 = w1 , e2 = w2 − w1 , e3 = −w2 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='7) and α is a vector on the root space given by αr1r2s1s2 = b � (1 − r1)w1 + (1 − r2)w2 � + 1 b � (1 − s1)w1 + (1 − s2)w2 � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='8) – 4 – with r1, r2, s1 and s2 positive integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' In the large central charge limit, we have that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='8) becomes α → −bj, j = m1w1 + m2w2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='9) As explained before, we are mostly interested in semi-degenerate fields (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='7) in the large central charge limit, for which the conformal dimension and the charge assume the values hα = −m1 − m2, qα = i 3 � 2 5(m2 − m1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='10) W3 module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' A W3 highest-weight vector |hα, qα⟩ given by |hα, qα⟩ = lim z→0 Φα(z) |0⟩ , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='11) satisfies the conditions L0 |hα, qα⟩ = hα |hα, qα⟩ , W0 |hα, qα⟩ = qα |hα, qα⟩ , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='12) Ln |hα, qα⟩ = Wn |hα, qα⟩ = 0 , n > 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='13) The W3 module associated with this highest-weight vector is spanned by the basis of descen- dant states L−I |hα, qα⟩ = L−i1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' L−imW−j1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' W−jn |hα, qα⟩ , I = {i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' , im;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' , jn} , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='14) with 1 ≤ i1 ≤ · · · ≤ im , 1 ≤ j1 ≤ · · · ≤ jn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='15) The sum N = � ia,jb∈I ia + jb (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='16) is called level of the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' Similarly, the sl3 module is defined when in the set I, the indices are restricted to i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' , im = 1 , j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' , jn = 1, 2 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='17) that is, when descendants are generated only by L−1, W−1 and W−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' For the basis states of W3 and sl3 modules, we introduce the following convenient notation L−I |hα, qα⟩ = |hα, qα, N⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='18) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='2 Brief review of 3d Chern-Simons gravity theory It is known [43] that the 2 + 1-dimensional Einstein-Hilbert action S with a negative cosmo- logical constant can be written in terms of the Chern-Simons action S = SCS[A] − SCS[ ¯A] , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='19) – 5 – where SCS[A] = k 4π � M3 Tr � A ∧ dA + 2 3A ∧ A ∧ A � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='20) The constant k is related to the 3-dimensional Newton constant G3 as k = r 4G3 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='21) with r the AdS3 radius, Tr stands for the invariant Killing form, M3 is the 3-dimensional space-time and A ( ¯A) is the (anti-) chiral gauge connection, that is, a one-form valued in the gauge group, given by a composition of vielbein and spin-connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' In the case we are interested in, that is the SL(N) × SL(N) Chern-Simons theory, A is an SL(N) connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' The equations of motion that follow from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='20) are given by (the same equations are valid for the anti-chiral connection) dA + A ∧ A = 0 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='22) which are the flatness conditions on the connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' It can be shown [44] that, by choosing the proper boundary conditions, in the local coordinates xµ = (ρ, z, ¯z) with ρ ≥ 0 the radial one, the solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='22) can be written as the gauge-transformed Ω: A = U −1ΩU + U −1dU , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='23) with Ω = � L1 − 2π6T(z) c L−1 � dz , U = eρL0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='24) Here T(z) is the holomorphic boundary energy-momentum tensor, while c is the central charge, defined through the Brown-Henneaux relation c = 3r 2G3 [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' L1, L−1 are two of the generators of the sl2 subalgebra of the gauge algebra slN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' We are interested in the space-time with periodic time conditions, that is to say, the thermal AdS3, in which case the stress-energy tensor is T(z) = − c 48π , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='25) and the connection Ω becomes Ω = � L1 + 1 4L−1 � dz , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='26) together with the identifications z ∼ z + 2π and z ∼ z + 2πτ, with iτ ∈ R<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' Finally, we define the Wilson line operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' Given a chiral connection A defined in a representation R of slN and a path l that connects two points x1, x2 ∈ M3, we define the Wilson line operator WR[l] as WR[l] = P exp � � l A � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='27) where P denotes the path-ordering operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' – 6 – 3 One-point conformal block from CFT In this section, we define the holomorphic W3 and sl3 one-point conformal blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' The result of the calculation of the W3 block will be given in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' The holomorphic W3 one-point conformal block F(α, α1, q) is given by F(α, α1, q) = 1 ⟨h0, q0| Φα1(z) |h0, q0⟩ � M,N=0 M=N ⟨h0, q0, M| qL0Φα1(z) |h0, q0, N⟩ G−1 MN , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='1) where hα := h0, qα := q0, the states |h0, q0, N⟩ belong to the W3 module (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='14), and G−1 MN is the inverse of the Shapavalov matrix GMN = ⟨h0, q0, M|h0, q0, N⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='2) For general Φα1, the matrix elements ⟨h0, q0, M| qL0Φα1(z) |h0, q0, N⟩ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='3) are not defined uniquely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' For this reason, we focus on the case where the external field Φα1(z) satisfies (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='8), from which it follows that − h2 1 5 = 9q2 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='4) This constraint is satisfied when the vector j1 corresponding to the field Φα1 is given by j1 = aw1 or j1 = aw2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' Without loss of generality, we choose j1 = aw1, which implies h1 = −a , q1 = − i 3 � 2 5a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='5) Similarly, the holomorphic sl3 one-point conformal block F(α, α1, q)sl3 is defined by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='1), but now the states |h0, q0, N⟩ belong to the sl3 module, obtained by restricting the set I (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='14) to the conditions (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' For the fields Φα, Φα1 with h’s and q’s given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='10) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='5) respectively, we compute the contribution to the sl3 one-point conformal block up to the second level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' The matrix elements involved in this computation are listed in appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' Here we present the result F(α, α1, q)sl3 = 1 + �18m1m2 − a2(m1 + m2) − 3a(m1 + m2) 9m1m2 � q + + 1 162 �(a + 6)(a + 3)a(a − 3) m2 − 1 + + � (a + 3)a − 18m1 �� (a + 3)a − 36(m1 − 1) � m1(m1 − 1) + − (a + 3)a � (a + 3)a(m1 − 1) + 36(m1 + 1) � (m1 + 1)m2 + + 2(a + 3)a(a − 3m1)(a + 3m1 + 3) (m1 + 1)m1(m1 + m2 + 1) � q2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='6) – 7 – As we see from the matrix elements in appendix A, the computation becomes quite tedious, even at the second level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' For this reason, it is clear that another method is needed to compute a general expression of the one-point conformal block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' Nevertheless, these first terms of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='6) will help us to compare with the computation from the Wilson lines approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' Zero-point Conformal Block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' In the case when Φα1 is given by the identity operator and Φα is fully degenerate, the sl3 one-point conformal block F(α, α1, q)sl3 reduces to the sl3 character χsl3 j of the finite-dimensional representation with highest-weight vector j = m1w1 +m2w2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' This character can be computed using the Weyl character formula (the details of this computation are given in appendix B), χsl3 j = q−m1−m2(−1 + q1+m1)(−1 + q1+m2)(−1 + q2+m1+m2) (−1 + q)3(1 + q) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='7) 4 One-point conformal blocks from AGT In this section, we move to compute the W3 one-point conformal block (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='1) in the limit c → ∞ up to the second level of the module (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' To this end we use the algorithm based on the AGT correspondence (notations and some details of the algorithm are given in appendix C),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' which leads to the following expression F(α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' α1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' q) = 1 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='�18m1m2 − a2(m1 + m2) − 3a(m1 + m2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='9m1m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='q + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='�(a + 3)2(a − 3m1 − 3)(a − 3m2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='81(m1 + 1)m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='+ (a + 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='a2(m1 + m2) + 3a(m1 + m2) − 9(a + 3)m1m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='9m1m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='− (a + 3)2(a − 3m1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='27m1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='− (a + 3)2(a − 3m2 − 3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='27(m2 + 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='+ 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='2(a2 − a − 2) + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='+ (a + 6)(a + 3)(a − 3m1 + 3)(a − 3m1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='162m1(m1 − 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='− (a + 3)(a + 3m2 + 3)(a − 3m2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='27m2(m2 + 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='+ (a + 6)(a + 3)(a − 3m2 + 3)(a − 3m2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='162m2(m2 − 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='+ (a + 3)(a − 3m1)(a + 3m1 + 3)(a − 3m1 − 3m2 − 3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='81m1(m1 + 1)(m1 + m2 + 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='18(a + 6)(a + 3) + a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='3 + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='q2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='1) Now, we propose the following formula that relates the W3 block to the sl3 one in the following way F(α, α1, q) F(α, α1, q)sl3 = χW3 χsl3 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='2) where χW3 and χsl3 are W3 and sl3 (for infinite-dimensional representations) characters re- spectively, given by – 8 – χW3 = 1 �∞ i=1(1 − qi)2 , χsl3 = 1 (1 − q)3(1 + q) , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='3) where the second line is obtained by fixing the proper normalization of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='7) consistent with the standard normalization of CB’s and then taking the limit (m1, m2 → ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' By substitut- ing (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='1) in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='2) and solving for F(α, α1, q)sl3, we recover the expression (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' This confirms, up to the second level, our proposal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' The motivation of this conjectured formula comes from the fact that a similar relation [42] holds for the Virasoro block FVir in the large central charge limit and sl2 block, namely FVir Fsl2 = χVir χsl2 = 1 − q �∞ i=1(1 − qi) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='4) The relation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='2) allows effectively to compute higher-level contributions to the sl3 block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' Note that the characters and blocks involved in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='2) refer to infinite-dimensional represen- tations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' Nevertheless, we can still use this formula for finite-dimensional representations, truncating the series up to an appropriate level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' 5 Conformal Blocks through Wilson line operators In [40], it was proved that the expectation value of the Wilson network operator computes the sl2 toroidal one-point conformal block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' This section aims to show that a similar result is obtained in the case of sl3 algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' In the case of the spherical topology and the sl3 algebra, in [39], it was given the Wilson lines description of the sl3 four-point block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' Here we will combine the ideas of [40] and [39] to proceed with our task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' Before introducing the object we start with some definitions and notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' In the subsequent, we will work with finite-dimensional representations R of sl3 that, in general, are characterized by a highest-weight vector j given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' We will label the finite-dimensional representations R of sl3 by the vectors α, α1, having in mind that they correspond to highest-weight vectors j, j1 according to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='9) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' The Wilson line operators in thermal AdS3 are given by Wα[x, y] = P exp � − � y x Ω � = exp � (x − y)(L1 + 1 4L−1) � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='1) where L1, L−1 are defined in the representation Rα of the sl3 algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' The Wilson network operator allows to compute the conformal block and contains a number of elements (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=', [40]) that are listed below for completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' Φ(hα,qα)(zα, ¯zα) on the boundary is attached to a bulk-to-boundary Wilson line operator Wα[zb, zα] acting in the representation Rα, which connects the boundary point zα to the bulk point zb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' – 9 – 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' Bulk-to-bulk Wilson line operator Wβ[z1, z2] in the representation Rβ, which connects two bulk points and acts trivially for a contractible loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' In the toroidal topology, there exists a non-trivial bulk-to-bulk Wilson loop operator Wβ[zb, zb + 2πτ], associated with a non-contractible cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' A vertex in the bulk, obtained when three Wilson line operators, associated with the representations Rα, Rβ and Rγ, meet each other in the same point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' This vertex is described by the trivalent intertwining operator Iα;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='β,γ : Rβ ⊗ Rγ −→ Rα , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='2) defined by the invariance property Iα;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='β,γUβUγ = UαIα;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='β,γ , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='3) where Ua, (a = α, β, γ), are elements of the gauge group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' With a proper choice of the gauge, we can simplify the expression of the Wilson loop operator [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' Due to the gauge covariance of the Wilson line operator, we can perform a gauge transformation of the connection Ω = U �ΩU −1 , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='4) where, if we choose as gauge element U = e− i 2 L−1eiL1e−i π 2 L0 , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='5) we obtain �Ω = −iL0 dz .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='6) With this particular choice, known as diagonal gauge, we have that the Wilson loop reads Wi[0, 2πτ] = e2πiτL0 = qL0 with q = e2πiτ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='7) The only element that is affected by this choice of the gauge is the external field, which has to transform accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' We consider as the boundary state the lowest-weight state of the representation that has to be transformed as |lw⟩α −→ |� lw⟩α ≡ U −1 α |lw⟩α , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='8) where Uα is the gauge group element (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='5) in the representation Rα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' – 10 – 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='1 Computation of the Wilson line operators We are interested in performing the computation of the Wilson network operator exploiting the tensor products of representations of the sl3 algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' This is based on the fact that every state in a given representation of the algebra can be represented as a symmetric traceless tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' Let us show how this procedure works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' The fundamental representation of sl3 is characterized by the highest-weight vector j = w1 = (1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' This representation contains three states given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='7), which for convenience, we denote as |e1⟩ = w1, |e2⟩ = w2 − w1, |e3⟩ = −w2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='9) Similarly, we denote the states of the anti-fundamental representation (0, 1) (the conjugated of the fundamental) |¯e1⟩ = −w1, |¯e2⟩ = w1 − w2, |¯e3⟩ = w2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='10) For a generic sl3 representation with a highest-weight vector j = m1w1 + m2w2 ≡ (m1, m2) , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='11) the whole vector space of the representation is obtained by applying the lowering generators L1, W1, W2 to the highest-weight vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' Each of these states can be written as the tensor product of the states of the fundamental representation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='9) and its conjugate (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='10) in the following way T k1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='.km2 i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='im1 = 1 m1!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='m2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' |e(i1⟩ ⊗ |ei2⟩ ⊗ · · · ⊗ |eim1)⟩ |¯e(k1⟩ ⊗ |¯ek2⟩ ⊗ · · · ⊗ |¯ekm2)⟩ = = 1 m1!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='m2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' |e(i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='eim1)¯e(k1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='¯ekm2)⟩ , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='12) where the parentheses of lower and upper indices denote the symmetrization, and the tensor T k1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='.km2 i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='im1 has to be traceless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' The highest-weight vector corresponds to the case when all indices i are equal to 1 for the fundamental representation, while for the anti-fundamental one, we require all the indices k to be equal to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' Below, we will need to compute the tensor product of the kind Rα ⊗ Rα1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' In particular, here we need to extract the representation Rα, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=', we need the projection P : Rα ⊗ Rα1 → Rα .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='13) The role of this projector is played by the intertwining operator Iα;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='α,α1 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' In the way this intertwining operator is defined, its matrix elements are given by the Clebsch-Gordan coefficients, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' ⟨a| Iα;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='α,α1 |b⟩ ⊗ |c⟩ = Cabc , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='14) where a, b ∈ Rα, c ∈ Rα1, and Cabc is the Clebsch-Gordan coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' However, in what follows, we are going to define this projector explicitly, since for sl3, the Clebsch-Gordan coefficients are not known for generic representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' – 11 – We want to show that the following expression holds, where F(α, α1, q)sl3 is the sl3 one- point conformal block introduced in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='6) Vα|α1(τ) := � |α⟩∈Rα ⟨α| Wα[zb, zb + 2πτ]Iα;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='α,α1 |α⟩ ⊗ Wα1[zb, z1] |lw⟩α1 = = �C(α, α1)F(α, α1, q)sl3 , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='15) where z1 is the point on the boundary of the solid torus that is a geometric representation of the thermal AdS3, zb is an arbitrary point in the bulk of AdS3, |lw⟩α1 is the lowest-weight state of the representation Rα1 with j1 = (a, 0) and �C(α, α1) is an overall constant irrelevant to our discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' By using the diagonal gauge (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='7) it can be shown that Vα|α1(τ) does not depend on zb and z1, and it can be expressed as Vα|α1(τ) = Trα ⟨α| qL0Iα;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='α,α1 |α⟩ ⊗ U −1 α1 |lw⟩α1 , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='16) where Trα = � |α⟩∈Rα and U −1 α1 is given by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='5) in the representation Rα1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' Now, properly inserting resolutions of identities 1 = � Rβ |β⟩ ⟨β| , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='17) where � Rβ stands for the sum over all the states in the representation Rβ, we can write, in a symbolic way, Vα|α1(τ) = � Rα � Rγ ⟨α| qL0 |α⟩ � ⟨α| Iα;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='α,γ |α⟩ ⊗ |γ⟩ � ⟨γ| U −1 α1 |lw⟩α1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='18) Let us see how to represent each of the factors in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' The first term can be represented in terms of the so-called Wigner D-matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' We will be interested in particular in the fundamental and anti-fundamental representations, where the D-matrices read respectively Dij := ⟨ei| qL0 |ej⟩ = � � � q 0 0 0 1 0 0 0 q−1 � � � , ¯Dij := ⟨¯ei| qL0 |¯ej⟩ = � � � q−1 0 0 0 1 0 0 0 q � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='19) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' The second factor in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='18), the one involving the intertwining operator, acts as a projector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' Its explicit action will be explained later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' The third term consists in the coordinates of the transformed lowest-weight state of the external representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' Again, focusing on the fundamental representation, we have that the transformed lowest-weight state is4 U −1 fund |e3⟩ ∝ |e1⟩ + √ 2 |e2⟩ + |e3⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='21) 4In the fundamental representation, we use the following forms of the generators of sl2 subalgebra of sl3 L1 = √ 2 � � � 0 0 0 1 0 0 0 1 0 � � � , L0 = � � � 1 0 0 0 0 0 0 0 −1 � � � , L−1 = √ 2 � � � 0 −1 0 0 0 −1 0 0 0 � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='20) – 12 – This means that, in general, defining pk := ⟨ek| U −1 fund |e3⟩ = δk,1 + √ 2δk,2 + δk,3 k = 1, 2, 3 , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='22) we have that ⟨ek1ek2 · · · eka| U −1 α1 |lw⟩α1 ∼ pk1pk2 · · · pka .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='23) At this stage, we state the main result of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' Even though we do not have, as of this writing, a general proof of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='15) for generic representations Rα, Rα1, we have tested it for some non-trivial representations given by the following values (m1, m2, a) = {(2, 2, 3), (2, 2, 6), (2, 3, 3), (2, 3, 6), (3, 3, 3), (3, 3, 6), (4, 4, 3), (4, 4, 6)} , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='24) obtaining that in all these cases, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='15) holds precisely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' Let us anticipate here that, to obtain a non-vanishing one-point toroidal conformal block, we have to require a ≡ 0 mod 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' The details of the case (2, 2, 3) are given in appendix D as an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' Furthermore, as a complementary motivation of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='15) for the case when a = 0 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=', when Rα1 is the trivial representation), we show explicitly that for representations Rα with highest-weight vector j = (m1, 0) or j = (0, m2), Vα|α1 in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='15) is equal to the character (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='1 Zero-point conformal block The simplest case of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='15) is when Rα1 is the trivial representation (one-dimensional), in which case we expect to find the character (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='7) (or zero-point conformal block) of the sl3 representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' Let us check this statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' We show this by choosing as highest-weight vector of Rα the vector j = (m1, 0)5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' Since Rα1 is the trivial representation, the intertwining operator Iα;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='α,0 (or the projector (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='13)) reduces to the identity operator, thus Vα|0(τ) ≡ Vα = � i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=',im1=1,2,3 ⟨ei1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' eim1| qL0 |e(i1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' eim1)⟩ = = � i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=',im1=1,2,3 Di1(i1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' Dim1im1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='25) Due to the symmetrization of the indices, the above expression can be written as Vα = m1 � m,n,k=0 m+n+k=m1 qmq−n(1)k , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='26) and after some algebra, we can show that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='26) gives Vα = m1 � r=1 � q−r + qr� � −M(m1 + r) + m1 + r 2 − r + 1 � − M(m1) + m1 2 + 1 , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='27) 5The same procedure holds in the case j = (0, m2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' – 13 – where M(r) = 1−(−1)r 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' By expanding χsl3 j=(m1,0) of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='7) in q, we can see that this character is equal to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' We would like to obtain the same result in the general case when m2 ̸= 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=', we want to show the following equation � i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=',im1=1,2,3 k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=',km2=1,2,3 ⟨¯ek1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' ¯ekm2 ei1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' eim1| qL0T k1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='km2 i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='im1 = χsl3 j , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='28) where χsl3 j is the character (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='7), and the tensor T k1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='.km2 i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='im1 is the symmetric (in lower and upper indices) and traceless tensor from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' We do not have a general proof of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' However, the numerical results we have obtained for many values of j confirm this equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='2 One-point conformal block In this section, we elaborate on the expression (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='16) (which we claimed that reproduces the sl3 one-point conformal block F(α, α1, q)sl3) for general representations Rα and Rα1 labelled by the highest-weight vectors j = (m1, m2) and j1 = (a, 0), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' The purpose is to simplify this expression in order to compare it with F(α, α1, q)sl3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' In what follows, we will not give too much attention to overall constants since they are generally ineffective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' For this reason, some of the following equalities have to be considered to hold up to a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' The idea is to represent (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='16) as a symmetric traceless tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' The first step will be constructing a tensor with the right structure to represent the one-point block but without the requirements of symmetry and tracelessness, which will be imposed at the end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' So, let us start defining the following tensor b1···bm2 a1···am1T r1···rm2 s1···sm1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='k1···ka := = ⟨¯er1 · · · ¯erm2 es1 · · · esm1| qL0 |ea1 · · · eam1 ¯eb1 · · · ¯ebm2⟩ ⟨ek1 · · · eka| U −1 α1 |lw⟩α1 , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='29) which, thanks to the definitions (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='19), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='22) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='23), can be rewritten as b1···bm2 a1···am1T r1···rm2 s1···sm1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='k1···ka = Da1s1 · · · Dam1sm1 ¯Db1r1 · · · ¯Dbm2rm2pk1 · · · pka .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='30) The tensor defined in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='29) resembles the quantity (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='16) we are interested in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' However, we need to insert the intertwining operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' In order to do that, let us recall that it acts as a projector (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='13), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=', it translates in the insertion of a tensor P, which allows us to define b1···bm2 a1···am1M c1···cm2 d1···dm1 = � P c1···cm2 d1···dm1 �s1···sm1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='k1···ka r1···rm2 b1···bm2 a1···am1T r1···rm2 s1···sm1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='k1···ka .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='31) In order to construct the tensor P, we follow the method described in [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' It has to be a product of sl3 invariant tensors, which are δi j , ϵijk , ϵijk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='32) – 14 – However, since the final result has to be traceless and symmetric, the only possible contrac- tions we are interested in are with δk r , ϵcsk , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='33) where we have labelled the indices coherently to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' If we suppose that P is made out of x contractions with δk r and y contractions with ϵcsk, since the remaining indices have to be the same in number as the expected ones, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=', m1 lower and m2 upper indices, a simple counting shows us that a has to satisfy the condition a = 3l , l ∈ Z≥0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='34) In this way, we obtain that the tensor that represents the intertwining operator is � P c1···cm2 d1···dm1 �s1···sm1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='k1···ka r1···rm2 = δs1 d1 · · · δ sm1−l dm1−lδk1 dm1−l+1 · · · δkl dm1δc1 r1 · · · δ cm2−l rm2−l× × δkl+1 rm2−l+1 · · · δk2l rm2ϵcm2−l+1sm1−l+1k2l+1 · · · ϵcm2sm1k3l .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='35) Inserting this projector into (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='31), we end up with b1···bm2 a1···am1M c1···cm2 d1···dm2 = Da1d1 · · · Dam1−ldm1−l ¯Db1c1 · · · ¯Dbm2−lcm2−lpdm1−l+1 · · · pdm1× × ¯pbm2−l+1 · · · ¯pbm2 A cm2−l+1 am1−l+1 · · · A cm2 am1, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='36) where we have introduced the new quantities ¯pb := ¯Dbrpr , Ac a := ϵcskDaspk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='37) We want to make this tensor symmetric, and we do this procedure just formally, remembering in what follows that we have to consider all the possible cyclic permutations of the indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' Hence we write b1···bm2 a1···am1M c1···cm2 d1···dm1 ≡ b1···bm2 a1···am1M (c1···cm2) (d1···dm1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='38) The last step consists in making this tensor traceless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' We do this by exploiting the procedure described in [39], where it is shown that a symmetric tensor can be made traceless as b1···bm2 a1···am1 � M c1···cm2 d1···dm1 = min � n=0 Cnδ(c1 (d1 · · · δcn dn b1···bm2 a1···am1M cn+1···cm2)f1···fn dn+1···dm1)f1···fn , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='39) where Cn = (−1)n (m1 + m2 − n + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' (m1 + m2 + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' (m1 − n)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' (m2 − n)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' , min = min(m1, m2) , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='40) and in each summand, we take the sums over repeated indices fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' Finally, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='15) can be written as b1···bm2 a1···am1 � M (c1···cm2) (d1···dm1)δb1c1 · · · δbm2cm2δa1d1 · · · δam1dm1 = �C(α, α1)F(α, α1, q)sl3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='41) As of the date of this work, we have not found a general proof of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='41), but many numerical computations (which are partially described in the appendix D) confirm this equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' – 15 – 6 Discussion In this work we studied the semiclassical limit of W3 CFT on the torus, with a focus on the sl3 one-point conformal blocks of fully degenerate primary fields for an external field satisfy- ing (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' We compute the W3 block in the light c → ∞ limit using AGT correspondence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' In order to obtain the sl3 block we propose the relation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' The utility of this relation is twofold since it serves as a check of the dual representation for sl3 conformal block and provides a computational method to calculate its higher-level contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' The dual description of sl3 one-point toroidal conformal block is given in terms of the Wilson lines in 3d Chern-Simons gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' The proposal summarizes in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='41).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' The dual approach seems to give more manageable computations compared to those based on the original definition, at least in its construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' The simplest test of this proposal is the case when the one-point block reduces to the zero-point block, which we confirmed analytically and numerically in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' Further, we have performed more general tests for specific cases of representations (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='24), obtaining that all these examples confirm (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' The absence of a general expression of sl3 Clebsch-Gordan coefficients forces us to use the tensor technique in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='1, the general proof of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='41) remains an open problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' From the Wilson line construction a natural generalization follows, that is the study of the dual description in the cases of higher-point conformal blocks, which show additional features, like the presence of different intermediate channels (usually called s and t), as shown for the sl2 case in [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' This generalization is straightforward in the context of the Wilson lines formulation of the conformal blocks, according to the construction of section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' Also, it would be worth to extend this approach to the whole W3 CFT, which would require to focus also on quantum corrections (for related consideration see [26]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' Another interesting question is how the Wilson line approach works if multiplicities in the representations under consideration are present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' From the dual construction it follows that in such cases we can obtain multiple nonequivalent expressions for the projector (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' For instance, if the external field is in the adjoint representation, there is a multiplicity two for the conformal blocks in the W3 CFT [46], which corresponds to the fact that Clebsch- Gordan coefficients are not uniquely defined [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' In this case, the multiplicity is present also in the definition of the projector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' It would be interesting to establish a precise correspondence between uncertainty related to the OPE multiplicities for conformal blocks and multiplicities arising in the definition of the projector in the dual construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' A Matrix Elements Here, we give the matrix elements that enter in the computation of the sl3 conformal block (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='6) at the first and second levels of sl3 module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' We use the algorithm described in [48] to compute these matrix elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' – 16 – First level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' From the sl3 commutation relations, one can find the following products ⟨h0, q0|L1L−1|h0, q0⟩ = 2h0 , ⟨h0, q0|W1L−1|h0, q0⟩ = 3q0 , ⟨h0, q0|W1W−1|h0, q0⟩ = −h0/5 , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='1) thus the Shapavalov matrix G at the first level is G = � 2h0 3q0 3q0 − h0 5 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='2) The matrix elements at the first level are6 ⟨h0, q0|L1Φα1L−1|h0, q0⟩ = 2h0 + (h1 − 1) h1, ⟨h0, q0|W1Φα1L−1|h0, q0⟩ = 3q0 + 1 2 (h1 − 1) q1, ⟨h0, q0|L1Φα1W−1|h0, q0⟩ = 3q0 − 1 2 (h1 − 1) q1, ⟨h0, q0|W1Φα1W−1|h0, q0⟩ = 1 5 (h1 − h0) − (h1 − 3) q2 1 4h1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='3) Second level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' At this level, we consider the basis L2 −1 , W2 −1 , L−1W−1 , W−2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='4) We obtain the following products ⟨h0, q0|L2 1L2 −1|h0, q0⟩ = 4h0 (2h0 + 1) , ⟨h0, q0|W2 1L2 −1|h0, q0⟩ = 18q2 0 − 6h0 5 , ⟨h0, q0|W1L1L2 −1|h0, q0⟩ = 6 (2h0 + 1) q0, ⟨h0, q0|W2L2 −1|h0, q0⟩ = 12q0, ⟨h0, q0|W2 1W2 −1|h0, q0⟩ = 1 25h0 (2h0 + 1) , ⟨h0, q0|W1L1W2 −1|h0, q0⟩ = 1 5(−3) (2h0 + 3) q0, ⟨h0, q0|W2W2 −1|h0, q0⟩ = 6q0 5 , ⟨h0, q0|W1L1L−1W−1|h0, q0⟩ = 9q2 0 − 2 5h0 (h0 + 1) , ⟨h0, q0|W2L−1W−1|h0, q0⟩ = −1 5 (4h0) , ⟨h0, q0|W2W−2|h0, q0⟩ = 8h0 5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='5) 6All the matrix elements are proportional to ⟨h0, q0|Φα1|h0, q0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' But, for the sake of brevity, we omit this factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' – 17 – Hence the Shapavalov matrix G at the second level is G = � � � � � 4h0 (2h0 + 1) 18q2 0 − 6h0 5 6 (2h0 + 1) q0 12q0 18q2 0 − 6h0 5 1 25h0 (2h0 + 1) 1 5(−3) (2h0 + 3) q0 6q0 5 6 (2h0 + 1) q0 1 5(−3) (2h0 + 3) q0 9q2 0 − 2 5h0 (h0 + 1) − 1 5 (4h0) 12q0 6q0 5 − 1 5 (4h0) 8h0 5 � � � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='6) The matrix elements at the second level are ⟨h0, q0|L2 1Φα1L2 −1|h0, q0⟩ = 8h2 0 + (8 (h1 − 1) h1 + 4) h0 + (h1 − 1) h1 ((h1 − 1) h1 + 2) , ⟨h0, q0|L2 1Φα1W2 −1|h0, q0⟩ = 18q2 0 − 6 (h1 − 1) q1q0 + � h3 1 − 7h1 + 6 � q2 1 4h1 − 1 5 (h1 − 1) h1 (h1 + 4) − 6h0 5 , ⟨h0, q0|L2 1Φα1L−1W−1|h0, q0⟩ = 6 (2h0 + (h1 − 1) h1 + 1) q0 − 1 2 (h1 − 1) (4h0 + (h1 − 1) h1 + 2) q1, ⟨h0, q0|L2 1Φα1W−2|h0, q0⟩ = 12q0 − 2 (h1 − 1) h1q1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='7) ⟨h0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' q0|W2 1Φα1L2 −1|h0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' q0⟩ = 18q2 0 + 6 (h1 − 1) q1q0 + � h3 1 − 7h1 − 30 � q2 1 4h1 − 1 5h1 (h1 (h1 + 3) − 2) − 6h0 5 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' ⟨h0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' q0|W2 1Φα1W2 −1|h0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' q0⟩ = (h1 − 6) (h1 − 3) (h1 + 3) q4 1 16h3 1 + (2h0 (h1 − 3) − 3 (h1 − 2) h1 + 12) q2 1 10h1 + 1 25 � 2h2 0 + (1 − 4h1) h0 + h1 (3h1 + 2) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' ⟨h0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' q0|W2 1Φα1L−1W−1|h0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' q0⟩ = 12 40 �15q2 1 h1 − 5q2 1 − 4h0 + 4h1 − 6 � q0+ + 1 40 � 12h2 1 − 8 (h0 + 1) h1 + 8h0 − 5 ((h1 − 3) (h1 − 1) h1 − 36) q2 1 h2 1 + 4 � q1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' ⟨h0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' q0|W2 1Φα1W−2|h0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' q0⟩ = −(h1 − 6) (h1 − 3) q3 1 2h2 1 + 2 5 (2h1 − 3) q1 + 6q0 5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='8) – 18 – ⟨h0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' q0|W1L1Φα1L2 −1|h0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' q0⟩ = 6 (2h0 + (h1 − 1) h1 + 1) q0+ + 1 2 (h1 − 1) (4h0 + (h1 − 1) h1 + 2) q1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' ⟨h0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' q0|W1L1Φα1W2 −1|h0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' q0⟩ = 12 40 �15q2 1 h1 − 5q2 1 − 4h0 + 4h1 − 6 � q0+ + 1 40 (h1 − 1) q1 � 5 (h1 − 6) (h1 + 3) q2 1 − 4h2 1 (−2h0 + 3h1 + 2) � h2 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' ⟨h0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' q0|W1L1Φα1L−1W−1|h0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' q0⟩ = 1 5 � −2h2 0 − (h1 − 2) (h1 − 1) h0 + 45q2 0 + h1 ((h1 − 1) h1 + 2) � − (h1 − 3) (2h0 + (h1 − 1) h1 + 2) q2 1 4h1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' ⟨h0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' q0|W1L1Φα1W−2|h0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' q0⟩ = 2 5 � h2 1 + h1 − 2h0 � − ((h1 − 6) h1 + 3) q2 1 h1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='9) ⟨h0, q0|W2Φα1L2 −1|h0, q0⟩ = 2 (6q0 + (h1 − 1) h1q1) , ⟨h0, q0|W2Φα1W2 −1|h0, q0⟩ = � h2 1 − 9 � q3 1 2h2 1 − 3 5 (h1 − 1) q1 + 6q0 5 , ⟨h0, q0|W2Φα1L−1W−1|h0, q0⟩ = 1 10 �5 (h1 (3 − 2h1) + 3) q2 1 h1 − 8h0 + 2h1 (h1 + 3) � , ⟨h0, q0|W2Φα1W−2|h0, q0⟩ = 8h0 5 − 4 (h1 − 3) q2 1 h1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='10) B Weyl character formula Here, we want to compute the character χsl3 j (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='7) of a finite-dimensional representation of sl3 with highest-weight vector given by j = m1w1 + m2w2, where w1,2 are the fundamental weights and m1, m2 are positive integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' Let us define some notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' The fundamental weights satisfy the following products (w1, w1) = (w2, w2) = 2 3, (w1, w2) = 1 3 , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='1) The Weyl vector is defined as ρ = w1 + w2 , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='2) and the Weyl group W of sl3 is formed by 6 elements, which are denoted as W = [1, s1, s2, s1s2, s2s1, s1s2s1] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='3) – 19 – and they act on the vector j as follows 1(j) = j , s1(j) = −m1w1 + (m1 + m2)w2 , s2(j) = (m1 + m2)w1 − m2w2 , s1s2(j) = −(m1 + m2)w1 + m1w2 , s2s1(j) = m2w1 − (m1 + m2)w2 , s1s2s1(j) = −m1w1 − m2w2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='4) The signature ϵ(w) of a element w ∈ W is defined by ϵ(w) = (−1)l(w) , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='5) where l(w) is the length of w, that is the number of si that w contains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' For example, the signatures of the elements (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='3) are respectively ϵ(W) = [1, −1, −1, 1, 1, −1] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='6) We compute the character χsl3 j from the Weyl character formula χsl3 j = � w∈W ϵ(w)e(w(j+ρ),2πiτρ) � w∈W ϵ(w)e(w(ρ),2πiτρ) , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='7) where we use q = e2πiτ, obtaining exactly (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' C AGT relation In this section, we comment on some important details of the AGT relation we used to compute the expression (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' According to the AGT relation, the W3 conformal block F(α, α1, q) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='1) can be computed by F(α, α1, q) = Zinst SU(3) ∞ � i=1 (1 − qi)1−2h1 , (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='1) where the infinite product is the so-called Heisenberg factor, and Zinst SU(3) is the SU(3) instanton partition function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' For the algorithm which we used to compute Zinst SU(3) we refer to [49], specifically the section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' Here, we want to clarify the relations between the parameters of F(α, α1, q) and Zinst SU(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' The partition function Zinst SU(3) is given by equation (28) of [49] and depends on the pa- rameters xi = (Q − α, ei), µ, ϵ1, ϵ2 which are related to our parameters as follows µ = a/3, ϵ1 = b, ϵ2 = 1/b, (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='2) while α is given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='9) and Q, ei are according to our notations above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' Finally, the expres- sion (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='1) is given in the limit c → ∞ (or equivalently b → 0) of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' – 20 – D Examples The simplest test7 of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='41) we can perform is the case j = 2w1 + 2w2 = (2, 2), j1 = (3, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' In this case the tensor (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='38) becomes b1b2 a1a2Mc1c2 d1d2 = 1 4 ¯pb2(Da1d1pd2 + Da1d2pd1)( ¯Db1c1Ac2 a2 + ¯Db1c2Ac1 a2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='1) The next step is to make this tensor traceless according to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='39) and then take the contraction according to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='41).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' We can show that there will be 16 terms proportional to C1, and all the terms proportional to C2 vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' To write the result, we use the notation of the matrix B Ba b = ¯papb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='2) By using this notation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' and denoting the trace of a generic matrix C by Tr[C] = [C],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' one can show that b1b2 a1a2 � M(c1c2) (d1d2)δb1c1δb2c2δa1d1δa2d2 = = 1 4 � C0 � [D]2[AB] + [D][ ¯DBA] + [ ¯D][DAB] + [DA ¯DB] � + C1 4 � [D ¯DBAT ] + [DAB ¯D] + [D ¯DBA] + [ ¯DD][AB] + [D][ ¯DBAT ] + [D][B ¯DA] + [ ¯DBDAT ] + [D ¯DBAB] + [ ¯D][DABT ] + [ ¯D][DAT B] + [DA ¯DBT ] + [D ¯DBAT BT ] + [ ¯D][D][AT B] + [D][ ¯DAT B] + [ ¯D][BDAT ] + [D ¯DAT B] � + (0)C2 � = = �C(2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' 3)(1 − 8 5q2 − 4 5q3 + 4 5q5 + 8 5q6 − q8) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='3) The firs three terms in q, (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' up to q2) of the above expression are exactly the terms obtained by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='6) in this case (m1, m2, a → (2, 2, 3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' This validates (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='41).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' It is clear that the difficulty of this “simple” problem increases substantially when m1, m2, a increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' We were able to work out the cases (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' Here we give the results and let us emphasize that all these cases confirm (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='41).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' Cases : (m1, m2, a) = (2, 2, 6) b1b2 a1a2 � M(b1b2) (a1a2) = �C(2, 2, 6)(1 − 4q + 4q2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' ) , (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='4) (m1, m2, a) = (2, 3, 3) b1b2b3 a1a2 � M(b1b2b3) (a1a2) = �C(2, 3, 3)(1 + q 3 − 7q2 9 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' ) , (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='5) 7Since the expression (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='6) contains contributions of two states to the first level and four states to the second level, then m1, m2 ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' – 21 – (m1, m2, a) = (2, 3, 6) b1b2b3 a1a2 � M(b1b2b3) (a1a2) = �C(2, 3, 6)(1 − 3q + q2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' ) , (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='6) (m1, m2, a) = (3, 3, 3) b1b2b3 a1a2a3 � M(b1b2b3) (a1a2a3) = �C(3, 3, 3)(1 + 2q 3 + 2q2 21 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' ) , (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='7) (m1, m2, a) = (3, 3, 6) b1b2b3 a1a2a3 � M(b1b2b3) (a1a2a3) = �C(3, 3, 6)(1 − 2q − 10q2 7 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' ) , (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='8) (m1, m2, a) = (4, 4, 3) b1b2b3b4 a1a2a3a4 � M(b1b2b3b4) (a1a2a3d4) = �C(4, 4, 3)(1 + q + q2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' ) , (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='9) (m1, m2, a) = (4, 4, 6) b1b2b3b4 a1a2a3a4 � M(b1b2b3b4) (a1a2a3a4) = �C(4, 4, 6)(1 − q − 5q2 3 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content='10) References [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' Belavin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' Polyakov and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE3T4oBgHgl3EQfiAoP/content/2301.04575v1.pdf'} +page_content=' B.' metadata={'source': 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In the year 2000, Eric Egge introduced the generalized Terwilliger algebra +T of a distance-regular graph Γ. For any vertex x of Γ, there is a surjective algebra +homomorphism ♮ from T to the Terwilliger algebra T (x). If Γ is complete, then ♮ is an +isomorphism. If Γ is not complete, then ♮ may or may not be an isomorphism, and in +general the details are unknown. We show that if Γ is a hypercube, then the algebra +homomorphism ♮ : T → T (x) is an isomorphism for all vertices x of Γ. +Keywords. Distance-regular graph; Terwilliger algebra; Hamming cube; hypercube. +2020 Mathematics Subject Classification. 05E30. +1. Introduction +In the topic of Algebraic Combinatorics, there is a family of finite undirected graphs said +to be distance-regular [3]. These graphs are heavily studied; see [1–3,7,16]. For a vertex +x of a distance-regular graph Γ, the associated Terwilliger algebra T(x) was introduced +in [17] (there called the subconstituent algebra). This algebra is finite-dimensional, semi- +simple, and noncommutative in general. Some notable papers about T(x) are [4–6,8,11, +12, 14, 15, 17–19]. There are some well-known relations in T(x) called the triple product +relations; the form of the triple product relations is independent of x (see [17, Lemma +3.2]). +In [9], Eric Egge introduced the generalized Terwilliger algebra T of Γ. This algebra +is defined by generators and relations; the main relations are the triple product relations. +By construction, for any vertex x of Γ there is a surjective algebra homomorphism ♮ : +T → T(x). If the graph Γ is complete, then ♮ is an isomorphism. If Γ is not complete, +then ♮ may or may not be an isomorphism, and in general the details are unknown. +There is a type of distance-regular graph called the hypercube. +The hypercube of +diameter d is often denoted Qd. +General information about Qd can be found in [3, +Chapters 1 and 9]. In [10], Junie Go showed that for any vertex x of Qd there is an +algebra isomorphism from T(x) to a direct sum of full matrix algebras: +T(x) → +� +0≤r≤⌊d/2⌋ +Md+1−2r(C). +In this paper, we investigate the algebra T associated with Qd. Our main result is that +for any vertex x of Qd, the algebra homomorphism ♮ : T → T(x) is an isomorphism. To +prove this result, we use the following strategy. +Writing Td for the generalized Terwilliger algebra associated with Qd, we will display +an algebra isomorphism +Td → Md+1(C) ⊕ Td−2 +Date: January 23, 2023. +1 + +2 +NATHAN NICHOLSON +for d ≥ 2. We will then argue by induction on d. +The paper is organized as follows. In Sections 2 and 3, we discuss the algebras T(x) +and T for any distance-regular graph. In Sections 4 and 5, we discuss T(x) and T for +the hypercube Qd. In Section 6, we discuss a certain central idempotent u0 of T that was +introduced in [9]. In Section 7, we prove the main result. +2. Distance-regular Graphs +In this section, we review some definitions and results concerning distance-regular +graphs and the Terwilliger algebra. For more information, we refer the reader to [3,16,17]. +Suppose X is a nonempty finite set. Let MX(C) denote the algebra consisting of the +matrices with rows and columns indexed by X and entries in C. Let CX denote the vector +space over C consisting of column vectors with coordinates indexed by X and entries in +C. The algebra MX(C) acts on CX by left multiplication. For any positive integer n, let +Mn(C) denote the algebra consisting of square n × n matrices with entries in C. +Let Γ = (X, R) denote a finite, undirected, connected graph, without loops or multiple +edges, with vertex set X, edge set R, and path-length distance function ∂. Let +d = max{∂(x, y) | x, y ∈ X}. +We call d the diameter of Γ. Vertices x, y ∈ X are said to be adjacent whenever they +form an edge. For any vertex x ∈ X, let Γ(x) denote the set of vertices that are adjacent +to x. +Define A ∈ MX(C) with (x, y)-entry +Axy = +� +1 +if x and y are adjacent; +0 +if x and y are not adjacent +(x, y ∈ X). +We call A the adjacency matrix of Γ. +We say that Γ is regular with valency k whenever k = |Γ(x)| for all x ∈ X. +We say that Γ is distance-regular whenever for any 0 ≤ h, i, j ≤ d, and any vertices +x, y ∈ X with ∂(x, y) = h, the number of vertices z ∈ X such that ∂(x, z) = i and +∂(y, z) = j is a constant depending only on h, i, j, but not on x and y. +We denote +this constant by ph +ij and call it an intersection number of Γ. +Observe that ph +ij = ph +ji +(0 ≤ h, i, j ≤ d). +For the rest of this section, assume Γ is distance-regular. Note that Γ is regular with +valency k = p0 +11 if d ≥ 1. +We now recall the Bose-Mesner algebra of Γ. For 0 ≤ i ≤ d, define Ai ∈ MX(C) with +(x, y)-entry +(Ai)xy = +� +1 +if ∂(x, y) = i; +0 +if ∂(x, y) ̸= i +(x, y ∈ X). +We call Ai the ith distance matrix of Γ. Note that A0 = I, where I ∈ MX(C) denotes the +identity matrix. Note that A1 = A, provided that d ≥ 1. The sum �d +i=0 Ai = J, where +J ∈ MX(C) denotes the matrix with every entry equal to 1. For notational convenience, +define +(2.1) +Ai = 0 +(i < 0 or i > d). + +THE GENERALIZED TERWILLIGER ALGEBRA OF THE HYPERCUBE +3 +By [3, p. 44], +(2.2) +AiAj = +d +� +h=0 +ph +ijAh +(0 ≤ i, j ≤ d). +Thus {Ai}d +i=0 forms a basis for a commutative subalgebra M of MX(C). The matrix A +generates M by [3, p. 127]. We call M the Bose-Mesner algebra of Γ. Note that M has +dimension d + 1. +For 0 ≤ i ≤ d, define ki = p0 +ii. For any x ∈ X, ki is equal to the number of vertices at +distance i from x. We call ki the ith valency of Γ. Note that k0 = 1. Moreover, k1 = k if +d ≥ 1. +The matrix A has d+1 distinct eigenvalues, because A generates M and M has dimen- +sion d+1. Denote these eigenvalues by θ0 > θ1 > · · · > θd. For 0 ≤ i ≤ d, let Ei ∈ MX(C) +be the matrix which acts as I on the θi-eigenspace of A and as 0 on all other eigenspaces +of A. The matrices {Ei}d +i=0 form a basis for M, and +(2.3) +d +� +i=0 +Ei = I, +EiEj = δijEi +(0 ≤ i, j ≤ d). +We call Ei the primitive idempotent of Γ associated with θi (0 ≤ i ≤ d). Define +(2.4) +Ei = 0 +(i < 0 or i > d). +By [3, p. 45], we have E0 = |X|−1J. By construction, +(2.5) +A = +d +� +i=0 +θiEi. +Next we recall the Krein parameters of Γ. For B, C ∈ MX(C), let B ◦ C denote their +entry-wise product. Note that +Ai ◦ Aj = δijAi +(0 ≤ i, j ≤ d). +Consequently, M is closed under ◦. Because {Ei}d +i=0 is a basis for M, there exist scalars +(ph +ij)∗ ∈ C (0 ≤ h, i, j ≤ d) such that +(2.6) +Ei ◦ Ej = |X|−1 +d +� +h=0 +(ph +ij)∗Eh +(0 ≤ i, j ≤ d). +The scalars (ph +ij)∗ are called the Krein parameters of Γ. By [3, p. 50], (ph +ij)∗ is real and +nonnegative for 0 ≤ h, i, j ≤ d. +For 0 ≤ i ≤ d, define k∗ +i = (p0 +ii)∗. We call k∗ +i the ith dual valency of Γ. +We next recall the dual Bose-Mesner algebras of Γ. Fix a vertex x ∈ X. For 0 ≤ i ≤ d, +let E∗ +i = E∗ +i (x) denote the diagonal matrix in MX(C) with (y, y)-entry +(E∗ +i )yy = +� +1 +if ∂(x, y) = i; +0 +if ∂(x, y) ̸= i +(y ∈ X). +We call E∗ +i the ith dual primitive idempotent of Γ with respect to x. Define +(2.7) +E∗ +i = 0 +(i < 0 or i > d). + +4 +NATHAN NICHOLSON +By construction, +d +� +i=0 +E∗ +i = I, +E∗ +i E∗ +j = δijE∗ +i +(0 ≤ i, j ≤ d). +It follows that {E∗ +i }d +i=0 forms a basis for a commutative subalgebra M∗ = M∗(x) of +MX(C). We call M∗ the dual Bose-Mesner algbebra of Γ with respect to x. +For 0 ≤ i ≤ d, we define a diagonal matrix A∗ +i = A∗ +i (x) ∈ MX(C) with (y, y)-entry +(A∗ +i )yy = |X|(Ei)xy +(y ∈ X). +Note that A∗ +0 = I. For d ≥ 1, we abbreviate A∗ = A∗ +1 and call this the dual adjacency +matrix of Γ with respect to x. Define +(2.8) +A∗ +i = 0 +(i < 0 or i > d). +By [16, Corollary 11.6], A∗ generates M∗. +By (2.6), +A∗ +i A∗ +j = +d +� +h=0 +(ph +ij)∗A∗ +h +(0 ≤ i, j ≤ d). +By [16, Lemma 5.8], the matrices {A∗ +i }d +i=0 form a basis for M∗. +We next recall the Terwilliger algebra of Γ. Let T = T(x) denote the subalgebra of +MX(C) generated by M and M∗. We call T the Terwilliger algebra of Γ with respect to +x. We remark that T is sometimes called the subconstituent algebra. +For d ≥ 1, since the matrices {E∗ +i }d +i=0 form a basis for M∗, there exist scalars θ∗ +i ∈ C +(0 ≤ i ≤ d) such that +A∗ = +d +� +i=0 +θ∗ +i E∗ +i . +Because {Ai}d +i=0 and {Ei}d +i=0 each constitute bases for M, there must exist scalars +pi(j), qi(j) ∈ C (0 ≤ i, j ≤ d) such that +Ai = +d +� +j=0 +pi(j)Ej, +Ei = |X|−1 +d +� +j=0 +qi(j)Aj +(0 ≤ i ≤ d). +(2.9) +Similarly, there must exist scalars p∗ +i (j), q∗ +i (j) ∈ C (0 ≤ i, j ≤ d) such that +A∗ +i = +d +� +j=0 +p∗ +i (j)E∗ +j , +E∗ +i = |X|−1 +d +� +j=0 +q∗ +i (j)A∗ +j +(0 ≤ i ≤ d). +By the construction of {E∗ +i }d +i=0 and {A∗ +i }d +i=0, we have +(2.10) +qi(j) = p∗ +i (j), +pi(j) = q∗ +i (j) +(0 ≤ i, j ≤ d). +For d ≥ 1, we have +(2.11) +θi = p1(i), +θ∗ +i = p∗ +1(i) +(0 ≤ i ≤ d). +By [3, Lemma 2.2.1], +(2.12) +p0(j) = 1, +q0(j) = 1 +(0 ≤ j ≤ d). + +THE GENERALIZED TERWILLIGER ALGEBRA OF THE HYPERCUBE +5 +Define matrices P, Q ∈ Md+1(C) with entries Pij = pj(i) and Qij = qj(i) (0 ≤ i, j ≤ d). +Then P is the change of basis matrix from {Ei}d +i=0 to {Ai}d +i=0, and |X|−1Q is the change +of basis matrix from {Ei}d +i=0 to {Ai}d +i=0. Moreover, P and |X|−1Q are inverses. +By (2.10), |X|−1P is the change of basis matrix from {A∗ +i }d +i=0 to {E∗ +i }d +i=0, and Q is the +change of basis matrix from {E∗ +i }d +i=0 to {A∗ +i }d +i=0. +By [17, Lemma 3.2], the following hold for 0 ≤ h, i, j ≤ d: +E∗ +hAiE∗ +j = 0 +iff +ph +ij = 0, +EhA∗ +i Ej = 0 +iff +(ph +ij)∗ = 0. +The above statements are referred to as the triple product relations. +To conclude this section, we recall the notion of self-duality. We say that Γ is self- +dual whenever ph +ij = (ph +ij)∗ (0 ≤ h, i, j ≤ d). In this case, we have ki = k∗ +i and θi = θ∗ +i +(0 ≤ i ≤ d), and we have pi(j) = qi(j) (0 ≤ i, j ≤ d). See [3, p. 49]. +3. The Generalized Terwilliger Algebra +In this section, we recall the generalized Terwilliger algebra and some related results +from [9]. Let Γ = (X, R) denote a distance regular graph with diameter d. We fix a vertex +x ∈ X and let T = T(x). +Definition 3.1. (See [9, Definition 4.1]) Let T denote the algebra with 1, with generators +x0, . . . , xd, x∗ +0, . . . , x∗ +d and relations (T1)–(T3∗). +(T1) +x0 = x∗ +0 = 1. +(T2) +xixj = +d +� +h=0 +ph +ijxh +(0 ≤ i, j ≤ d). +(T2∗) +x∗ +i x∗ +j = +d +� +h=0 +(ph +ij)∗x∗ +h +(0 ≤ i, j ≤ d). +(T3) +e∗ +hxie∗ +j = 0 if ph +ij = 0 +(0 ≤ h, i, j ≤ d). +(T3∗) +ehx∗ +i ej = 0 if (ph +ij)∗ = 0 +(0 ≤ h, i, j ≤ d). +In the above lines, we define +(3.1) +ei = |X|−1 +d +� +j=0 +qi(j)xj, +e∗ +i = |X|−1 +d +� +j=0 +q∗ +i (j)x∗ +j +(0 ≤ i ≤ d). +We call T the generalized Terwilliger algebra associated with Γ. +For notational convenience, define +xi = 0, +x∗ +i = 0, +ei = 0, +e∗ +i = 0 +(i < 0 or i > d). +(3.2) +Remark 3.2. If d = 0, the algebra T is isomorphic to C. +Remark 3.3. In [9, Definition 4.1], the relation (T1) is given as x0 = x∗ +0. In [9, Propo- +sition 5.1], a proof is given that x0 = x∗ +0 = 1. However the proof is not correct, and this +can be seen by considering the case d = 0. Thus we adjusted our statement of (T1) to +incorporate the author’s assumption that x0 = x∗ +0 = 1. + +6 +NATHAN NICHOLSON +Next we recall some results about T . +Lemma 3.4. (See [9, p. 3]) There exists an algebra homomorphism ♮ : T → T which +sends +xi �→ Ai, +x∗ +i �→ A∗ +i , +ei �→ Ei, +e∗ +i �→ E∗ +i , +for 0 ≤ i ≤ d. Moreover, ♮ is surjective. +The map ♮ in Lemma 3.4 is not an isomorphism in general. However we do have the +following results. +Lemma 3.5. (See [9, Propositions 5.4 and 10.2]) The following (i)–(iii) hold. +(i) The elements {xi}d +i=0 form a basis for a commutative subalgebra C of T . +(ii) The elements {ei}d +i=0 form a basis for C. +(iii) The restriction of ♮ to C induces an algebra isomorphism C → M. +Lemma 3.6. (See [9, Propositions 5.7 and 10.3]) The following (i)–(iii) hold. +(i) The elements {x∗ +i }d +i=0 form a basis for a commutative subalgebra C∗ of T . +(ii) The elements {e∗ +i }d +i=0 form a basis for C∗. +(iii) The restriction of ♮ to C∗ induces an algebra isomorphism C∗ → M∗. +The next lemmas are consequences of Lemmas 3.5 and 3.6. +Lemma 3.7. (See [9, Propositions 5.3 and 5.6]) The following (i), (ii) hold in T . +(i) �d +i=0 ei = 1. +(ii) �d +i=0 e∗ +i = 1. +Proof. Follows from (2.3) and Lemmas 3.5 and 3.6. +□ +Lemma 3.8. (See [9, Propositions 5.3 and 5.6]) For 0 ≤ i, j ≤ d, the following (i), (ii) +hold in T . +(i) eiej = δijei. +(ii) e∗ +i e∗ +j = δije∗ +j. +Proof. Follows from (2.3) and Lemmas 3.5 and 3.6. +□ +Lemma 3.9. (See [9, Propositions 5.3 and 5.6]) For d ≥ 1, the following (i), (ii) hold in +T . +(i) x1 = �d +i=0 θiei. +(ii) x∗ +1 = �d +i=0 θ∗ +i e∗ +i . +Proof. Follows from (2.5) and Lemmas 3.5 and 3.6. +□ +Lemma 3.10. The following (i), (ii) hold in T . +(i) e0 = |X|−1 �d +i=0 xi. +(ii) e∗ +0 = |X|−1 �d +i=0 x∗ +i . +Proof. Follows from (2.10), (2.12), and (3.1). +□ + +THE GENERALIZED TERWILLIGER ALGEBRA OF THE HYPERCUBE +7 +4. The Hypercube Qd +In this section, we recall a family of distance-regular graphs called the hybercubes, and +we review some results related to the associated Terwilliger Algebra. For more informa- +tion, we refer the reader to [3,10,13]. +Definition 4.1. Assume d ≥ 0. Let Qd denote the graph with vertex set X consisting of +d-tuples (a1, a2, . . . , ad) such that ai ∈ {−1, 1} (1 ≤ i ≤ d). Two vertices are adjacent in +Qd whenever they differ in exactly one coordinate. The graph Qd is called the d-cube or +a hypercube. +The graph Qd has 2d vertices. Furthermore, Qd is distance-regular (see [3, Proposition +1.12.1]). +From now through Lemma 4.17, we consider the distance-regular graph Γ = Qd with +d ≥ 1. We now recall the intersection numbers of Qd. +Proposition 4.2. (See [13, p. 238]) For 0 ≤ h, i, j ≤ d we have +ph +ij = +� +0 +if h + i + j is odd; +� +h +(i−j+h)/2 +�� +d−h +(i+j−h)/2 +� +if h + i + j is even. +In the lines above, we interpret +�n +m +� += 0 if m < 0 or m > n. +We mention some consequences of Proposition 4.2. +Corollary 4.3. For 0 ≤ i ≤ d we have ki = +�d +i +� +. +Proof. Follows from Proposition 4.2. +□ +Corollary 4.4. For 0 ≤ h, i ≤ d we have +ph +1i = + + + + + +d − h +if h = i − 1; +h +if h = i + 1; +0 +if h ̸= i ± 1. +Proof. Follows from Proposition 4.2. +□ +Corollary 4.5. For 0 ≤ i ≤ d we have +AAi = (i + 1)Ai+1 + (d − i + 1)Ai−1, +where we recall (2.1). +Proof. Follows from (2.2) and Corollary 4.4. +□ +Let z denote an indeterminate, and let C[z] denote the algebra of polynomials in z that +have coefficients in C. Motivated by Corollary 4.5, we now define some polynomials in +C[z]. +Definition 4.6. Let {Fi}d+1 +i=0 denote polynomials in C[z] such that F0 = 1, F1 = z, and +zFi = (i + 1)Fi+1 + (d − i + 1)Fi−1 +(1 ≤ i ≤ d). +We remark that the polynomial Fi has degree i and leading coefficient 1 +i! (0 ≤ i ≤ d+1). +Lemma 4.7. We have Fi(A) = Ai (0 ≤ i ≤ d), and Fd+1(A) = 0. +Furthermore, +pi(j) = Fi(θj) (0 ≤ i, j ≤ d). + +8 +NATHAN NICHOLSON +Proof. By construction, F0(A) = I = A0 and F1(A) = A = A1. To see that Ai = Fi(A) +(2 ≤ i ≤ d) and Fd+1(A) = 0, use induction and compare Corollary 4.5 with Definition 4.6. +To see that Fi(θj) = pi(j), apply Fi to both sides of (2.5) and simplify with (2.3) to +obtain +Fi(A) = +d +� +j=0 +Fi(θj)Ej. +Comparing the above with (2.9), it follows that Fi(θj) = pi(j). +□ +We next consider the eigenvalues of Qd. +Lemma 4.8. (See [3, Proposition 9.2.1]) For 0 ≤ i ≤ d we have θi = d − 2i. +The following definition is for notational convenience. +Definition 4.9. Define Φd ∈ C[z] by +(4.1) +Φd = +d +� +i=0 +� +z − (d − 2i) +� +. +Lemma 4.10. The minimal polynomial of A is equal to Φd. +Proof. Immediate from Lemma 4.8 and Definition 4.9. +□ +Lemma 4.11. We have Φd = (d + 1)!Fd+1. +Proof. By Lemma 4.7, Fd+1(A) = 0. +By Lemma 4.10, Φd must divide Fd+1 in C[z]. +Because both Φd and Fd+1 have degree d + 1, one must be a scalar multiple of the other. +Comparing the leading coefficients, the result follows. +□ +We have a comment. +Lemma 4.12. (See [3, p. 194]) The hypercube Qd is self-dual. In other words, ph +ij = (ph +ij)∗ +(0 ≤ h, i, j ≤ d). +Corollary 4.13. For 0 ≤ i ≤ d we have k∗ +i = +�d +i +� +. +Proof. Follows from Corollary 4.3 and Lemma 4.12. +□ +Next we state some corollaries of Lemma 4.12. For the rest of this section, fix a vertex +x of Qd, and let T = T(x). +Corollary 4.14. For 0 ≤ i ≤ d we have +A∗A∗ +i = (i + 1)A∗ +i+1 + (d − i + 1)A∗ +i−1, +where we recall (2.8). +Proof. Similar to the proof of Corollary 4.5. +□ +Corollary 4.15. We have Fi(A∗) = A∗ +i (0 ≤ i ≤ d), and Fd+1(A∗) = 0. Furthermore, +p∗ +i (j) = Fi(θ∗ +j) (0 ≤ i, j ≤ d). +Proof. Similar to the proof of Lemma 4.7. +□ +Corollary 4.16. For 0 ≤ i ≤ d we have θ∗ +i = d − 2i. + +THE GENERALIZED TERWILLIGER ALGEBRA OF THE HYPERCUBE +9 +Proof. Follows from Lemma 4.12 and the discussion of self-duality at the end of Section 2. +□ +Lemma 4.17. The minimal polynomial of A∗ is equal to Φd. +Proof. Follows from Corollary 4.16. +□ +From now until the end of the section, we assume Γ = Qd with d ≥ 0. +In the next result, we consider the triples h, i, j such that ph +ij and (ph +ij)∗ are nonzero. +Corollary 4.18. For 0 ≤ h, i, j ≤ d, the intersection number ph +ij is nonzero if and only if +the Krein parameter (ph +ij)∗ is nonzero if and only if the following (i)–(iii) hold. +(i) h, i, j satisfy the triangle inequality: +h ≤ i + j, +i ≤ h + j, +j ≤ h + i. +(ii) h + i + j ≤ 2d. +(iii) h + i + j is even. +Proof. The case of d = 0 is trivial. The case of d ≥ 1 follows upon inspection of Proposi- +tion 4.2 and Corollary 4.12. +□ +Definition 4.19. Let Pd denote the set consisting of the 3-tuples of integers (h, i, j) such +that 0 ≤ h, i, j ≤ d which satisfy (i)–(iii) of Corollary 4.18. +Lemma 4.20. For 0 ≤ h, i, j ≤ d, the following (i)–(iii) are equivalent. +(i) ph +ij ̸= 0. +(ii) (ph +ij)∗ ̸= 0. +(iii) (h, i, j) ∈ Pd. +Proof. Compare Corollary 4.18 with Definition 4.19. +□ +We conclude this section with a brief definition and a comment about the Terwilliger +algebra T. +Definition 4.21. Assume x is a vertex of Qd. Let Td = Td(x) denote the Terwilliger +algebra of Qd with respect to x. +Proposition 4.22. (See [10, Theorem 14.14]) There exists an algebra isomorphism +Td → +� +0≤r≤⌊d/2⌋ +Md+1−2r(C). +5. The Generalized Terwilliger Algebra for Qd +In Definition 3.1, we described the generalized Terwilliger algebra for a distance-regular +graph. In this section, we consider this algebra for the graph Qd. +Definition 5.1. For d ≥ 0, let Td denote the generalized Terwilliger algebra associated +with Qd. + +10 +NATHAN NICHOLSON +By Remark 3.2, the algebra T0 is ismorphic to C. For the rest of this section, we restrict +our attention to the algebra Td with d ≥ 1. +We next observe some analogues of results from Section 4. +Lemma 5.2. For 0 ≤ i ≤ d and with reference to (3.2), the following (i), (ii) hold in Td. +(i) x1xi = (i + 1)xi+1 + (d − i + 1)xi−1. +(ii) x∗ +1x∗ +i = (i + 1)x∗ +i+1 + (d − i + 1)x∗ +i−1. +Proof. Follows from Lemmas 3.5 and 3.6 and Corollaries 4.5 and 4.14. +□ +Lemma 5.3. For 0 ≤ i ≤ d, the following (i), (ii) hold in Td. +(i) xi = Fi(x1). +(ii) x∗ +i = Fi(x∗ +1). +Moreover, Td is generated by x1 and x∗ +1. +Proof. Follows from Lemmas 3.5 and 3.6 and Corollaries 4.7 and 4.15. +□ +Lemma 5.4. The polynomial Φd is equal to the minimal polynomial of both x1 and x∗ +1. +Proof. Follows from Lemmas 3.5, 3.6, 4.10, and 4.17. +□ +Now that we have Lemmas 5.3 and 5.4, some of the relations in Definition 3.1 become +redundant, giving us the following, simpler presentation Td. +Proposition 5.5. The algebra Td is isomorphic to the algebra with 1, with generators +x1, x∗ +1 and relations (1)–(4). +(1) Φd(x1) = 0. +(2) Φd(x∗ +1) = 0. +(3) e∗ +hxie∗ +j = 0 if (h, i, j) /∈ Pd +(0 ≤ h, i, j ≤ d). +(4) ehx∗ +i ej = 0 if (h, i, j) /∈ Pd +(0 ≤ h, i, j ≤ d). +In the above lines, we define +x0 = 1, +x∗ +0 = 1, +xi = Fi(x1), +x∗ +i = Fi(x∗ +1) +(2 ≤ i ≤ d), +ei = 2−d +d +� +j=0 +qi(j)xj, +e∗ +i = 2−d +d +� +j=0 +q∗ +i (j)x∗ +j +(0 ≤ i ≤ d). +Proof. Compare Definition 3.1 with Lemmas 5.3 and 5.4. +□ +We would like to provide another presentation for Td. To do this, we first define the +following algebra. +Definition 5.6. Let T ′ +d denote the algebra with 1, with generators e0, . . . , ed, e∗ +0, . . . , e∗ +d +and relations (1)–(4). +(1) �d +i=0 ei = 1. +(2) �d +i=0 e∗ +i = 1. +(3) e∗ +hxie∗ +j = 0 if (h, i, j) /∈ Pd +(0 ≤ h, i, j ≤ d). +(4) ehx∗ +i ej = 0 if (h, i, j) /∈ Pd +(0 ≤ h, i, j ≤ d). + +THE GENERALIZED TERWILLIGER ALGEBRA OF THE HYPERCUBE +11 +In the above lines, we define +x0 = 1, +x∗ +0 = 1, +(5.1) +x1 = +d +� +i=0 +(d − 2i)ei, +x∗ +1 = +d +� +i=0 +(d − 2i)e∗ +i , +(5.2) +xi = Fi(x1), +x∗ +i = Fi(x∗ +1) +(2 ≤ i ≤ d). +(5.3) +We will soon show that the algebra Td is isomorphic to T ′ +d. We first give some lemmas +about T ′ +d. +Lemma 5.7. For 0 ≤ i ≤ d, the following (i), (ii) hold in T ′ +d. +(i) eiej = δijei. +(ii) e∗ +i e∗ +j = δije∗ +i . +Proof. (i) First assume that i ̸= j. The triple (i, 0, j) violates Corollary 4.18 (i), thus +eix∗ +0ej = 0 by relation (4) of Definition 5.6. As x∗ +0 = 1, we have eiej = 0. +Next assume that i = j. We will show that e2 +i = ei. By relation (1) of Definition 5.6 +and the previous paragraph, +ei = ei +d +� +ℓ=0 +eℓ = e2 +i . +(ii) Similar to the proof of (i). +□ +Note that by (5.2) and Lemma 5.7, the following equations hold in T ′ +d : +(5.4) +� +x1 − (d − 2i) +� +ei = 0, +� +x∗ +1 − (d − 2i) +� +e∗ +i = 0 +(0 ≤ i ≤ d). +Lemma 5.8. The following (i), (ii) hold in T ′ +d. +(i) Φd(x1) = 0. +(ii) Φd(x∗ +1) = 0. +Proof. (i) First note that the term +� +z −(d −2i) +� +is a factor of the right-hand side of (4.1) +(0 ≤ i ≤ d). Thus by Definition 4.9 and (5.4), +Φd(x1)ei = 0 +(0 ≤ i ≤ d). +Hence by relation (1) of Definition 5.6, +Φd(x1) = Φd(x1) +d +� +i=0 +ei = 0. +(ii) Similar to the proof of (i). +□ +Lemma 5.9. For 0 ≤ i ≤ d, the following (i), (ii) hold in T ′ +d. +(i) ei = 2−d �d +j=0 qi(j)xj. +(ii) e∗ +i = 2−d �d +j=0 q∗ +i (j)x∗ +j. +Proof. (i) By (5.3) and Lemma 5.7, +xi = Fi(x1) = +d +� +j=0 +Fi(θj)ej. + +12 +NATHAN NICHOLSON +Hence by Lemma 4.7, +(5.5) +xi = +d +� +j=0 +pi(j)ej. +The result follows from (5.5) and the fact that the matrices P, 2−dQ from below (2.11) +are inverses. +(ii) Similar to the proof of (i). +□ +We now show that the algebra Td is isomorphic to T ′ +d. +Proposition 5.10. There exists a unique algebra isomorphism Td → T ′ +d that sends +x1 �→ x1, +x∗ +1 �→ x∗ +1. +Moreover, this map sends +xi �→ xi, +x∗ +i �→ x∗ +i , +ei �→ ei, +e∗ +i �→ e∗ +i , +for 0 ≤ i ≤ d. +Proof. We will first show that there exists an algebra homomorphism σ : Td → T ′ +d which +sends x1 �→ x1 and x∗ +1 �→ x∗ +1. We will then show that there exists an algebra homomorphism +τ : T ′ +d → Td which sends ei �→ ei and e∗ +i �→ e∗ +i (0 ≤ i ≤ d). We will next show that σ +and τ are inverses, and hence algebra isomorphisms. We will last show that σ(xi) = xi, +σ(x∗ +i ) = x∗ +i , σ(ei) = ei, and σ(e∗ +i ) = e∗ +i (0 ≤ i ≤ d). +We begin by showing that σ exists. For 0 ≤ i ≤ d, let fi = 2−d �d +j=0 qi(j)xj ∈ T ′ +d and +f ∗ +i = 2−d �d +j=0 q∗ +i (j)x∗ +j ∈ T ′ +d. To show that σ exists, it is sufficient to show that in T ′ +d, +Φd(x1) = 0, +Φd(x∗ +1) = 0, +(5.6) +f ∗ +hxif ∗ +j = 0, +fhx∗ +i fj = 0, +(5.7) +for 0 ≤ h, i, j ≤ d such that (h, i, j) /∈ Pd. +Lemma 5.8 implies (5.6). Lemma 5.9 implies that fi = ei and f ∗ +i = e∗ +i (0 ≤ i ≤ d). +Hence (5.7) follows by relations (3) and (4) of Definition 5.6. +We next show τ exists. Let y1 = �d +i=0 θiei ∈ Td and y∗ +1 = �d +i=0 θ∗ +i e∗ +i ∈ Td. To show +that τ exists, it is sufficient to show that in Td, +d +� +ℓ=0 +eℓ = 1, +d +� +ℓ=0 +e∗ +ℓ = 1, +(5.8) +e∗ +hFi(y1)e∗ +j = 0, +ehFi(y∗ +1)ej = 0, +(5.9) +for 0 ≤ h, i, j ≤ d such that (h, i, j) /∈ Pd. +Lemma 3.7 implies (5.8). +Lemma 3.9 implies that x1 = y1 and x∗ +1 = y∗ +1. +Hence +Fi(y1) = xi and Fi(y∗ +1) = x∗ +i (0 ≤ i ≤ d) by Lemma 5.3. Thus (5.9) follows by relations +(3) and (4) of Proposition 5.5. +We now show that σ and τ are inverses. By Lemma 3.9, τ(x1) = x1 and τ(x∗ +1) = x∗ +1. +Thus τ ◦ σ : Td → Td is the identity map. By Lemma 5.9, σ(ei) = ei and σ(e∗ +i ) = e∗ +i +(0 ≤ i ≤ d). Thus σ ◦ τ : T ′ +d → T ′ +d is the identity map. Therefore σ and τ are inverses, +and hence algebra isomorphisms. + +THE GENERALIZED TERWILLIGER ALGEBRA OF THE HYPERCUBE +13 +By construction, σ(x1) = x1 and σ(x∗ +1) = x∗ +1. Thus σ(xi) = xi and σ(x∗ +i ) = x∗ +i (0 ≤ +i ≤ d). As noted previously, σ(ei) = ei and σ(e∗ +i ) = e∗ +i (0 ≤ i ≤ d). This completes the +proof. +□ +For the rest of the paper, we identify the algebras Td and T ′ +d via the isomorphism in +Proposition 5.10. +We next define a free algebra. +Definition 5.11. Let Td denote the free algebra with generators e0, . . . , ed, e∗ +0, . . . , e∗ +d. +Define x0 = 1, x1 = �d +i=0 θiei, and xi = Fi(x1) (2 ≤ i ≤ d). Similarly, define x∗ +0 = 1, +x∗ +1 = �d +i=0 θ∗ +i e∗ +i , and x∗ +i = Fi(x∗ +1) (2 ≤ i ≤ d). +For notational convenience, define +xi = 0, +x∗ +i = 0, +ei = 0, +e∗ +i = 0 +(i < 0 or i > d). +(5.10) +Definition 5.12. Let Sd denote the two-sided ideal of Td generated by the following +(1)–(4). +(1) �d +i=0 ei − 1, +(2) �d +i=0 e∗ +i − 1, +(3) e∗ +hxie∗ +j such that (h, i, j) /∈ Pd +(0 ≤ h, i, j ≤ d), +(4) ehx∗ +i ej such that (h, i, j) /∈ Pd +(0 ≤ h, i, j ≤ d). +Remark 5.13. Because Td is free, there exists an algebra homomorphism ψd : Td → Td +that sends +ei �→ ei, +e∗ +i �→ e∗ +i +(0 ≤ i ≤ d). +Comparing Definitions 5.6 and 5.12, it follows that the map ψd is surjective with kernel +Sd, and that +xi �→ xi, +x∗ +i �→ x∗ +i +(0 ≤ i ≤ d). +To end this section, we define an algebra homomorphism that will be useful later in the +paper. +Definition 5.14. Consider the quotient algebra Td/Sd. With reference to Remark 5.13, +the algebra homomorphism ψd induces an algebra isomorphism Td/Sd → Td. We denote +the inverse of this map by pd. +6. The Primary Central Idempotent of Td +We continue our discussion of the algebra Td from Definition 5.1. In [9], Egge defines +a certain element u0 ∈ Td called the primary central idempotent. Later in the paper, we +will use u0 to compute the dimension of Td. In this section, we recall the definition of u0 +and develop some basic facts about it. + +14 +NATHAN NICHOLSON +Lemma 6.1. (See [9, Propositions 11.1 and 11.4]) For d ≥ 0, following holds in Td: +(6.1) +2d +d +� +i=0 +k−1 +i e∗ +i e0e∗ +i = 2d +d +� +i=0 +(k∗ +i )−1eie∗ +0ei. +This element is central and idempotent. +Definition 6.2. (See [9, Proposition 11.1]) Referring to Lemma 6.1, we define u0 to be +the common value expressed in (6.1). We call u0 the primary central idempotent of Td. +Proposition 6.3. (See [9, Proposition 11.5 and Theorem 12.5]) For d ≥ 0, the following +(i)–(iii) hold. +(i) The sum Td = Tdu0 + Td(1 − u0) is direct. +(ii) Tdu0 and Td(1 − u0) are both two-sided ideals of Td. +(iii) The algebra Tdu0 is isomorphic to Md+1(C). +Corollary 6.4. For d ≥ 0, the algebra Td is isomorphic to the direct sum +Md+1(C) ⊕ Td(1 − u0). +Proof. Follows from Proposition 6.3. +□ +Corollary 6.5. There exists an algebra isomorphism Td/Tdu0 → Td(1 − u0) that sends +ei + Tdu0 �→ ei(1 − u0), +e∗ +i + Tdu0 �→ e∗ +i (1 − u0) +(0 ≤ i ≤ d). +Proof. Follows from Proposition 6.3 (i). +□ +We have some comments about u0. +Lemma 6.6. For d ≥ 0, the following (i)–(iv) hold in Td. +(i) u0e0 = e0. +(ii) u0ed = ed. +(iii) u0e∗ +0 = e∗ +0. +(iv) u0e∗ +d = e∗ +d. +Proof. (i) By Lemma 3.8 (i), Lemma 3.10 (ii), Corollary 4.13, and Definition 6.2, +u0e0 = 2d +� +d +� +r=0 +(k∗ +r)−1ere∗ +0er +� +e0 = 2de0e∗ +0e0 = e0 +� +d +� +i=0 +x∗ +i +� +e0. +For 1 ≤ i ≤ d, the triple (0, i, 0) does not satisfy Corollary 4.18 (i), thus e0x∗ +i e0 = 0 by +relation (4) of Definition 5.6. Hence by relation (T1) of Definition 3.1, +e0 +� +d +� +i=0 +x∗ +i +� +e0 = e0x∗ +0e0 = e2 +0 = e0. +Therefore u0e0 = e0. +(ii) By Lemma 3.8 (i), Lemma 3.10 (ii), Corollary 4.13, and Definition 6.2, +u0ed = 2d +� +d +� +r=0 +(k∗ +r)−1ere∗ +0er +� +ed = 2dede∗ +0ed = ed +� +d +� +i=0 +x∗ +i +� +ed. + +THE GENERALIZED TERWILLIGER ALGEBRA OF THE HYPERCUBE +15 +For 1 ≤ i ≤ d, the triple (d, i, d) does not satisfy Corollary 4.18 (ii), thus edx∗ +i ed = 0 by +relation (4) of Definition 5.6. Hence by relation (T1) of Definition 3.1, +ed +� +d +� +i=0 +x∗ +i +� +ed = edx∗ +0ed = e2 +d = ed. +Therefore u0ed = ed. +(iii) Similar to the proof of (i). +(iv) Similar to the proof of (ii). +□ +We finish this section with a comment about the case d = 1. +Proposition 6.7. For d = 1, the element u0 = 1. Moreover, the algebra T1 is isomorphic +to M2(C). +Proof. Because d = 1, Lemmas 3.7 and 6.6 imply that +u0 = u0(e0 + e1) = e0 + e1 = 1. +Therefore the algebra T1 is isomorphic to M2(C) by Proposition 6.3 part (iii). +□ +7. The Main Result +Recall the algebra homomorphism ♮ : Td → Td from Lemma 3.4. In this section, we +prove that ♮ is an algebra isomorphism. +Recall the free algebra Td from Definition 5.11. +Definition 7.1. Assume d ≥ 2. Let ϕd : Td → Td−2 be the algebra homomorphism that +sends +e0 �→ 0, +e∗ +0 �→ 0, +ei �→ ei−1, +e∗ +i �→ e∗ +i−1 +(1 ≤ i ≤ d − 1), +ed �→ 0, +e∗ +d �→ 0. +Referring to Definition 7.1 and using (5.10), we see that ϕd sends +(7.1) +ei �→ ei−1, +e∗ +i �→ e∗ +i−1 +(0 ≤ i ≤ d). +Definition 7.2. Assume d ≥ 2. Let Kd denote the two-sided ideal of Td generated by +e0, ed, e∗ +0, e∗ +d. +Lemma 7.3. Assume d ≥ 2. The map ϕd is surjective with kernel Kd. +Proof. Routine consequence of Definitions 7.1 and 7.2. +□ +Lemma 7.4. Assume d ≥ 2. For 0 ≤ i ≤ d, the following (i), (ii) hold. +(i) ϕd(xi) = + + + + + + + + + +xi +if i = 0 or i = 1; +xi − xi−2 +if 2 ≤ i ≤ d − 2; +Φd−2(x1) +(d−1)! − xi−2 +if i = d − 1; +x1Φd−2(x1) +d! +− xi−2 +if i = d. + +16 +NATHAN NICHOLSON +(ii) ϕd(x∗ +i ) = + + + + + + + + + +x∗ +i +if i = 0 or i = 1; +x∗ +i − x∗ +i−2 +if 2 ≤ i ≤ d − 2; +Φd−2(x∗ +1) +(d−1)! − x∗ +i−2 +if i = d − 1; +x∗ +1Φd−2(x∗ +1) +d! +− x∗ +i−2 +if i = d. +Proof. (i) We begin with a comment. Note that by Definitions 4.6 and 5.11, the following +holds in Td: +(7.2) +jxj = x1xj−1 − (d − j + 2)xj−2 +(2 ≤ j ≤ d). +We now consider the cases for i. +First, assume i = 0. The result holds, because x0 = 1. Next, assume i = 1. Then by +Lemma 4.8, Definition 5.11, and Definition 7.1, +ϕd(x1) = ϕd +� +d +� +j=0 +(d − 2j)ej +� += +d−1 +� +j=1 +(d − 2j)ej−1 += +d−2 +� +j=0 +(d − 2 − 2j)ej += x1. +By (5.10), it is correct to say that ϕd sends xi �→ xi − xi−2 for i = 0 and i = 1. This +allows us to use induction for 2 ≤ i ≤ d − 2. We proceed by induction on i. +Assume 2 ≤ i ≤ d−2. Setting j = i in (7.2), applying ϕd to both sides, using induction, +and dividing by i, we obtain +(7.3) +ϕd(xi) = x1(xi−1 − xi−3) − (d − i + 2)(xi−2 − xi−4) +i +. +Using Definitions 4.6 and 5.11, we find that in Td−2, +(7.4) +x1xi−1 = ixi + (d − i)xi−2, +x1xi−3 = (i − 2)xi−2 + (d − i + 2)xi−4. +In equation (7.3), we distribute terms in the numerator, then eliminate x1xi−1 and x1xi−3 +via (7.4). This yields ϕd(xi) = xi − xi−2. +Next, assume i = d − 1. Setting j = d − 1 in (7.2), applying ϕd to the result, using +induction, and dividing by d − 1 yields +(7.5) +ϕd(xd−1) = x1(xd−2 − xd−4) − 3(xd−3 − xd−5) +d − 1 +. +Using Definitions 4.6 and 5.11, we find that in Td−2, +(7.6) +x1xd−4 = (d − 3)xd−3 + 3xd−5. +In (7.5), we distribute terms in the numerator and eliminate x1xd−4 via (7.6). This yields +(7.7) +ϕd(xd−1) = x1xd−2 − xd−3 +d − 1 +− xd−3. + +THE GENERALIZED TERWILLIGER ALGEBRA OF THE HYPERCUBE +17 +By Lemma 4.11, +(7.8) +x1xd−2 − xd−3 = Φd−2(x1) +(d − 2)! . +We use (7.8) to eliminate the numerator in the right-hand side of (7.7). This yields +ϕd(xd−1) = Φd−2(x1) +(d − 1)! − xd−3. +For the rest of this proof, assume i = d. Setting j = d in (7.2), applying ϕd to the +result, using induction, and dividing by d yields +ϕd(xd) = +x1 +�Φd−2(x1) +(d−1)! − xd−3 +� +− 2(xd−2 − xd−4) +d +. +(7.9) +Using Definitions 4.6 and 5.11, we find that in Td−2, +(7.10) +x1xd−3 = (d − 2)xd−2 + 2xd−4. +In (7.9), we distribute terms in the numerator and eliminate x1xd−3 via (7.10). This yields +ϕd(xd) = +x1Φd−2(x1) +(d−1)! +− dxd−2 +d += x1Φd−2(x1) +d! +− xd−2. +(ii) Similar to the proof of (i). +□ +Recall the ideal Sd ⊆ Td from Definition 5.12. Our next general goal is to show that +ϕd(Sd) = Sd−2. To do that, we will show that ϕd(Sd) ⊆ Sd−2 and Sd−2 ⊆ ϕd(Sd). +Lemma 7.5. For d ≥ 2, the following (i), (ii) hold. +(i) ϕd +� �d +i=0 ei − 1 +� += �d−2 +i=0 ei − 1. +(ii) ϕd +� �d +i=0 e∗ +i − 1 +� += �d−2 +i=0 e∗ +i − 1. +Proof. (i) By Definition 7.1, +ϕd +� +d +� +i=0 +ei − 1 +� += +d−1 +� +i=1 +ei−1 − 1 = +d−2 +� +i=0 +ei − 1. +(ii) Similar to the proof of (i). +□ +Lemma 7.6. Assume d ≥ 2. For 0 ≤ h, i, j ≤ d such that (h, i, j) /∈ Pd, the following (i), +(ii) hold. +(i) (h − 1, i, j − 1) /∈ Pd−2. +(ii) (h − 1, i − 2, j − 1) /∈ Pd−2. +Proof. We consider the three cases in Corollary 4.18. For convenience, we consider them +in the order (ii), (iii), (i). + +18 +NATHAN NICHOLSON +First, assume h + i + j > 2d. Then +(h − 1) + i + (j − 1) > 2(d − 2), +(h − 1) + (i − 2) + (j − 1) > 2(d − 2). +Thus (i) and (ii) hold. +Next, assume h+i+j is odd. Then (h−1)+i+(j−1) is odd and (h−1)+(i−2)+(j−1) +is odd. Thus (i) and (ii) hold. +For the rest of this proof, assume h, i, j fail the triangle inequality. This leaves two +subcases: +i > h + j, +i < |h − j|. +First, assume i > h + j. Then +i > (h − 1) + (j − 1), +i − 2 > (h − 1) + (j − 1). +Hence h − 1, i, j − 1 and h − 1, i − 2, j − 1 fail the triangle inequality. Thus (i) and (ii) +hold. +Lastly, assume i < |h − j|. Then +i < |(h − 1) − (j − 1)|, +i − 2 < |(h − 1) − (j − 1)|. +Hence h − 1, i, j − 1 and h − 1, i − 2, j − 1 fail the triangle inequality. Thus (i) and (ii) +hold. +□ +Lemma 7.7. Assume d ≥ 2. For 0 ≤ h, i, j ≤ d such that (h, i, j) /∈ Pd, the following +(i), (ii) hold. +(i) ϕd(e∗ +hxie∗ +j) ∈ Sd−2. +(ii) ϕd(ehx∗ +i ej) ∈ Sd−2. +Proof. (i) First note that if h = 0 or h = d or j = 0 or j = d, then ϕd(e∗ +hxie∗ +j) = 0 by +Definition 7.1. Thus for the remainder of this proof, we assume 1 ≤ h, j ≤ d − 1. +We consider the cases from Lemma 7.4 (i). +First, assume i = 0 or i = 1. By Definition 7.1 and Lemma 7.4, +ϕd(e∗ +hxie∗ +j) = e∗ +h−1xie∗ +j−1. +By Lemma 7.6, (h − 1, i, j − 1) /∈ Pd−2. Hence by Definition 5.12, +e∗ +h−1xie∗ +j−1 ∈ Sd−2. +Next, assume 2 ≤ i ≤ d − 2. Then +ϕd(e∗ +hxie∗ +j) = e∗ +h−1xie∗ +j−1 − e∗ +h−1xi−2e∗ +j−1. +By Lemma 7.6, (h − 1, i, j − 1) /∈ Pd−2 and (h − 1, i − 2, j − 1) /∈ Pd−2. +Hence by +Definition 5.12, +e∗ +h−1xie∗ +j−1 − e∗ +h−1xi−2e∗ +j−1 ∈ Sd−2. +Next, assume i = d − 1. Then +ϕd(e∗ +hxie∗ +j) = e∗ +h−1Φd−2(x1)e∗ +j−1 +(d − 1)! +− e∗ +h−1xi−2e∗ +j−1. + +THE GENERALIZED TERWILLIGER ALGEBRA OF THE HYPERCUBE +19 +By Lemma 5.8, Φd−2(x1) ∈ Sd−2. By Definition 5.12 and Lemma 7.6, e∗ +h−1xi−2e∗ +j−1 ∈ Sd−2. +Thus +e∗ +h−1Φd−2(x1)e∗ +j−1 +(d − 1)! +− e∗ +h−1xi−2e∗ +j−1 ∈ Sd−2. +For the remainder of this proof, assume i = d. Then +ϕd(e∗ +hxiej∗) = e∗ +h−1x1Φd−2(x1)e∗ +j−1 +d! +− e∗ +h−1xi−2e∗ +j−1. +By Lemma 5.8, Φd−2(x1) ∈ Sd−2. By Definition 5.12 and Lemma 7.6, e∗ +h−1xi−2e∗ +j−1 ∈ Sd−2. +Thus +e∗ +h−1x1Φd−2(x1)e∗ +j−1 +d! +− e∗ +h−1xi−2e∗ +j−1 ∈ Sd−2. +(ii) Similar to the proof of (i). +□ +We have now shown that ϕd(Sd) ⊆ Sd−2. Next we show that Sd−2 ⊆ ϕd(Sd). To that +end, we include the following technical results. +Lemma 7.8. Assume d ≥ 2. The following (i), (ii) hold. +(i) Φd−2(x1)(1 − e0 − ed) ∈ Sd. +(ii) Φd−2(x∗ +1)(1 − e∗ +0 − e∗ +d) ∈ Sd. +Proof. (i) Observe that +(7.11) +Φd−2(x1)(1 − e0 − ed) = Φd−2(x1) +� +1 − +d +� +i=0 +ei +� ++ Φd−2(x1) +d−1 +� +i=1 +ei. +By Definition 5.12, 1 − �d +i=0 ei ∈ Sd. Hence +(7.12) +Φd−2(x1) +� +1 − +d +� +i=0 +ei +� +∈ Sd. +By Lemma 4.8 and Definition 4.9, +Φd−2(x1) = +d−1 +� +i=1 +(x1 − θi). +By (5.4) and Remark 5.13, (x1 − θi)ei ∈ Sd (1 ≤ i ≤ d − 1). Hence +(7.13) +Φd−2(x1) +d−1 +� +i=1 +ei ∈ Sd. +It follows from (7.11), (7.12), and (7.13) that +Φd−2(x1)(1 − e0 − ed) ∈ Sd. +(ii) Similar to the proof of (i). +□ +Corollary 7.9. Assume d ≥ 2. The following (i), (ii) hold. +(i) Φd−2(x1) ∈ ϕd(Sd). +(ii) Φd−2(x∗ +1) ∈ ϕd(Sd). + +20 +NATHAN NICHOLSON +Proof. (i) By Lemma 7.8 (i), Φd−2(x1)(1−e0−ed) ∈ Sd. By Definition 7.1 and Lemma 7.4, +ϕd +� +Φd−2(x1)(1 − e0 − ed) +� += Φd−2(x1). +(ii) Similar to the proof of (i). +□ +Lemma 7.10. Assume d ≥ 2. For 0 ≤ h, i, j ≤ d − 2 such that (h, i, j) /∈ Pd−2, +(7.14) +(h + 1, i − 2r, j + 1) /∈ Pd +(0 ≤ r ≤ ⌊i/2⌋), +or +(7.15) +(h + 1, i + 2r, j + 1) /∈ Pd +(1 ≤ r ≤ ⌊(d − i)/2⌋). +Proof. We consider the three cases in Corollary 4.18. For convenience, we consider these +cases in order (ii), (iii), (i). +First, assume that h + i + j > 2(d − 2). Then +(h + 1) + (i + 2r) + (j + 1) > 2d +(1 ≤ r ≤ ⌊(d − i)/2⌋). +Thus (7.15) holds. +Next, assume that h+i+j is odd. Then (h+1)+(i−2r)+(j+1) is odd (0 ≤ r ≤ ⌊i/2⌋). +Thus (7.14) holds. +For the rest of this proof, assume that h, i, j fail the triangle inequality. This leaves two +subcases: +i > h + j, +i < |h − j|. +First, assume i > h + j. Then +i + 2r > (j + 1) + (h + 1) +(1 ≤ r ≤ ⌊(d − i)/2⌋). +Hence h + 1, i + 2r, j + 1 fail the triangle inequality (1 ≤ r ≤ ⌊(d − i)/2⌋). Thus (7.15) +holds. +Lastly, assume i < |h − j|. Then +i − 2r < |(h + 1) − (j + 1)| +(0 ≤ r ≤ ⌊i/2⌋). +Hence h+1, i−2r, j+1 fail the triangle inequality (0 ≤ r ≤ ⌊i/2⌋). Thus (7.14) holds. +□ +Lemma 7.11. Assume d ≥ 2. For 0 ≤ h, i, j ≤ d − 2 such that (h, i, j) /∈ Pd−2, the +following (i), (ii) hold. +(i) e∗ +hxie∗ +j ∈ ϕd(Sd). +(ii) ehx∗ +i ej ∈ ϕd(Sd). +Proof. (i) We consider the two cases in Lemma 7.10. +First, assume (7.14) holds. Then by Definition 5.12, +e∗ +h+1xi−2re∗ +j+1 ∈ Sd, +(0 ≤ r ≤ ⌊i/2⌋). +By Definition 7.1 and Lemma 7.4, +ϕd +� ⌊i/2⌋ +� +r=0 +e∗ +h+1xi−2re∗ +j+1 +� += +⌊i/2⌋−1 +� +r=0 +(e∗ +hxi−2re∗ +j − e∗ +hxi−2r−2e∗ +j) + e∗ +hxi−2⌊i/2⌋e∗ +j. +(7.16) +After expanding the sum and cancelling terms, the right-hand side of (7.16) becomes +e∗ +hxie∗ +j. Thus e∗ +hxie∗ +j ∈ ϕd(Sd). + +THE GENERALIZED TERWILLIGER ALGEBRA OF THE HYPERCUBE +21 +For the rest of this proof, assume (7.15) holds. Then by Definition 5.12, +e∗ +h+1xi+2re∗ +j+1 ∈ Sd +(1 ≤ r ≤ ⌊(d − i)/2⌋). +For notational convenience, define a polynomial g ∈ C[z] by +g = +� +1 +(d−1)! +if d − i is odd; +z +d! +if d − i is even. +We have defined g such that by Lemma 7.4, +ϕd(xi+2⌊(d−i)/2⌋) = g(x1)Φd−2(x1) − xi+2⌊(d−i)/2⌋−2. +Thus by Definition 7.1 and Lemma 7.4, +ϕd +� ⌊(d−i)/2⌋ +� +r=1 +e∗ +h+1xi+2re∗ +j+1 +� += +⌊(d−i)/2⌋−1 +� +r=1 +(e∗ +hxi+2re∗ +j − e∗ +hxi+2r−2e∗ +j) + e∗ +hg(x1)Φd−2(x1)e∗ +j − e∗ +hxi+2⌊(d−i)/2⌋−2e∗ +j. +(7.17) +After expanding the sum and cancelling terms, the right-hand side of (7.17) becomes +−e∗ +hxie∗ +j + e∗ +hg(x1)Φd−2(x1)e∗ +j. Hence +(7.18) +− e∗ +hxie∗ +j + e∗ +hg(x1)Φd−2(x1)e∗ +j ∈ ϕd(Sd). +By Corollary 7.9 (i) and the surjectivity of ϕd, +(7.19) +e∗ +hg(x1)Φd−2(x1)e∗ +j ∈ ϕd(Sd). +Therefore by (7.18) and (7.19), e∗ +hxie∗ +j ∈ ϕd(Sd). +(ii) Similar to the proof of (i). +□ +We have Sd−2 ⊆ ϕd(Sd) by Lemmas 7.5 and 7.11 together with the surjectivity of ϕd. +Proposition 7.12. Assume d ≥ 2. Then ϕd(Sd) = Sd−2. +Proof. We mentioned below Lemma 7.7 that ϕd(Sd) ⊆ Sd−2, and we mentioned below +Lemma 7.11 that Sd−2 ⊆ ϕd(Sd). +□ +We next consider how ϕd induces an algebra homomorphism from Td → Td−2. +Proposition 7.13. Assume d ≥ 2. There exists an algebra homomorphism ϕ′ +d : Td → Td−2 +that sends +e0 �→ 0, +e∗ +0 �→ 0, +ei �→ ei−1, +e∗ +i �→ e∗ +i−1 +(1 ≤ i ≤ d − 1), +ed �→ 0, +e∗ +d �→ 0. +Moreover, ϕ′ +d is surjective, and ker(ϕ′ +d) = ψd(Kd). +Proof. We first consider the existence of ϕ′ +d. By Lemma 5.13, Lemma 7.3, and Proposi- +tion 7.12, we have a surjective algebra homomorphism ψd−2 ◦ ϕd : Td → Td−2 with kernel +equal to Sd + Kd. This map induces an algebra isomorphism from the quotient algebra +Td/(Sd + Kd) → Td−2; we say this isomorphism is canonical. + +22 +NATHAN NICHOLSON +Let q : Td/Sd → Td/(Sd + Kd) denote the quotient map, which we recall is an algebra +homomorphism. +Recall the algebra isomorphism pd : Td → Td/Sd from Definition 5.14. +The following composition gives an algebra homomorphism from Td → Td−2: +(7.20) +ϕ′ +d : +Td +Td/Sd +Td/(Sd + Kd) +Td−2. +pd +q +can +We have shown that ϕ′ +d exists. With reference to (3.2), one routinely check that ϕ′ +d sends +ei �→ ei−1 and e∗ +i �→ e∗ +i−1 (0 ≤ i ≤ d). +We next show that ϕ′ +d is surjective. This follows because each of the composition factors +in (7.20) is surjective. +Lastly, we consider the kernel of ϕ′ +d. Inspection of (7.20) shows that ker(ϕ′ +d) = p−1 +d (Kd+ +Sd). By the construction of pd, p−1 +d (Kd + Sd) = ψd(Kd). Hence ker(ϕ′ +d) = ψd(Kd). +□ +Proposition 7.14. Assume d ≥ 2. The ideal Tdu0 is equal to ψd(Kd). Moreover, Tdu0 = +ker(ϕ′ +d). +Proof. We first consider the first assertion. +Because ψd is surjective, ψd(Kd) is equal +to the two-sided ideal of Td generated by e0, ed, e∗ +0, e∗ +d. Thus by Lemma 6.6, the ideal +ψd(Kd) ⊆ Tdu0. +Recall that u0 = 2d �d +r=0 k−1 +r e∗ +re0e∗ +r. Thus +Tdu0 ⊆ Tde0 ⊆ ψd(Kd). +This proves the first assertion. +The second assertion follows by the first, together with Proposition 7.13. +□ +Corollary 7.15. Assume d ≥ 2. There exists an algebra isomorphism Td(1 − u0) → Td−2 +that sends +e0(1 − u0) �→ 0, +e∗ +0(1 − u0) �→ 0, +ei(1 − u0) �→ ei−1, +e∗ +i (1 − u0) �→ e∗ +i−1 +(1 ≤ i ≤ d − 1), +ed(1 − u0) �→ 0, +e∗ +d(1 − u0) �→ 0. +Proof. With reference to (3.2), Proposition 7.13 implies that there exists an induced +algebra isomorphism Td/ ker(ϕ′ +d) → Td−2 which sends +ei + ker(ϕ′ +d) �→ ei−1, +e∗ +i + ker(ϕ′ +d) �→ e∗ +i−1 +(0 ≤ i ≤ d). +By Proposition 7.14, we know that ker(ϕ′ +d) = Tdu0. +Identifying the quotient algebra +Td/Tdu0 with Td(1 − u0) via the isomorphism in Corollary 6.5, the result follows. +□ +Corollary 7.16. Assume d ≥ 2. There exists an algebra isomorphism +Td → Md+1(C) ⊕ Td−2. +Proof. Follows from Corollaries 6.4 and 7.15. +□ +Proposition 7.17. Assume d ≥ 0. Then there exists an algebra isomorphism +Td → +� +0≤r≤⌊d/2⌋ +Md+1−2r(C). +Moreover, the algebra Td is isomorphic to Td. + +THE GENERALIZED TERWILLIGER ALGEBRA OF THE HYPERCUBE +23 +Proof. We first consider the first assertion. We proceed by induction on d. The base cases +of d = 0 and d = 1 are addressed in Remark 3.2 and Proposition 6.7. Now assume d ≥ 2. +By Corollary 7.16 and induction, we have algebra isomorphisms +Td → Md+1(C) ⊕ Td−2 → +� +0≤r≤d/2 +Md+1−2r(C). +This completes the proof of the first assertion. +To prove the second assertion, compare this result to Proposition 4.22. +□ +We conclude with the main result of this paper. +Theorem 7.18. Assume d ≥ 0. The map ♮ : Td → Td from Lemma 3.4 is an algebra +isomorphism. +Proof. By Proposition 7.17, Td and Td have the same dimension as algebras. Because ♮ is +a surjective algebra homomorphism between two algebras of the same dimension, it is an +algebra isomorphism. +□ +8. Acknowledgments +The author is presently a graduate student at the University of Wisconsin–Madison. +He would like to thank his advisor, Paul Terwilliger, for suggesting this project, for his +hours of mentoring, and for giving many valuable suggestions for this manuscript. +References +[1] E. Bannai and T. Ito. Algebraic Combinatorics I: Association Schemes. Benjamin/Cummings, Lon- +don, 1984. +[2] E. Bannai, E. Bannai, T. Ito, R. Tanaka. Algebraic Combinatorics. De Gruyter Series in Discrete +Math and Applications 5. De Gruyter, 2021. +[3] A. E. Brouwer, A. M. Cohen, and A. Neumaier. Distance-Regular Graphs. Springer-Verlag, Berlin, +1989. +[4] J. S. Caughman IV. The Terwilliger algebras of bipartite P- and Q-polynomial association schemes. +Discrete Math. 196 (1999) 65–95. +[5] B. Curtin. Bipartite distance-regular graphs I. Graphs Combin. 15 (1999) 143-158. +[6] B. Curtin. Bipartite distance-regular graphs II. Graphs Combin. 15 (1999) 377-391. +[7] E.R. van Dam, J. H. Koolen, H. Tanaka. Distance-regular graphs. Electron. J. Combin. (2016) DS22; +arXiv: 1410.6294. +[8] G. Dickie and P. Terwilliger. A note on thin P-polynomial and dual-thin Q-polynomial symmetric +association schemes. J. Algebraic Combin. 7 (1998) 5–15. +[9] E. Egge. A generalization of the Terwilliger algebra. J. Algebra 233 (2000) 213–252. +[10] J. T. Go. The Terwilliger algebra of the hypercube. European J. Combin. 23 (2002) 399–429. +[11] S. A. Hobart and T. Ito. The structure of nonthin irreducible T-modules: ladder bases and classical +parameters. J. Algebraic Combin. 7 (1998) 53–75. +[12] A. A. Pascasio. On the multiplicities of the primitive idempotents of a Q-polynomial distance-regular +graph. European J. Combin. 23 (2002) 1073–1078. +[13] N. J. A. Sloane. An introduction to association schemes and coding theory. Theory and application +of special functions (Proc. Advanced Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., +1975), pp. 225–260. Math. Res. Center, Univ. Wisconsin, Publ. No. 35, Academic Press, New York, +1975. +[14] K. Tanabe. The irreducible modules of the Terwilliger algebras of Doob schemes. J. Algebraic Com- +bin. 6 (1997) 173–195. + +24 +NATHAN NICHOLSON +[15] H. Tanaka and T. Wang. The Terwilliger algebra of the twisted Grassmann graph: the thin case. +Electron. J. Combin. 27 (2020) Paper No. 4.15, 22 pp. +[16] P. Terwilliger. Distance-regular Graphs, the Subconstituent Algebra, and the Q-polynomial Property. +Preprint arXiv (2022). +[17] P. Terwilliger. The subconstituent algebra of an association scheme I. J. Algebraic Combin. 1 (1992) +363–388. +[18] P. Terwilliger. The subconstituent algebra of an association scheme II. J. Algebraic Combin. 2 (1993) +73–103. +[19] P. Terwilliger. The subconstituent algebra of an association scheme III. J. Algebraic Combin. +2 +(1993) 177–210. +Nathan Nicholson +Department of Mathematics +University of Wisconsin +480 Lincoln Drive +Madison, WI 53706-1388 USA +email: nlnicholson@wisc.edu + diff --git a/mNE_T4oBgHgl3EQf6xyS/content/tmp_files/load_file.txt b/mNE_T4oBgHgl3EQf6xyS/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..43592fb9fa17d2338eade5edaad682ad8e8110f5 --- /dev/null +++ b/mNE_T4oBgHgl3EQf6xyS/content/tmp_files/load_file.txt @@ -0,0 +1,1312 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf,len=1311 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='08366v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='CO] 20 Jan 2023 THE GENERALIZED TERWILLIGER ALGEBRA OF THE HYPERCUBE NATHAN NICHOLSON Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' In the year 2000, Eric Egge introduced the generalized Terwilliger algebra T of a distance-regular graph Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' For any vertex x of Γ, there is a surjective algebra homomorphism ♮ from T to the Terwilliger algebra T (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' If Γ is complete, then ♮ is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' If Γ is not complete, then ♮ may or may not be an isomorphism, and in general the details are unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' We show that if Γ is a hypercube, then the algebra homomorphism ♮ : T → T (x) is an isomorphism for all vertices x of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Keywords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Distance-regular graph;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Terwilliger algebra;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Hamming cube;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' hypercube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' 05E30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Introduction In the topic of Algebraic Combinatorics, there is a family of finite undirected graphs said to be distance-regular [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' These graphs are heavily studied;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' see [1–3,7,16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' For a vertex x of a distance-regular graph Γ, the associated Terwilliger algebra T(x) was introduced in [17] (there called the subconstituent algebra).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' This algebra is finite-dimensional, semi- simple, and noncommutative in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Some notable papers about T(x) are [4–6,8,11, 12, 14, 15, 17–19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' There are some well-known relations in T(x) called the triple product relations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' the form of the triple product relations is independent of x (see [17, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' In [9], Eric Egge introduced the generalized Terwilliger algebra T of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' This algebra is defined by generators and relations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' the main relations are the triple product relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' By construction, for any vertex x of Γ there is a surjective algebra homomorphism ♮ : T → T(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' If the graph Γ is complete, then ♮ is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' If Γ is not complete, then ♮ may or may not be an isomorphism, and in general the details are unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' There is a type of distance-regular graph called the hypercube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' The hypercube of diameter d is often denoted Qd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' General information about Qd can be found in [3, Chapters 1 and 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' In [10], Junie Go showed that for any vertex x of Qd there is an algebra isomorphism from T(x) to a direct sum of full matrix algebras: T(x) → � 0≤r≤⌊d/2⌋ Md+1−2r(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' In this paper, we investigate the algebra T associated with Qd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Our main result is that for any vertex x of Qd, the algebra homomorphism ♮ : T → T(x) is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' To prove this result, we use the following strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Writing Td for the generalized Terwilliger algebra associated with Qd, we will display an algebra isomorphism Td → Md+1(C) ⊕ Td−2 Date: January 23, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' 1 2 NATHAN NICHOLSON for d ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' We will then argue by induction on d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' In Sections 2 and 3, we discuss the algebras T(x) and T for any distance-regular graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' In Sections 4 and 5, we discuss T(x) and T for the hypercube Qd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' In Section 6, we discuss a certain central idempotent u0 of T that was introduced in [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' In Section 7, we prove the main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Distance-regular Graphs In this section, we review some definitions and results concerning distance-regular graphs and the Terwilliger algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' For more information, we refer the reader to [3,16,17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Suppose X is a nonempty finite set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Let MX(C) denote the algebra consisting of the matrices with rows and columns indexed by X and entries in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Let CX denote the vector space over C consisting of column vectors with coordinates indexed by X and entries in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' The algebra MX(C) acts on CX by left multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' For any positive integer n, let Mn(C) denote the algebra consisting of square n × n matrices with entries in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Let Γ = (X, R) denote a finite, undirected, connected graph, without loops or multiple edges, with vertex set X, edge set R, and path-length distance function ∂.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Let d = max{∂(x, y) | x, y ∈ X}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' We call d the diameter of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Vertices x, y ∈ X are said to be adjacent whenever they form an edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' For any vertex x ∈ X, let Γ(x) denote the set of vertices that are adjacent to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Define A ∈ MX(C) with (x, y)-entry Axy = � 1 if x and y are adjacent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' 0 if x and y are not adjacent (x, y ∈ X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' We call A the adjacency matrix of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' We say that Γ is regular with valency k whenever k = |Γ(x)| for all x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' We say that Γ is distance-regular whenever for any 0 ≤ h, i, j ≤ d, and any vertices x, y ∈ X with ∂(x, y) = h, the number of vertices z ∈ X such that ∂(x, z) = i and ∂(y, z) = j is a constant depending only on h, i, j, but not on x and y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' We denote this constant by ph ij and call it an intersection number of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Observe that ph ij = ph ji (0 ≤ h, i, j ≤ d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' For the rest of this section, assume Γ is distance-regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Note that Γ is regular with valency k = p0 11 if d ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' We now recall the Bose-Mesner algebra of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' For 0 ≤ i ≤ d, define Ai ∈ MX(C) with (x, y)-entry (Ai)xy = � 1 if ∂(x, y) = i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' 0 if ∂(x, y) ̸= i (x, y ∈ X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' We call Ai the ith distance matrix of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Note that A0 = I, where I ∈ MX(C) denotes the identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Note that A1 = A, provided that d ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' The sum �d i=0 Ai = J, where J ∈ MX(C) denotes the matrix with every entry equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' For notational convenience, define (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='1) Ai = 0 (i < 0 or i > d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' THE GENERALIZED TERWILLIGER ALGEBRA OF THE HYPERCUBE 3 By [3, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' 44], (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='2) AiAj = d � h=0 ph ijAh (0 ≤ i, j ≤ d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Thus {Ai}d i=0 forms a basis for a commutative subalgebra M of MX(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' The matrix A generates M by [3, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' 127].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' We call M the Bose-Mesner algebra of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Note that M has dimension d + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' For 0 ≤ i ≤ d, define ki = p0 ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' For any x ∈ X, ki is equal to the number of vertices at distance i from x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' We call ki the ith valency of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Note that k0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Moreover, k1 = k if d ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' The matrix A has d+1 distinct eigenvalues, because A generates M and M has dimen- sion d+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Denote these eigenvalues by θ0 > θ1 > · · · > θd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' For 0 ≤ i ≤ d, let Ei ∈ MX(C) be the matrix which acts as I on the θi-eigenspace of A and as 0 on all other eigenspaces of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' The matrices {Ei}d i=0 form a basis for M, and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='3) d � i=0 Ei = I, EiEj = δijEi (0 ≤ i, j ≤ d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' We call Ei the primitive idempotent of Γ associated with θi (0 ≤ i ≤ d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Define (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='4) Ei = 0 (i < 0 or i > d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' By [3, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' 45], we have E0 = |X|−1J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' By construction, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='5) A = d � i=0 θiEi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Next we recall the Krein parameters of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' For B, C ∈ MX(C), let B ◦ C denote their entry-wise product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Note that Ai ◦ Aj = δijAi (0 ≤ i, j ≤ d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Consequently, M is closed under ◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Because {Ei}d i=0 is a basis for M, there exist scalars (ph ij)∗ ∈ C (0 ≤ h, i, j ≤ d) such that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='6) Ei ◦ Ej = |X|−1 d � h=0 (ph ij)∗Eh (0 ≤ i, j ≤ d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' The scalars (ph ij)∗ are called the Krein parameters of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' By [3, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' 50], (ph ij)∗ is real and nonnegative for 0 ≤ h, i, j ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' For 0 ≤ i ≤ d, define k∗ i = (p0 ii)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' We call k∗ i the ith dual valency of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' We next recall the dual Bose-Mesner algebras of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Fix a vertex x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' For 0 ≤ i ≤ d, let E∗ i = E∗ i (x) denote the diagonal matrix in MX(C) with (y, y)-entry (E∗ i )yy = � 1 if ∂(x, y) = i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' 0 if ∂(x, y) ̸= i (y ∈ X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' We call E∗ i the ith dual primitive idempotent of Γ with respect to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Define (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='7) E∗ i = 0 (i < 0 or i > d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' 4 NATHAN NICHOLSON By construction, d � i=0 E∗ i = I, E∗ i E∗ j = δijE∗ i (0 ≤ i, j ≤ d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' It follows that {E∗ i }d i=0 forms a basis for a commutative subalgebra M∗ = M∗(x) of MX(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' We call M∗ the dual Bose-Mesner algbebra of Γ with respect to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' For 0 ≤ i ≤ d, we define a diagonal matrix A∗ i = A∗ i (x) ∈ MX(C) with (y, y)-entry (A∗ i )yy = |X|(Ei)xy (y ∈ X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Note that A∗ 0 = I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' For d ≥ 1, we abbreviate A∗ = A∗ 1 and call this the dual adjacency matrix of Γ with respect to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Define (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='8) A∗ i = 0 (i < 0 or i > d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' By [16, Corollary 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='6], A∗ generates M∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='6), A∗ i A∗ j = d � h=0 (ph ij)∗A∗ h (0 ≤ i, j ≤ d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' By [16, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='8], the matrices {A∗ i }d i=0 form a basis for M∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' We next recall the Terwilliger algebra of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Let T = T(x) denote the subalgebra of MX(C) generated by M and M∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' We call T the Terwilliger algebra of Γ with respect to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' We remark that T is sometimes called the subconstituent algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' For d ≥ 1, since the matrices {E∗ i }d i=0 form a basis for M∗, there exist scalars θ∗ i ∈ C (0 ≤ i ≤ d) such that A∗ = d � i=0 θ∗ i E∗ i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Because {Ai}d i=0 and {Ei}d i=0 each constitute bases for M, there must exist scalars pi(j), qi(j) ∈ C (0 ≤ i, j ≤ d) such that Ai = d � j=0 pi(j)Ej, Ei = |X|−1 d � j=0 qi(j)Aj (0 ≤ i ≤ d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='9) Similarly, there must exist scalars p∗ i (j), q∗ i (j) ∈ C (0 ≤ i, j ≤ d) such that A∗ i = d � j=0 p∗ i (j)E∗ j , E∗ i = |X|−1 d � j=0 q∗ i (j)A∗ j (0 ≤ i ≤ d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' By the construction of {E∗ i }d i=0 and {A∗ i }d i=0, we have (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='10) qi(j) = p∗ i (j), pi(j) = q∗ i (j) (0 ≤ i, j ≤ d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' For d ≥ 1, we have (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='11) θi = p1(i), θ∗ i = p∗ 1(i) (0 ≤ i ≤ d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' By [3, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='1], (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='12) p0(j) = 1, q0(j) = 1 (0 ≤ j ≤ d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' THE GENERALIZED TERWILLIGER ALGEBRA OF THE HYPERCUBE 5 Define matrices P, Q ∈ Md+1(C) with entries Pij = pj(i) and Qij = qj(i) (0 ≤ i, j ≤ d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Then P is the change of basis matrix from {Ei}d i=0 to {Ai}d i=0, and |X|−1Q is the change of basis matrix from {Ei}d i=0 to {Ai}d i=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Moreover, P and |X|−1Q are inverses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='10), |X|−1P is the change of basis matrix from {A∗ i }d i=0 to {E∗ i }d i=0, and Q is the change of basis matrix from {E∗ i }d i=0 to {A∗ i }d i=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' By [17, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='2], the following hold for 0 ≤ h, i, j ≤ d: E∗ hAiE∗ j = 0 iff ph ij = 0, EhA∗ i Ej = 0 iff (ph ij)∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' The above statements are referred to as the triple product relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' To conclude this section, we recall the notion of self-duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' We say that Γ is self- dual whenever ph ij = (ph ij)∗ (0 ≤ h, i, j ≤ d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' In this case, we have ki = k∗ i and θi = θ∗ i (0 ≤ i ≤ d), and we have pi(j) = qi(j) (0 ≤ i, j ≤ d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' See [3, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' 49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' The Generalized Terwilliger Algebra In this section, we recall the generalized Terwilliger algebra and some related results from [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Let Γ = (X, R) denote a distance regular graph with diameter d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' We fix a vertex x ∈ X and let T = T(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (See [9, Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='1]) Let T denote the algebra with 1, with generators x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' , xd, x∗ 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' , x∗ d and relations (T1)–(T3∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (T1) x0 = x∗ 0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (T2) xixj = d � h=0 ph ijxh (0 ≤ i, j ≤ d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (T2∗) x∗ i x∗ j = d � h=0 (ph ij)∗x∗ h (0 ≤ i, j ≤ d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (T3) e∗ hxie∗ j = 0 if ph ij = 0 (0 ≤ h, i, j ≤ d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (T3∗) ehx∗ i ej = 0 if (ph ij)∗ = 0 (0 ≤ h, i, j ≤ d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' In the above lines, we define (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='1) ei = |X|−1 d � j=0 qi(j)xj, e∗ i = |X|−1 d � j=0 q∗ i (j)x∗ j (0 ≤ i ≤ d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' We call T the generalized Terwilliger algebra associated with Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' For notational convenience, define xi = 0, x∗ i = 0, ei = 0, e∗ i = 0 (i < 0 or i > d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='2) Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' If d = 0, the algebra T is isomorphic to C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' In [9, Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='1], the relation (T1) is given as x0 = x∗ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' In [9, Propo- sition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='1], a proof is given that x0 = x∗ 0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' However the proof is not correct, and this can be seen by considering the case d = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Thus we adjusted our statement of (T1) to incorporate the author’s assumption that x0 = x∗ 0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' 6 NATHAN NICHOLSON Next we recall some results about T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (See [9, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' 3]) There exists an algebra homomorphism ♮ : T → T which sends xi �→ Ai, x∗ i �→ A∗ i , ei �→ Ei, e∗ i �→ E∗ i , for 0 ≤ i ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Moreover, ♮ is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' The map ♮ in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='4 is not an isomorphism in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' However we do have the following results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (See [9, Propositions 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='4 and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='2]) The following (i)–(iii) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (i) The elements {xi}d i=0 form a basis for a commutative subalgebra C of T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (ii) The elements {ei}d i=0 form a basis for C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (iii) The restriction of ♮ to C induces an algebra isomorphism C → M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (See [9, Propositions 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='7 and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='3]) The following (i)–(iii) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (i) The elements {x∗ i }d i=0 form a basis for a commutative subalgebra C∗ of T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (ii) The elements {e∗ i }d i=0 form a basis for C∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (iii) The restriction of ♮ to C∗ induces an algebra isomorphism C∗ → M∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' The next lemmas are consequences of Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='5 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (See [9, Propositions 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='3 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='6]) The following (i), (ii) hold in T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (i) �d i=0 ei = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (ii) �d i=0 e∗ i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='3) and Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='5 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (See [9, Propositions 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='3 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='6]) For 0 ≤ i, j ≤ d, the following (i), (ii) hold in T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (i) eiej = δijei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (ii) e∗ i e∗ j = δije∗ j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='3) and Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='5 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (See [9, Propositions 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='3 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='6]) For d ≥ 1, the following (i), (ii) hold in T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (i) x1 = �d i=0 θiei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (ii) x∗ 1 = �d i=0 θ∗ i e∗ i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='5) and Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='5 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' The following (i), (ii) hold in T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (i) e0 = |X|−1 �d i=0 xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (ii) e∗ 0 = |X|−1 �d i=0 x∗ i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='10), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='12), and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' □ THE GENERALIZED TERWILLIGER ALGEBRA OF THE HYPERCUBE 7 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' The Hypercube Qd In this section, we recall a family of distance-regular graphs called the hybercubes, and we review some results related to the associated Terwilliger Algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' For more informa- tion, we refer the reader to [3,10,13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Assume d ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Let Qd denote the graph with vertex set X consisting of d-tuples (a1, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' , ad) such that ai ∈ {−1, 1} (1 ≤ i ≤ d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Two vertices are adjacent in Qd whenever they differ in exactly one coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' The graph Qd is called the d-cube or a hypercube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' The graph Qd has 2d vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Furthermore, Qd is distance-regular (see [3, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' From now through Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='17, we consider the distance-regular graph Γ = Qd with d ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' We now recall the intersection numbers of Qd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (See [13, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' 238]) For 0 ≤ h, i, j ≤ d we have ph ij = � 0 if h + i + j is odd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' � h (i−j+h)/2 �� d−h (i+j−h)/2 � if h + i + j is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' In the lines above, we interpret �n m � = 0 if m < 0 or m > n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' We mention some consequences of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' For 0 ≤ i ≤ d we have ki = �d i � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Follows from Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' □ Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' For 0 ≤ h, i ≤ d we have ph 1i = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 d − h if h = i − 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' h if h = i + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' 0 if h ̸= i ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Follows from Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' □ Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' For 0 ≤ i ≤ d we have AAi = (i + 1)Ai+1 + (d − i + 1)Ai−1, where we recall (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='2) and Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' □ Let z denote an indeterminate, and let C[z] denote the algebra of polynomials in z that have coefficients in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Motivated by Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='5, we now define some polynomials in C[z].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Let {Fi}d+1 i=0 denote polynomials in C[z] such that F0 = 1, F1 = z, and zFi = (i + 1)Fi+1 + (d − i + 1)Fi−1 (1 ≤ i ≤ d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' We remark that the polynomial Fi has degree i and leading coefficient 1 i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (0 ≤ i ≤ d+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' We have Fi(A) = Ai (0 ≤ i ≤ d), and Fd+1(A) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Furthermore, pi(j) = Fi(θj) (0 ≤ i, j ≤ d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' 8 NATHAN NICHOLSON Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' By construction, F0(A) = I = A0 and F1(A) = A = A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' To see that Ai = Fi(A) (2 ≤ i ≤ d) and Fd+1(A) = 0, use induction and compare Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='5 with Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' To see that Fi(θj) = pi(j), apply Fi to both sides of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='5) and simplify with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='3) to obtain Fi(A) = d � j=0 Fi(θj)Ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Comparing the above with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='9), it follows that Fi(θj) = pi(j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' □ We next consider the eigenvalues of Qd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (See [3, Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='1]) For 0 ≤ i ≤ d we have θi = d − 2i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' The following definition is for notational convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Define Φd ∈ C[z] by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='1) Φd = d � i=0 � z − (d − 2i) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' The minimal polynomial of A is equal to Φd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Immediate from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='8 and Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' □ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' We have Φd = (d + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='Fd+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='7, Fd+1(A) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='10, Φd must divide Fd+1 in C[z].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Because both Φd and Fd+1 have degree d + 1, one must be a scalar multiple of the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Comparing the leading coefficients, the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' □ We have a comment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (See [3, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' 194]) The hypercube Qd is self-dual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' In other words, ph ij = (ph ij)∗ (0 ≤ h, i, j ≤ d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' For 0 ≤ i ≤ d we have k∗ i = �d i � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Follows from Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='3 and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' □ Next we state some corollaries of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' For the rest of this section, fix a vertex x of Qd, and let T = T(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' For 0 ≤ i ≤ d we have A∗A∗ i = (i + 1)A∗ i+1 + (d − i + 1)A∗ i−1, where we recall (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Similar to the proof of Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' □ Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' We have Fi(A∗) = A∗ i (0 ≤ i ≤ d), and Fd+1(A∗) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Furthermore, p∗ i (j) = Fi(θ∗ j) (0 ≤ i, j ≤ d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Similar to the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' □ Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' For 0 ≤ i ≤ d we have θ∗ i = d − 2i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' THE GENERALIZED TERWILLIGER ALGEBRA OF THE HYPERCUBE 9 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Follows from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='12 and the discussion of self-duality at the end of Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' □ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' The minimal polynomial of A∗ is equal to Φd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Follows from Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' □ From now until the end of the section, we assume Γ = Qd with d ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' In the next result, we consider the triples h, i, j such that ph ij and (ph ij)∗ are nonzero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' For 0 ≤ h, i, j ≤ d, the intersection number ph ij is nonzero if and only if the Krein parameter (ph ij)∗ is nonzero if and only if the following (i)–(iii) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (i) h, i, j satisfy the triangle inequality: h ≤ i + j, i ≤ h + j, j ≤ h + i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (ii) h + i + j ≤ 2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (iii) h + i + j is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' The case of d = 0 is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' The case of d ≥ 1 follows upon inspection of Proposi- tion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='2 and Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' □ Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Let Pd denote the set consisting of the 3-tuples of integers (h, i, j) such that 0 ≤ h, i, j ≤ d which satisfy (i)–(iii) of Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' For 0 ≤ h, i, j ≤ d, the following (i)–(iii) are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (i) ph ij ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (ii) (ph ij)∗ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (iii) (h, i, j) ∈ Pd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Compare Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='18 with Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' □ We conclude this section with a brief definition and a comment about the Terwilliger algebra T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Assume x is a vertex of Qd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Let Td = Td(x) denote the Terwilliger algebra of Qd with respect to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (See [10, Theorem 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='14]) There exists an algebra isomorphism Td → � 0≤r≤⌊d/2⌋ Md+1−2r(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' The Generalized Terwilliger Algebra for Qd In Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='1, we described the generalized Terwilliger algebra for a distance-regular graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' In this section, we consider this algebra for the graph Qd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' For d ≥ 0, let Td denote the generalized Terwilliger algebra associated with Qd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' 10 NATHAN NICHOLSON By Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='2, the algebra T0 is ismorphic to C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' For the rest of this section, we restrict our attention to the algebra Td with d ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' We next observe some analogues of results from Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' For 0 ≤ i ≤ d and with reference to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='2), the following (i), (ii) hold in Td.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (i) x1xi = (i + 1)xi+1 + (d − i + 1)xi−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (ii) x∗ 1x∗ i = (i + 1)x∗ i+1 + (d − i + 1)x∗ i−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Follows from Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='5 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='6 and Corollaries 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='5 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' □ Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' For 0 ≤ i ≤ d, the following (i), (ii) hold in Td.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (i) xi = Fi(x1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (ii) x∗ i = Fi(x∗ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Moreover, Td is generated by x1 and x∗ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Follows from Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='5 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='6 and Corollaries 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='7 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' □ Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' The polynomial Φd is equal to the minimal polynomial of both x1 and x∗ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Follows from Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='5, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='6, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='10, and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' □ Now that we have Lemmas 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='3 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='4, some of the relations in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='1 become redundant, giving us the following, simpler presentation Td.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' The algebra Td is isomorphic to the algebra with 1, with generators x1, x∗ 1 and relations (1)–(4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (1) Φd(x1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (2) Φd(x∗ 1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (3) e∗ hxie∗ j = 0 if (h, i, j) /∈ Pd (0 ≤ h, i, j ≤ d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (4) ehx∗ i ej = 0 if (h, i, j) /∈ Pd (0 ≤ h, i, j ≤ d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' In the above lines, we define x0 = 1, x∗ 0 = 1, xi = Fi(x1), x∗ i = Fi(x∗ 1) (2 ≤ i ≤ d), ei = 2−d d � j=0 qi(j)xj, e∗ i = 2−d d � j=0 q∗ i (j)x∗ j (0 ≤ i ≤ d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Compare Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='1 with Lemmas 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='3 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' □ We would like to provide another presentation for Td.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' To do this, we first define the following algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Let T ′ d denote the algebra with 1, with generators e0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' , ed, e∗ 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' , e∗ d and relations (1)–(4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (1) �d i=0 ei = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (2) �d i=0 e∗ i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (3) e∗ hxie∗ j = 0 if (h, i, j) /∈ Pd (0 ≤ h, i, j ≤ d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (4) ehx∗ i ej = 0 if (h, i, j) /∈ Pd (0 ≤ h, i, j ≤ d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' THE GENERALIZED TERWILLIGER ALGEBRA OF THE HYPERCUBE 11 In the above lines, we define x0 = 1, x∗ 0 = 1, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='1) x1 = d � i=0 (d − 2i)ei, x∗ 1 = d � i=0 (d − 2i)e∗ i , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='2) xi = Fi(x1), x∗ i = Fi(x∗ 1) (2 ≤ i ≤ d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='3) We will soon show that the algebra Td is isomorphic to T ′ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' We first give some lemmas about T ′ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' For 0 ≤ i ≤ d, the following (i), (ii) hold in T ′ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (i) eiej = δijei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (ii) e∗ i e∗ j = δije∗ i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (i) First assume that i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' The triple (i, 0, j) violates Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='18 (i), thus eix∗ 0ej = 0 by relation (4) of Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' As x∗ 0 = 1, we have eiej = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Next assume that i = j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' We will show that e2 i = ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' By relation (1) of Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='6 and the previous paragraph, ei = ei d � ℓ=0 eℓ = e2 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (ii) Similar to the proof of (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' □ Note that by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='2) and Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='7, the following equations hold in T ′ d : (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='4) � x1 − (d − 2i) � ei = 0, � x∗ 1 − (d − 2i) � e∗ i = 0 (0 ≤ i ≤ d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' The following (i), (ii) hold in T ′ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (i) Φd(x1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (ii) Φd(x∗ 1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (i) First note that the term � z −(d −2i) � is a factor of the right-hand side of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='1) (0 ≤ i ≤ d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Thus by Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='9 and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='4), Φd(x1)ei = 0 (0 ≤ i ≤ d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Hence by relation (1) of Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='6, Φd(x1) = Φd(x1) d � i=0 ei = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (ii) Similar to the proof of (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' □ Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' For 0 ≤ i ≤ d, the following (i), (ii) hold in T ′ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (i) ei = 2−d �d j=0 qi(j)xj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (ii) e∗ i = 2−d �d j=0 q∗ i (j)x∗ j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (i) By (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='3) and Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='7, xi = Fi(x1) = d � j=0 Fi(θj)ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' 12 NATHAN NICHOLSON Hence by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='7, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='5) xi = d � j=0 pi(j)ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' The result follows from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='5) and the fact that the matrices P, 2−dQ from below (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='11) are inverses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (ii) Similar to the proof of (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' □ We now show that the algebra Td is isomorphic to T ′ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' There exists a unique algebra isomorphism Td → T ′ d that sends x1 �→ x1, x∗ 1 �→ x∗ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Moreover, this map sends xi �→ xi, x∗ i �→ x∗ i , ei �→ ei, e∗ i �→ e∗ i , for 0 ≤ i ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' We will first show that there exists an algebra homomorphism σ : Td → T ′ d which sends x1 �→ x1 and x∗ 1 �→ x∗ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' We will then show that there exists an algebra homomorphism τ : T ′ d → Td which sends ei �→ ei and e∗ i �→ e∗ i (0 ≤ i ≤ d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' We will next show that σ and τ are inverses, and hence algebra isomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' We will last show that σ(xi) = xi, σ(x∗ i ) = x∗ i , σ(ei) = ei, and σ(e∗ i ) = e∗ i (0 ≤ i ≤ d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' We begin by showing that σ exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' For 0 ≤ i ≤ d, let fi = 2−d �d j=0 qi(j)xj ∈ T ′ d and f ∗ i = 2−d �d j=0 q∗ i (j)x∗ j ∈ T ′ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' To show that σ exists, it is sufficient to show that in T ′ d, Φd(x1) = 0, Φd(x∗ 1) = 0, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='6) f ∗ hxif ∗ j = 0, fhx∗ i fj = 0, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='7) for 0 ≤ h, i, j ≤ d such that (h, i, j) /∈ Pd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='8 implies (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='9 implies that fi = ei and f ∗ i = e∗ i (0 ≤ i ≤ d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Hence (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='7) follows by relations (3) and (4) of Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' We next show τ exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Let y1 = �d i=0 θiei ∈ Td and y∗ 1 = �d i=0 θ∗ i e∗ i ∈ Td.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' To show that τ exists, it is sufficient to show that in Td, d � ℓ=0 eℓ = 1, d � ℓ=0 e∗ ℓ = 1, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='8) e∗ hFi(y1)e∗ j = 0, ehFi(y∗ 1)ej = 0, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='9) for 0 ≤ h, i, j ≤ d such that (h, i, j) /∈ Pd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='7 implies (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='9 implies that x1 = y1 and x∗ 1 = y∗ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Hence Fi(y1) = xi and Fi(y∗ 1) = x∗ i (0 ≤ i ≤ d) by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Thus (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='9) follows by relations (3) and (4) of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' We now show that σ and τ are inverses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='9, τ(x1) = x1 and τ(x∗ 1) = x∗ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Thus τ ◦ σ : Td → Td is the identity map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='9, σ(ei) = ei and σ(e∗ i ) = e∗ i (0 ≤ i ≤ d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Thus σ ◦ τ : T ′ d → T ′ d is the identity map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Therefore σ and τ are inverses, and hence algebra isomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' THE GENERALIZED TERWILLIGER ALGEBRA OF THE HYPERCUBE 13 By construction, σ(x1) = x1 and σ(x∗ 1) = x∗ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Thus σ(xi) = xi and σ(x∗ i ) = x∗ i (0 ≤ i ≤ d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' As noted previously, σ(ei) = ei and σ(e∗ i ) = e∗ i (0 ≤ i ≤ d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' □ For the rest of the paper, we identify the algebras Td and T ′ d via the isomorphism in Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' We next define a free algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Let Td denote the free algebra with generators e0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' , ed, e∗ 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' , e∗ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Define x0 = 1, x1 = �d i=0 θiei, and xi = Fi(x1) (2 ≤ i ≤ d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Similarly, define x∗ 0 = 1, x∗ 1 = �d i=0 θ∗ i e∗ i , and x∗ i = Fi(x∗ 1) (2 ≤ i ≤ d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' For notational convenience, define xi = 0, x∗ i = 0, ei = 0, e∗ i = 0 (i < 0 or i > d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='10) Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Let Sd denote the two-sided ideal of Td generated by the following (1)–(4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (1) �d i=0 ei − 1, (2) �d i=0 e∗ i − 1, (3) e∗ hxie∗ j such that (h, i, j) /∈ Pd (0 ≤ h, i, j ≤ d), (4) ehx∗ i ej such that (h, i, j) /∈ Pd (0 ≤ h, i, j ≤ d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Because Td is free, there exists an algebra homomorphism ψd : Td → Td that sends ei �→ ei, e∗ i �→ e∗ i (0 ≤ i ≤ d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Comparing Definitions 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='6 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='12, it follows that the map ψd is surjective with kernel Sd, and that xi �→ xi, x∗ i �→ x∗ i (0 ≤ i ≤ d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' To end this section, we define an algebra homomorphism that will be useful later in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Consider the quotient algebra Td/Sd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' With reference to Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='13, the algebra homomorphism ψd induces an algebra isomorphism Td/Sd → Td.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' We denote the inverse of this map by pd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' The Primary Central Idempotent of Td We continue our discussion of the algebra Td from Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' In [9], Egge defines a certain element u0 ∈ Td called the primary central idempotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Later in the paper, we will use u0 to compute the dimension of Td.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' In this section, we recall the definition of u0 and develop some basic facts about it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' 14 NATHAN NICHOLSON Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (See [9, Propositions 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='1 and 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='4]) For d ≥ 0, following holds in Td: (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='1) 2d d � i=0 k−1 i e∗ i e0e∗ i = 2d d � i=0 (k∗ i )−1eie∗ 0ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' This element is central and idempotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (See [9, Proposition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='1]) Referring to Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='1, we define u0 to be the common value expressed in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' We call u0 the primary central idempotent of Td.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (See [9, Proposition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='5 and Theorem 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='5]) For d ≥ 0, the following (i)–(iii) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (i) The sum Td = Tdu0 + Td(1 − u0) is direct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (ii) Tdu0 and Td(1 − u0) are both two-sided ideals of Td.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (iii) The algebra Tdu0 is isomorphic to Md+1(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' For d ≥ 0, the algebra Td is isomorphic to the direct sum Md+1(C) ⊕ Td(1 − u0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Follows from Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' □ Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' There exists an algebra isomorphism Td/Tdu0 → Td(1 − u0) that sends ei + Tdu0 �→ ei(1 − u0), e∗ i + Tdu0 �→ e∗ i (1 − u0) (0 ≤ i ≤ d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Follows from Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='3 (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' □ We have some comments about u0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' For d ≥ 0, the following (i)–(iv) hold in Td.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (i) u0e0 = e0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (ii) u0ed = ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (iii) u0e∗ 0 = e∗ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (iv) u0e∗ d = e∗ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (i) By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='8 (i), Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='10 (ii), Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='13, and Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='2, u0e0 = 2d � d � r=0 (k∗ r)−1ere∗ 0er � e0 = 2de0e∗ 0e0 = e0 � d � i=0 x∗ i � e0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' For 1 ≤ i ≤ d, the triple (0, i, 0) does not satisfy Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='18 (i), thus e0x∗ i e0 = 0 by relation (4) of Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Hence by relation (T1) of Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='1, e0 � d � i=0 x∗ i � e0 = e0x∗ 0e0 = e2 0 = e0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Therefore u0e0 = e0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (ii) By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='8 (i), Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='10 (ii), Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='13, and Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='2, u0ed = 2d � d � r=0 (k∗ r)−1ere∗ 0er � ed = 2dede∗ 0ed = ed � d � i=0 x∗ i � ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' THE GENERALIZED TERWILLIGER ALGEBRA OF THE HYPERCUBE 15 For 1 ≤ i ≤ d, the triple (d, i, d) does not satisfy Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='18 (ii), thus edx∗ i ed = 0 by relation (4) of Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Hence by relation (T1) of Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='1, ed � d � i=0 x∗ i � ed = edx∗ 0ed = e2 d = ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Therefore u0ed = ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (iii) Similar to the proof of (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (iv) Similar to the proof of (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' □ We finish this section with a comment about the case d = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' For d = 1, the element u0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Moreover, the algebra T1 is isomorphic to M2(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Because d = 1, Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='7 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='6 imply that u0 = u0(e0 + e1) = e0 + e1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Therefore the algebra T1 is isomorphic to M2(C) by Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='3 part (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' □ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' The Main Result Recall the algebra homomorphism ♮ : Td → Td from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' In this section, we prove that ♮ is an algebra isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Recall the free algebra Td from Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Assume d ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Let ϕd : Td → Td−2 be the algebra homomorphism that sends e0 �→ 0, e∗ 0 �→ 0, ei �→ ei−1, e∗ i �→ e∗ i−1 (1 ≤ i ≤ d − 1), ed �→ 0, e∗ d �→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Referring to Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='1 and using (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='10), we see that ϕd sends (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='1) ei �→ ei−1, e∗ i �→ e∗ i−1 (0 ≤ i ≤ d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Assume d ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Let Kd denote the two-sided ideal of Td generated by e0, ed, e∗ 0, e∗ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Assume d ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' The map ϕd is surjective with kernel Kd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Routine consequence of Definitions 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='1 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' □ Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Assume d ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' For 0 ≤ i ≤ d, the following (i), (ii) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (i) ϕd(xi) = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f3 xi if i = 0 or i = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' xi − xi−2 if 2 ≤ i ≤ d − 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Φd−2(x1) (d−1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' − xi−2 if i = d − 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' x1Φd−2(x1) d!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' − xi−2 if i = d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' 16 NATHAN NICHOLSON (ii) ϕd(x∗ i ) = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f3 x∗ i if i = 0 or i = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' x∗ i − x∗ i−2 if 2 ≤ i ≤ d − 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Φd−2(x∗ 1) (d−1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' − x∗ i−2 if i = d − 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' x∗ 1Φd−2(x∗ 1) d!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' − x∗ i−2 if i = d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (i) We begin with a comment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Note that by Definitions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='6 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='11, the following holds in Td: (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='2) jxj = x1xj−1 − (d − j + 2)xj−2 (2 ≤ j ≤ d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' We now consider the cases for i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' First, assume i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' The result holds, because x0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Next, assume i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Then by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='8, Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='11, and Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='1, ϕd(x1) = ϕd � d � j=0 (d − 2j)ej � = d−1 � j=1 (d − 2j)ej−1 = d−2 � j=0 (d − 2 − 2j)ej = x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' By (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='10), it is correct to say that ϕd sends xi �→ xi − xi−2 for i = 0 and i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' This allows us to use induction for 2 ≤ i ≤ d − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' We proceed by induction on i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Assume 2 ≤ i ≤ d−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Setting j = i in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='2), applying ϕd to both sides, using induction, and dividing by i, we obtain (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='3) ϕd(xi) = x1(xi−1 − xi−3) − (d − i + 2)(xi−2 − xi−4) i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Using Definitions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='6 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='11, we find that in Td−2, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='4) x1xi−1 = ixi + (d − i)xi−2, x1xi−3 = (i − 2)xi−2 + (d − i + 2)xi−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' In equation (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='3), we distribute terms in the numerator, then eliminate x1xi−1 and x1xi−3 via (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' This yields ϕd(xi) = xi − xi−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Next, assume i = d − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Setting j = d − 1 in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='2), applying ϕd to the result, using induction, and dividing by d − 1 yields (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='5) ϕd(xd−1) = x1(xd−2 − xd−4) − 3(xd−3 − xd−5) d − 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Using Definitions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='6 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='11, we find that in Td−2, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='6) x1xd−4 = (d − 3)xd−3 + 3xd−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' In (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='5), we distribute terms in the numerator and eliminate x1xd−4 via (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' This yields (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='7) ϕd(xd−1) = x1xd−2 − xd−3 d − 1 − xd−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' THE GENERALIZED TERWILLIGER ALGEBRA OF THE HYPERCUBE 17 By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='11, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='8) x1xd−2 − xd−3 = Φd−2(x1) (d − 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' We use (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='8) to eliminate the numerator in the right-hand side of (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' This yields ϕd(xd−1) = Φd−2(x1) (d − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' − xd−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' For the rest of this proof, assume i = d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Setting j = d in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='2), applying ϕd to the result, using induction, and dividing by d yields ϕd(xd) = x1 �Φd−2(x1) (d−1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' − xd−3 � − 2(xd−2 − xd−4) d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='9) Using Definitions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='6 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='11, we find that in Td−2, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='10) x1xd−3 = (d − 2)xd−2 + 2xd−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' In (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='9), we distribute terms in the numerator and eliminate x1xd−3 via (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' This yields ϕd(xd) = x1Φd−2(x1) (d−1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' − dxd−2 d = x1Φd−2(x1) d!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' − xd−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (ii) Similar to the proof of (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' □ Recall the ideal Sd ⊆ Td from Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Our next general goal is to show that ϕd(Sd) = Sd−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' To do that, we will show that ϕd(Sd) ⊆ Sd−2 and Sd−2 ⊆ ϕd(Sd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' For d ≥ 2, the following (i), (ii) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (i) ϕd � �d i=0 ei − 1 � = �d−2 i=0 ei − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (ii) ϕd � �d i=0 e∗ i − 1 � = �d−2 i=0 e∗ i − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (i) By Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='1, ϕd � d � i=0 ei − 1 � = d−1 � i=1 ei−1 − 1 = d−2 � i=0 ei − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (ii) Similar to the proof of (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' □ Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Assume d ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' For 0 ≤ h, i, j ≤ d such that (h, i, j) /∈ Pd, the following (i), (ii) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (i) (h − 1, i, j − 1) /∈ Pd−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (ii) (h − 1, i − 2, j − 1) /∈ Pd−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' We consider the three cases in Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' For convenience, we consider them in the order (ii), (iii), (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' 18 NATHAN NICHOLSON First, assume h + i + j > 2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Then (h − 1) + i + (j − 1) > 2(d − 2), (h − 1) + (i − 2) + (j − 1) > 2(d − 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Thus (i) and (ii) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Next, assume h+i+j is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Then (h−1)+i+(j−1) is odd and (h−1)+(i−2)+(j−1) is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Thus (i) and (ii) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' For the rest of this proof, assume h, i, j fail the triangle inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' This leaves two subcases: i > h + j, i < |h − j|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' First, assume i > h + j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Then i > (h − 1) + (j − 1), i − 2 > (h − 1) + (j − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Hence h − 1, i, j − 1 and h − 1, i − 2, j − 1 fail the triangle inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Thus (i) and (ii) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Lastly, assume i < |h − j|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Then i < |(h − 1) − (j − 1)|, i − 2 < |(h − 1) − (j − 1)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Hence h − 1, i, j − 1 and h − 1, i − 2, j − 1 fail the triangle inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Thus (i) and (ii) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' □ Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Assume d ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' For 0 ≤ h, i, j ≤ d such that (h, i, j) /∈ Pd, the following (i), (ii) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (i) ϕd(e∗ hxie∗ j) ∈ Sd−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (ii) ϕd(ehx∗ i ej) ∈ Sd−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (i) First note that if h = 0 or h = d or j = 0 or j = d, then ϕd(e∗ hxie∗ j) = 0 by Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Thus for the remainder of this proof, we assume 1 ≤ h, j ≤ d − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' We consider the cases from Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='4 (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' First, assume i = 0 or i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' By Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='1 and Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='4, ϕd(e∗ hxie∗ j) = e∗ h−1xie∗ j−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' By Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='6, (h − 1, i, j − 1) /∈ Pd−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Hence by Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='12, e∗ h−1xie∗ j−1 ∈ Sd−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Next, assume 2 ≤ i ≤ d − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Then ϕd(e∗ hxie∗ j) = e∗ h−1xie∗ j−1 − e∗ h−1xi−2e∗ j−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' By Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='6, (h − 1, i, j − 1) /∈ Pd−2 and (h − 1, i − 2, j − 1) /∈ Pd−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Hence by Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='12, e∗ h−1xie∗ j−1 − e∗ h−1xi−2e∗ j−1 ∈ Sd−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Next, assume i = d − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Then ϕd(e∗ hxie∗ j) = e∗ h−1Φd−2(x1)e∗ j−1 (d − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' − e∗ h−1xi−2e∗ j−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' THE GENERALIZED TERWILLIGER ALGEBRA OF THE HYPERCUBE 19 By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='8, Φd−2(x1) ∈ Sd−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' By Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='12 and Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='6, e∗ h−1xi−2e∗ j−1 ∈ Sd−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Thus e∗ h−1Φd−2(x1)e∗ j−1 (d − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' − e∗ h−1xi−2e∗ j−1 ∈ Sd−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' For the remainder of this proof, assume i = d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Then ϕd(e∗ hxiej∗) = e∗ h−1x1Φd−2(x1)e∗ j−1 d!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' − e∗ h−1xi−2e∗ j−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='8, Φd−2(x1) ∈ Sd−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' By Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='12 and Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='6, e∗ h−1xi−2e∗ j−1 ∈ Sd−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Thus e∗ h−1x1Φd−2(x1)e∗ j−1 d!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' − e∗ h−1xi−2e∗ j−1 ∈ Sd−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (ii) Similar to the proof of (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' □ We have now shown that ϕd(Sd) ⊆ Sd−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Next we show that Sd−2 ⊆ ϕd(Sd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' To that end, we include the following technical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Assume d ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' The following (i), (ii) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (i) Φd−2(x1)(1 − e0 − ed) ∈ Sd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (ii) Φd−2(x∗ 1)(1 − e∗ 0 − e∗ d) ∈ Sd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (i) Observe that (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='11) Φd−2(x1)(1 − e0 − ed) = Φd−2(x1) � 1 − d � i=0 ei � + Φd−2(x1) d−1 � i=1 ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' By Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='12, 1 − �d i=0 ei ∈ Sd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Hence (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='12) Φd−2(x1) � 1 − d � i=0 ei � ∈ Sd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='8 and Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='9, Φd−2(x1) = d−1 � i=1 (x1 − θi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' By (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='4) and Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='13, (x1 − θi)ei ∈ Sd (1 ≤ i ≤ d − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Hence (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='13) Φd−2(x1) d−1 � i=1 ei ∈ Sd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' It follows from (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='11), (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='12), and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='13) that Φd−2(x1)(1 − e0 − ed) ∈ Sd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (ii) Similar to the proof of (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' □ Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Assume d ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' The following (i), (ii) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (i) Φd−2(x1) ∈ ϕd(Sd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (ii) Φd−2(x∗ 1) ∈ ϕd(Sd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' 20 NATHAN NICHOLSON Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (i) By Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='8 (i), Φd−2(x1)(1−e0−ed) ∈ Sd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' By Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='1 and Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='4, ϕd � Φd−2(x1)(1 − e0 − ed) � = Φd−2(x1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (ii) Similar to the proof of (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' □ Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Assume d ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' For 0 ≤ h, i, j ≤ d − 2 such that (h, i, j) /∈ Pd−2, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='14) (h + 1, i − 2r, j + 1) /∈ Pd (0 ≤ r ≤ ⌊i/2⌋), or (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='15) (h + 1, i + 2r, j + 1) /∈ Pd (1 ≤ r ≤ ⌊(d − i)/2⌋).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' We consider the three cases in Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' For convenience, we consider these cases in order (ii), (iii), (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' First, assume that h + i + j > 2(d − 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Then (h + 1) + (i + 2r) + (j + 1) > 2d (1 ≤ r ≤ ⌊(d − i)/2⌋).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Thus (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='15) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Next, assume that h+i+j is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Then (h+1)+(i−2r)+(j+1) is odd (0 ≤ r ≤ ⌊i/2⌋).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Thus (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='14) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' For the rest of this proof, assume that h, i, j fail the triangle inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' This leaves two subcases: i > h + j, i < |h − j|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' First, assume i > h + j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Then i + 2r > (j + 1) + (h + 1) (1 ≤ r ≤ ⌊(d − i)/2⌋).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Hence h + 1, i + 2r, j + 1 fail the triangle inequality (1 ≤ r ≤ ⌊(d − i)/2⌋).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Thus (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='15) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Lastly, assume i < |h − j|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Then i − 2r < |(h + 1) − (j + 1)| (0 ≤ r ≤ ⌊i/2⌋).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Hence h+1, i−2r, j+1 fail the triangle inequality (0 ≤ r ≤ ⌊i/2⌋).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Thus (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='14) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' □ Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Assume d ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' For 0 ≤ h, i, j ≤ d − 2 such that (h, i, j) /∈ Pd−2, the following (i), (ii) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (i) e∗ hxie∗ j ∈ ϕd(Sd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (ii) ehx∗ i ej ∈ ϕd(Sd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (i) We consider the two cases in Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' First, assume (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='14) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Then by Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='12, e∗ h+1xi−2re∗ j+1 ∈ Sd, (0 ≤ r ≤ ⌊i/2⌋).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' By Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='1 and Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='4, ϕd � ⌊i/2⌋ � r=0 e∗ h+1xi−2re∗ j+1 � = ⌊i/2⌋−1 � r=0 (e∗ hxi−2re∗ j − e∗ hxi−2r−2e∗ j) + e∗ hxi−2⌊i/2⌋e∗ j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='16) After expanding the sum and cancelling terms, the right-hand side of (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='16) becomes e∗ hxie∗ j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Thus e∗ hxie∗ j ∈ ϕd(Sd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' THE GENERALIZED TERWILLIGER ALGEBRA OF THE HYPERCUBE 21 For the rest of this proof, assume (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='15) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Then by Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='12, e∗ h+1xi+2re∗ j+1 ∈ Sd (1 ≤ r ≤ ⌊(d − i)/2⌋).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' For notational convenience, define a polynomial g ∈ C[z] by g = � 1 (d−1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' if d − i is odd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' z d!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' if d − i is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' We have defined g such that by Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='4, ϕd(xi+2⌊(d−i)/2⌋) = g(x1)Φd−2(x1) − xi+2⌊(d−i)/2⌋−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Thus by Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='1 and Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='4, ϕd � ⌊(d−i)/2⌋ � r=1 e∗ h+1xi+2re∗ j+1 � = ⌊(d−i)/2⌋−1 � r=1 (e∗ hxi+2re∗ j − e∗ hxi+2r−2e∗ j) + e∗ hg(x1)Φd−2(x1)e∗ j − e∗ hxi+2⌊(d−i)/2⌋−2e∗ j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='17) After expanding the sum and cancelling terms, the right-hand side of (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='17) becomes −e∗ hxie∗ j + e∗ hg(x1)Φd−2(x1)e∗ j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Hence (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='18) − e∗ hxie∗ j + e∗ hg(x1)Φd−2(x1)e∗ j ∈ ϕd(Sd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' By Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='9 (i) and the surjectivity of ϕd, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='19) e∗ hg(x1)Φd−2(x1)e∗ j ∈ ϕd(Sd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Therefore by (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='18) and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='19), e∗ hxie∗ j ∈ ϕd(Sd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' (ii) Similar to the proof of (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' □ We have Sd−2 ⊆ ϕd(Sd) by Lemmas 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='5 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='11 together with the surjectivity of ϕd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Assume d ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Then ϕd(Sd) = Sd−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' We mentioned below Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='7 that ϕd(Sd) ⊆ Sd−2, and we mentioned below Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='11 that Sd−2 ⊆ ϕd(Sd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' □ We next consider how ϕd induces an algebra homomorphism from Td → Td−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Assume d ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' There exists an algebra homomorphism ϕ′ d : Td → Td−2 that sends e0 �→ 0, e∗ 0 �→ 0, ei �→ ei−1, e∗ i �→ e∗ i−1 (1 ≤ i ≤ d − 1), ed �→ 0, e∗ d �→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Moreover, ϕ′ d is surjective, and ker(ϕ′ d) = ψd(Kd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' We first consider the existence of ϕ′ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='13, Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='3, and Proposi- tion 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='12, we have a surjective algebra homomorphism ψd−2 ◦ ϕd : Td → Td−2 with kernel equal to Sd + Kd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' This map induces an algebra isomorphism from the quotient algebra Td/(Sd + Kd) → Td−2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' we say this isomorphism is canonical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' 22 NATHAN NICHOLSON Let q : Td/Sd → Td/(Sd + Kd) denote the quotient map, which we recall is an algebra homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Recall the algebra isomorphism pd : Td → Td/Sd from Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' The following composition gives an algebra homomorphism from Td → Td−2: (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='20) ϕ′ d : Td Td/Sd Td/(Sd + Kd) Td−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' pd q can We have shown that ϕ′ d exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' With reference to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='2), one routinely check that ϕ′ d sends ei �→ ei−1 and e∗ i �→ e∗ i−1 (0 ≤ i ≤ d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' We next show that ϕ′ d is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' This follows because each of the composition factors in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='20) is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Lastly, we consider the kernel of ϕ′ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Inspection of (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='20) shows that ker(ϕ′ d) = p−1 d (Kd+ Sd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' By the construction of pd, p−1 d (Kd + Sd) = ψd(Kd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Hence ker(ϕ′ d) = ψd(Kd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' □ Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Assume d ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' The ideal Tdu0 is equal to ψd(Kd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Moreover, Tdu0 = ker(ϕ′ d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' We first consider the first assertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Because ψd is surjective, ψd(Kd) is equal to the two-sided ideal of Td generated by e0, ed, e∗ 0, e∗ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Thus by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='6, the ideal ψd(Kd) ⊆ Tdu0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Recall that u0 = 2d �d r=0 k−1 r e∗ re0e∗ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Thus Tdu0 ⊆ Tde0 ⊆ ψd(Kd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' This proves the first assertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' The second assertion follows by the first, together with Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' □ Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Assume d ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' There exists an algebra isomorphism Td(1 − u0) → Td−2 that sends e0(1 − u0) �→ 0, e∗ 0(1 − u0) �→ 0, ei(1 − u0) �→ ei−1, e∗ i (1 − u0) �→ e∗ i−1 (1 ≤ i ≤ d − 1), ed(1 − u0) �→ 0, e∗ d(1 − u0) �→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' With reference to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='2), Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='13 implies that there exists an induced algebra isomorphism Td/ ker(ϕ′ d) → Td−2 which sends ei + ker(ϕ′ d) �→ ei−1, e∗ i + ker(ϕ′ d) �→ e∗ i−1 (0 ≤ i ≤ d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' By Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='14, we know that ker(ϕ′ d) = Tdu0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Identifying the quotient algebra Td/Tdu0 with Td(1 − u0) via the isomorphism in Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='5, the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' □ Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Assume d ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' There exists an algebra isomorphism Td → Md+1(C) ⊕ Td−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Follows from Corollaries 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='4 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' □ Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Assume d ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Then there exists an algebra isomorphism Td → � 0≤r≤⌊d/2⌋ Md+1−2r(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Moreover, the algebra Td is isomorphic to Td.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' THE GENERALIZED TERWILLIGER ALGEBRA OF THE HYPERCUBE 23 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' We first consider the first assertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' We proceed by induction on d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' The base cases of d = 0 and d = 1 are addressed in Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='2 and Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Now assume d ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' By Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='16 and induction, we have algebra isomorphisms Td → Md+1(C) ⊕ Td−2 → � 0≤r≤d/2 Md+1−2r(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' This completes the proof of the first assertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' To prove the second assertion, compare this result to Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' □ We conclude with the main result of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Assume d ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' The map ♮ : Td → Td from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='4 is an algebra isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' By Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='17, Td and Td have the same dimension as algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Because ♮ is a surjective algebra homomorphism between two algebras of the same dimension, it is an algebra isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' □ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Acknowledgments The author is presently a graduate student at the University of Wisconsin–Madison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' He would like to thank his advisor, Paul Terwilliger, for suggesting this project, for his hours of mentoring, and for giving many valuable suggestions for this manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' References [1] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Bannai and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Ito.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Algebraic Combinatorics I: Association Schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Benjamin/Cummings, Lon- don, 1984.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' 15 (1999) 143-158.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' [6] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Curtin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Bipartite distance-regular graphs II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Graphs Combin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' 15 (1999) 377-391.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Distance-regular Graphs, the Subconstituent Algebra, and the Q-polynomial Property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Preprint arXiv (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' [17] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Terwilliger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' The subconstituent algebra of an association scheme I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Algebraic Combin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' 1 (1992) 363–388.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' [18] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Terwilliger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' The subconstituent algebra of an association scheme II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Algebraic Combin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' 2 (1993) 73–103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' [19] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Terwilliger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' The subconstituent algebra of an association scheme III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Algebraic Combin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' 2 (1993) 177–210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content=' Nathan Nicholson Department of Mathematics University of Wisconsin 480 Lincoln Drive Madison, WI 53706-1388 USA email: nlnicholson@wisc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} +page_content='edu' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE_T4oBgHgl3EQf6xyS/content/2301.08366v1.pdf'} diff --git a/mNFIT4oBgHgl3EQfsyss/content/tmp_files/2301.11337v1.pdf.txt b/mNFIT4oBgHgl3EQfsyss/content/tmp_files/2301.11337v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..1c6e63ec82768340a5f04936e91f054212c21e11 --- /dev/null +++ b/mNFIT4oBgHgl3EQfsyss/content/tmp_files/2301.11337v1.pdf.txt @@ -0,0 +1,3468 @@ +New critical states induced by measurement +Xinyu Sun,1 Hong Yao,1, ∗ and Shao-Kai Jian2, † +1Institute for Advanced Study, Tsinghua University, Beijing 100084, China +2Department of Physics and Engineering Physics, +Tulane University, New Orleans, Louisiana, 70118, USA +(Dated: January 30, 2023) +Finding new critical states of matter is an important subject in modern many-body physics. Here +we study the effect of measurement and postselection on the critical ground state of a Luttinger liquid +theory and show that it can lead to qualitatively new critical states. Depending on the Luttinger +parameter K, the effect of measurement is irrelevant (relevant) at K > 1 (K < 1). +We reveal +that this causes an entanglement transition between two phases, one with logarithmic entanglement +entropy for a subregion (K > 1), and the other an algebraic entanglement entropy (K < 1). At +the critical point K = 1, the measurement is marginal, and we find new critical states whose +entanglement entropy exhibits a logarithmic behavior with a continuous effective central charge as +a function of measurement strength. We also performed numerical density matrix renormalization +group and fermionic Gaussian state simulations to support our results. We believe that our work +provides a promising and feasible route to experimentally realize new critical states. +Introduction.—Critical state underlies various inter- +esting physics in phase transition, hydrodynamics, and +even quantum gravity according to AdS/CFT correspon- +dence. While conformal field theory (CFT) describes a +huge class of critical states, it is of impact to find critical +states beyond the description of CFT. Recently, stud- +ies on quantum trajectory with local measurement re- +veal a critical point separating two quantum phases with +distinct entanglement structures [1–5]. It is natural to +study the effect of measurement in CFT. The effect of +local projective measurement in CFT is well described +by boundary CFT [6–10], where (a region of) the critical +state is projected onto a Cardy state [8] with zero spatial +entanglement. Apart from the projective measurement +in boundary CFT, less is known about general measure- +ment, such as local weak measurement. +In this paper, we are interested in the effect of weak +measurement on the ground state of a Luttinger liquid. +Different from the constant monitoring in free fermion +systems studied previously [11–15], we consider perform- +ing weak measurement to the critical ground state of the +Luttinger liquid without time evolution. +Ref. [16] re- +ports a transition as a function of Luttinger parameter +between two phases with algebraic correlation functions +of distinct power laws after weak measurement, neverthe- +less, the entanglement property after weak measurement +is left unanswered. Since measurement can change entan- +glement radically, the scaling of entanglement entropy in +these two phases is not immediately obvious. +We study in depth the entanglement properties of the +(spinless) Luttinger liquid theory after weak measure- +ment and postselection. A schematic representation of +the model and the phase diagram are shown in Fig. 1(a) +and (b), respectively. Depending on the Luttinger pa- +rameter K, we find that for K > 1 the measurement +is irrelevant, and the state retains logarithmic entangle- +ment entropy with a central charge c = 1 for a subregion. +(a) +(b) +FIG. 1. (a) A schematic plot of the spinless fermion chain. t +and V denote the hopping and the nearest-neighbor interac- +tion, respectively, in Eq. (1) W is the measurement strength +in Eq. (3). (b) Phase diagram of the Luttinger liquid after +weak measurement. K and v ∝ W denote the Luttinger pa- +rameter and effective measurement strength, respectively. For +K > 1, the measurement is irrelevant, and the entanglement +entropy of a subregion A with length xA satisfies a log-law +with central charge c = 1. For K < 1, the measurement is +relevant, and changes the entanglement entropy from a log- +law to an area law with a subleading algebraic correction. At +K = 1, there is a continuous critical line, at which the mea- +surement is marginal. The entanglement entropy satisfies a +log-law with an effective central charge ceff = 1/ cosh2 v. +While for K < 1, the measurement becomes relevant, +and we find that the state exhibits an area law with a +(subleading) algebraic entanglement entropy. The alge- +braic power-law is obtained in the dual theory by tak- +ing advantage of a strong-weak duality. We further per- +form a large-scale density matrix renormalization group +arXiv:2301.11337v1 [quant-ph] 26 Jan 2023 + +V +t +MV +1 +SA +log xA +3cosh2 +V +SA = So + xA +-2/K+2 +logxA +3 +0 +1/2 +1 +K2 +(DMRG) calculation of an equivalent XXZ model to sup- +port our findings. In particular, the (subleading) alge- +braic entropy is consistent with our prediction over a wide +range of K. +Furthermore, at the critical point K = 1, the measure- +ment is marginal. We identify a critical line for different +measurement strength, on which the entanglement en- +tropy exhibits a logarithmic behavior with an effective +central charge continuously changing with the measure- +ment strength. Since at K = 1 the theory is equivalent to +a free fermion model, the effective central charge is cal- +culated by a spacetime rotation of the low-energy Dirac +fermion theory. We also perform a fermionic Gaussian +state simulation to calculate the half-chain entanglement +entropy, and the effective central charge extracted from +the data verifies our prediction for a wide range of mea- +surement strength. +Model.— We consider spinless fermions in a 1D chain +with the Hamiltonian +H = −t +� +i +(c† +ici+1 + h.c.) + V +� +i +(ni − 1 +2)(ni+1 − 1 +2),(1) +where c† +i (ci) denotes the fermion creation (annihilation) +operator at site i, and ni = c† +ici is the density operator. +t and V denote the hopping and the interaction between +nearest neighbor sites, respectively. We define ∆ = V/t, +and set t = 1 without loss of generality (i.e., energy is +measured in unit of t). +It is well known [17] that for +|∆| < 1 the ground state is described by a free compact +boson with the Luttinger parameter [see Eq. (5) below] +K = +π +2(π − arccos ∆). +(2) +We now consider a weak measurement to the ground +state of such a Luttinger liquid. For the model we consid- +ered, there is one qubit (given by the occupation number +of a spinless fermion) at each site. +The measurement +at site i is described by the following Kraus operator +{e−W Pi, +√ +1 − e−2W Pi}, where W ≥ 0 is the measure- +ment strength, and P2i−1 = 1 − n2i−1, P2i = n2i. +A +physical implementation of this Kraus operator is given +in Ref. [16]. After each measurement, the first outcome +is post-selected. Because measurement operators at dif- +ferent sites commute, we arrive at the total measurement +operator (up to an unimportant constant) +M = e−W � +i(−1)ini, +(3) +and the post-selected density matrix +ρm = +MρM † +Tr[MρM †], +(4) +where ρ = limβ→∞ e−βH is an unnormalized projection +onto the ground state. +We are interested in the entanglement properties of ρm. +For weak measurement strength W ≪ 1, we expect the +low energy bosonized theory is still valid, and a nontrivial +critical state can be obtained. +To this end, using the +path integral representation (see Supplemental Material +for the derivation), the post-selected density matrix is +proportional to ⟨˜φ(x)|MρM †|˜φ′(x)⟩ = +� +b.c. Dφe−S, with +the action +S = +� +dτdx +� +1 +2πK [(∂τφ)2 + (∂xφ)2] + δ(τ)v cos 2φ +� +,(5) +where φ is the boson field, and the boundary condition +φ(x, 0−) = ˜φ(x), φ(x, 0+) = ˜φ′(x). Here, |˜φ⟩ is the state +with field configuration given by ˜φ, while φ(x, τ) is the +compact boson field. Here v ∝ W, and the Delta func- +tion takes care of the measurement. The last term should +be considered as a sum of two terms at an infinitesimal +positive and negative τ, respectively. +It is worth not- +ing that while the model in Eq. (1) is at half-filling, our +main results is still valid for general filling (having Fermi +momentum kF ) with the effective measurement operator +having a 2kF momentum. +On the other hand, at strong measurement strength +W ≫ 1, the post-selected state is close to a product +state. In the following, we will first study the weak mea- +surement case in depth, which leads to qualitatively new +critical states, and defer the strong measurement case to +the discussion in the end. +Measurement induced transition.— To characterize the +critical state at weak measurement, we are particularly +interested in its entanglement entropy. It can be calcu- +lated via replica trick as +SA = −Tr[ρA log ρA] = lim +n→1 +1 +1 − nTr[ρn +A], +(6) +where ρA = Tr ¯ +A[ρm] is the reduced density matrix in +the subregion A (here +¯A denotes the complement of +A). +This amounts to replicate the theory ci → ci,a, +a = 1, ..., n is the replica index. +Because the mea- +surement operator is bilinear in the fermionic operator, +different replica momenta will decouple after we make +a Fourier transform w.r.t. +the replica index. +More +explicitly, the measurement operator in the replicated +theory is M = e−W � +i,a(−1)ini,a (with abuse of nota- +tion, we use the same symbol M). The replica Fourier +transform is defined by ci,k = +1 +√n +� +a ci,aei 2πka +n , with +k the replica momentum. +In the replica momentum +basis, M = � n−1 +2 +k=− n−1 +2 +Mk, and Mk = e−W � +i(−1)ini,k +can be straightforwardly bosonized to get δ(τ)v cos 2φk. +Combining the quadratic term originated from the pre- +measured Hamiltonian, we arrive at the following decou- +pled action for each replica momentum k (see Supple- +mental Material for details), +sk = +1 +πK +� dq +2π |q||φk(q)|2 + v +� +dx cos 2φk(x). +(7) +Here we have further integrated over the time direction. + +3 +q denotes the momentum from Fourier transform φk(q) = +� +dxφk(x)eiqx, and φk(x) = φk(x, τ = 0). +The entanglement entropy SA of compact bosons boils +down to the expectation value of the twist operator [18, +19] TA = � n−1 +2 +k=− n−1 +2 +TA,k with +TA,k = e−i k +n +√ +4 +K (φk(xA)−φk(0)), +(8) +where we assume that the interval A = {x|x ∈ (0, xA)}, +and take the replica limit. +Before we present results of the entanglement entropy, +we discuss the renormalization group (RG) flow at low +energies to determine the phase diagram. +In Eq. (7), +the momentum k is a dummy index because different +replica momenta decouple; therefore, the RG equation +for v is the same as that of a single replica. It is given +by [16, 20, 21] +dv +dl = (1 − K)v. +(9) +On the other hand, K is exactly marginal ( dK +dl = 0) be- +cause the first term in Eq. (7) is non-analytical that does +not receive a correction from RG process. Heuristically, +the measurement term is only present at τ = 0, so it +cannot renormalize K that is present in 1+1D. Combin- +ing these two facts, the flow of v is simple: it is relevant +(irrelevant) for K < 1 (K > 0), and marginal at K = 1. +For K > 1, v is irrelevant, and in the lowest energy +scale, the entanglement entropy of A after weak measure- +ment reduces to that of a free boson with central charge +c = 1, +SA = 1 +3 log xA. +(10) +While for K < 1, v is relevant, we will show in the follow- +ing that its entanglement property changes qualitatively +after weak measurement. Thus, there is an entanglement +transition at K = 1. +To study the entanglement entropy for K < 1, because +v is relevant, it is easier to work with the dual field [17, 21] +θ, defined via [∂xφ(x), θ(x′)] = iπδ(x−x′). We give a de- +tailed derivation of the dual field theory in Supplemental +Material. The dual action reads +sk = K +4π +� dq +2π |q||θk(q)|2 + γ +� +dx cos θk(x), +(11) +where γ = 2e−av−4√v, (a is a constant, whose expression +is given in Supplemental Material), and k denotes the +replica momentum. When v flows to a large number, it +means γ is small, so we can perform a perturbation cal- +culation. Thanks to decoupling between different replica +momenta, ⟨TA⟩ = � +k⟨TA,k⟩, where ⟨·⟩ = Tr[·ρ⊗n +m ]. At +the leading order, we arrive at (see Supplemental Mate- +rial for detail) ⟨TA,k⟩ = exp +� +γ2fk(K)x +− 2 +K +2 +A +� +. Finally, +W=0.0 +W=0.6 +10 +20 +50 +100 +200 +1.0 +1.5 +2.0 +2.5 +L +Sent +Δ=-0.6 +(a) +W=0.0 +50 +100 +150 +200 +1.2 +1.4 +1.6 +1.8 +2.0 +2.2 +L +Sent +Δ=0.6 +(b) +W=0.6 +50 +100 +150 +200 +0.275 +0.285 +0.295 +0.305 +L +Sent +Δ=0.6 +(c) +W=0.6 +0.57 +0.6 +0.63 +0.66 +0.69 +0.72 +0.8 +1.0 +1.2 +1.4 +K +power +(d) +FIG. 2. The half-chain entanglement entropy as a function +of different sizes at (a) ∆ = −0.6, (b,c) ∆ = 0.6. (a) The +blue (orange) curve is given by measurement strength W = 0 +(W = 0.6). +The same slope indicates the effective central +charges are the same. +(b) shows the data at measurement +strength W = 0. It shows a logarithmic function with central +charge c = 1. (c) shows the data at measurement strength +W = 0.6. It shows an algebraic function with power 0.77. (d) +The algebraic power as a function of different K for K < 1. +The measurement strength is W = 0.6. The black dots show +the power fitted by numerical data. The orange curve is our +prediction 2/K − 2. The numerical calculation is obtained +with bond dimension χ = 100. +taking the replica limit, the entanglement entropy for +K < 1 reads +SA = γ2 � +S0 + f(K)x +− 2 +K +2 +A +� +. +(12) +Here, the expressions for fk(K) and f(K) are given in +Supplemental Material. The first term is a non-universal +constant piece that accounts for the leading area-law +contribution. +Therefore, the entanglement entropy for +K < 1 shows an area law with a subleading power-law +behavior. +With Eq. (10) at K > 1 and Eq. (12) at K < 1, we +can conclude a measurement-induced entanglement tran- +sition between two phases with a logarithmic entangle- +ment and a (subleading) algebraic entanglement, respec- +tively. +To further support our conclusion, we perform +DMRG calculation [22] of entanglement entropy for ρm +in Eq. (4). Details of the simulation can be found in +Supplemental Material. For K > 1, Fig. 2(a) shows the +entanglement entropy for W = 0 (blue) and W = 0.6 +(orange). The same slope 1/3 indicates that the central +charge of both cases is c = 1, and the measurement is +irrelevant. For K < 1, Fig. 2 shows that the entangle- +ment entropy w.r.t system sizes is a logarithmic function +for W = 0 (b), an algebraic function for W = 0.6 (c). It + +4 +demonstrates that the measurement is relevant for K < 1 +and changes the entanglement entropy from a logarithm +law without measurement to an area law with sublead- +ing algebraic correction with measurement. In Fig. 2 (d), +the black dots represent the powers of the entanglement +entropy fitted from numerical data for W > 0 and the or- +ange curve is our prediction −2/K + 2. We can see that +they are consistent. We further perform data collapse to +demonstrate this measurement-induced transition in the +Supplemental Material. +Critical point.— We now discuss the entanglement en- +tropy at the critical point K = 1, where the measurement +parameter v is marginal. In this case, the interaction V +vanishes, and the model reduces to a free fermion the- +ory with measurement (3) at τ = 0. For a free fermion +theory, the twist operator is simply given by [23, 24] +TA = +� +k +ei 2πk +n QA, +QA = +� +x∈A +dxψ†(x)ψ(x). (13) +As we are interested in the long-wavelength behavior of +entanglement entropy, we have taken a continuum limit. +Here ψ(x) = ψ(x, 0) denotes the fermion operator in the +continuum limit. Because different replica momenta de- +couple in free fermion theory, we omit the dummy index k +in QA. Performing the product over replica momentum, +the entanglement entropy is SA = π2 +3 ⟨Q2 +A⟩ [24]. +Now consider the effect of measurement. The measure- +ment induces a scattering between left and right movers, +and it is not hard to deduce the low-energy theory +L = ¯ψ(/∂ + vδ(τ))ψ, +(14) +where ¯ψ = ψ†γ0, /∂ ≡ ∂µγµ is the Dirac operator, with +γ0 = σx, γ1 = −σy. For simplicity, we scale the coor- +dinate to set Fermi velocity to one, and v ∝ W is de- +termined by the measurement strength. Since the Dirac +operator has a rotational symmetry in the Euclidean sig- +nature, we can make a space-time rotation such that +the measurement operator becomes a defect located at +x = 0 [16, 20], with the corresponding Hamiltonian +H′ = −i∂xσz + vδ(x)σx. +(15) +Moreover, the entanglement entropy becomes the corre- +lation function in the time direction +SA = π2 +3 +� +t,t′⟨ψ†(0, t)ψ(0, t)ψ†(0, t′)ψ(0, t′)⟩ +(16) += π2 +3 +� +t,t′ +� +dµ|⟨ΨE|ψ†(0)ψ(0)|GS⟩|2e−iE(t−t′), +where +� +t,t′ = +� xA +0 +dt +� xA +0 +dt′, |GS⟩ denotes the ground +state whose energy is set to be zero without loss of gen- +erality, and |ΨE⟩ is a complete eigenbasis with energy E. +The integral over +� +dµ denotes the integral over this basis +that includes energy E and parity r, as we will discuss +W=1.0 +5 +10 +50 +100 +0.10 +0.15 +0.20 +0.25 +L +Sent +Δ=0 +(a) +0 +1 +2 +3 +4 +5 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +W +ceff +Δ=0 +(b) +FIG. 3. (a) The half-chain entanglement entropy at the crit- +ical point as a function of different system sizes L. The pa- +rameter is chosen to be t = 1, V = 0, W = 1. The black dots +represent the numerical data, and the orange line is a fitting +with 0.04 log L + 0.0435. (b) The effective central charge as +a function of different measurement strength W. The black +dots represent effective central charge that is extracted from +fitting the numerical data. The orange curve is our prediction +ceff = 1/ cosh2(αW), where α = 1.7 is a constant. +in the following. In the second step, we used the Heisen- +berg evolution ψ(0, t) = eiH′tψ(0)e−iH′t, and inserted a +complete basis +� +dµ|ΨE⟩⟨ΨE| = 1. +It is worth noting a reflecting symmetry for Eq. (15), +i.e., x → −x with conjugation of σx. +In the single- +particle eigenstate basis of Eq. (15), ψE,r(x), that satis- +fies H′ψE,r(x) = EψE,r(x), r = ±1, and σxψE,±(−x) = +±ψE,±(x), we have +ψ(x) = +� +r +� +dEψE,r(x)cE,r, +(17) +where cE,r denotes the annihilation operator for this +eigenstate. In particular, at the symmetric point, x = 0, +the eigenstate has the form +ψE,±(0) = φE,± +√ +2 (1, ±1)T , +(18) +where φE,± are two scalars. Using the eigenstate expan- +sion, we can obtain +|⟨ΨE|ψ†(0)ψ(0)|GS⟩|2 = +(19) +� +r +� +dE1dE2|φ∗ +E2,rφE1,r|2|⟨ΨE|c† +E2,rcE1,r|GS⟩|2. +To derive the above expression, it should be noted that +|ΨE⟩ is uniquely determined by a particle-hole excita- +tion above the ground state c† +E2,rcE1,r|GS⟩. +Given an +energy E, we have a constraint E = E2 + E1, i.e., +|⟨ΨE|c† +E2,rcE1,r|GS⟩|2 = δ(E − E1 − E2). Thus, the en- +tanglement entropy can be simplified as +SA = π2 +3 +� +t,t′ +� +r=± +� ∞ +0 +dE +� E +0 +dE′|φ∗ +E′,rφE−E′,r|2e−iE(t−t′), +which depends only on φE,r. +To calculate of single-particle eigenstate, we regularize +the Delta potential by a square potential barrier, vδ(x) = + +5 +limL→0 UL(x) with UL(x) = v +LΘ(L/2 − |x|). Θ(x) is the +step function [Θ(x > 0) = 1, Θ(x < 0) = 0]. At low +energy, we obtain φE,± as (see Supplemental Material +for a derivation) +φE,± = +1 +√ +2π cosh v +eiπ/4, +(20) +which is independent of the energy and the parity. Plug- +ging this into the entanglement entropy, it gives +SA = +� +t,t′ +� ∞ +0 +dE Ee−iE(t−t′) +6 cosh2 v += +log xA +3 cosh2 v , +(21) +and this shows that the effective central charge after mea- +surement is given by ceff = +1 +cosh2 v. +At v = 0, it cor- +rectly reduces to the free fermion case. For nontrivial v, +we see that it corresponds to a critical line separating +two phases, as shown in Fig. 1(b), where the effective +central charge is a continuous function of v. Note that +this is entirely different from the conventional Berezin- +skii–Kosterlitz–Thouless transition and the previous en- +tanglement transition in monitoring free fermion dynam- +ics [5, 11, 13, 14]. +To support our analytical calculation, we perform a +Gaussian state simulation [25] to calculate half-chain en- +tanglement entropy for different size. +Fig. 3(a) shows +the half-chain entanglement entropy at critical point, +t = 1, V = 0, W = 1, for different size L. It exhibits +a logarithmic behavior with an effective central charge +that deviates from one. In Fig. 3(b), we show the effec- +tive central charge extracted from fitting the numerical +data in black dots, and our prediction in an orange curve. +Note that the effective parameter v and the microscopic +parameter W are related by a constant factor. +Discussion and outlook.— We have performed thor- +ough analysis of the resulting state after weak measure- +ment W ≪ 1. When the measurement is strong, W ≫ 1, +it is expected to approach a projective measurement at +W → ∞. In this case, naively the Luttinger liquid theory +is not an appropriate starting point, since measurement +can insert high energy into the state. Remarkably, for +K < 1, in the strong-weak duality, the strong measure- +ment strength indicates γ → 0, our formula Eq. (12) +predicts vanishing entanglement entropy, which is con- +sistent with a product state. At K = 1, our result for +the critical point works well for large W, as shown in +Fig. 3. We leave a detailed study of strong measurement +to the future work. +In the end, we mention a few open questions. +At +∆ = 1, the theory has a larger SU(2) symmetry, though +the measurement explicitly breaks it. +It would be an +interesting future question to investigate the effect of +measurement at this special point. +Criticality under +measurement in higher dimensions is a natural exten- +sion [26]. Moreover, it would be interesting to investigate +the resulting state after measurement without postselec- +tion [16, 27]. +Finally, measurement effect is currently +under investigation in the context of holographic dual- +ity [28–33]. It would also be interesting to develop a holo- +graphic description for general measurement. We leave +such an investigation to future works. +Acknowledgement.— This work is supported in part by +the MOSTC under Grant No. 2021YFA1400100 and by +NSFC under Grant No. 11825404 (X.S. and H.Y.). 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Lett. 123, +170606 (2019). +SUPPLEMENTAL MATERIAL +A. +Twist operator in 1+1D free fermion theory and compact boson theory +In this section, we discuss the twist operator in 1+1D free fermion theory and compact boson theory, which is used +to calculate the entanglement entropy. Recall that the entanglement entropy can be evaluated by replica trick, e.g., +the entanglement entropy of subregion A reads +SA = −Tr[ρA log ρA] = lim +n→1 +1 +1 − nTr[ρn +A], +(S1) +where ρA = Tr ¯ +A[ρ] is the reduced density matrix of A. ¯A denotes the complement of subregion A. In the path integral +approach, we make n replicas of the original systems, a = 1, ..., n, and insert twist operators in the subregion A to +change the boundary condition. Then we can evaluate the path integral for n after which, a continuation of n to +a real number is taken followed by the replica limit n → 1. In the following two subsections, we discuss the twist +operators in free fermion theory and compact boson theory, respectively. +1. +Free fermion theory +Let ci,a, c† +i,a be the spinless fermion operator at site i and replica a. It satisfies {ci,a, c† +j,b} = δijδab The twist +operator for a subregion A is defined as +T † +Aci,aTA = +� +� +� +� +� +ci,a +i ∈ ¯A +ci,a+1 +i ∈ A, a < n +(−1)n+1ci,1 +i ∈ A, a = n +. +(S2) +The sign (−1)n+1 is due to the anticommutation of fermion operators [23]. This sign can be accounted by making the +transformation ci,a → (−1)aci,a. Then the twist operator TA is similar to a translation in the replica space. Therefore, +we can make a Fourier transform in the replica space +ci,k = +1 +√n +n +� +a=1 +ci,ae−i 2πka +n , +(S3) +with k denotes the replica momentum. In the replica momentum basis, the action of twist operator is diagonal, +namely, +T † +Aci,kTA = +� +ei 2πk +n ci,k +i ∈ A +ci,k +i ∈ ¯A . +(S4) + +7 +It is then not hard to deduce the twist operator: +TA = +(n−1)/2 +� +k=−(n−1)/2 +TA,k, +TA,k = ei 2πk +n QA,k, +QA,k = +� +i∈A +c† +i,kci,k. +(S5) +In free fermion theory, different replica momenta decouple, and they are described by a same theory. In this case, +we can omit the replica momentum index in QA,k. Then taking the continuum limit, we arrive at +QA,k = +� +x∈A +dxψ† +k(x)ψk(x). +(S6) +If we omit the dummy replica momentum index in free fermion theory, we arrive at (13) in the main text. +2. +Free compact boson theory +In 1+1D boson field theory, the entanglement entropy of an interval is related to the two-point correlation function +of branch-point twist operator [19, 36, 37]. In the following, we simply call the branch-point twist operator, denoted +as Tn(x, τ), the twist operator. Similar to (S3), we make a Fourier transformation in replica space which diagonalizes +the twist operator, +φk(x) = +1 +√n +n +� +a=1 +φa(x)e−i 2πka +n , +(S7) +where φa(x) is the boson operator at replica a. Denote the twist operator at replica momentum k as Tnk, which +satisfies +Tnk(u, 0)φk(v) = +� +ei 2πk +n φk(v), +u < v +φk(v), +u > v , +(S8) +the twist operator is Tn = � +k Tnk. Note that we use a tilde to denote the branch-point twist operator. It is a +local operator, which should be contrasted with the twist operator TA defined in (S4). +But they are related by +TA = Tnk(0, 0)Tnk(x0, 0) with A = {x|x ∈ (0, x0)}. Notice that in the Supplement Material, we use x0 instead of xA +to denote the length of subregion A. We will see this relation more explicitly in the following. +In the free boson theory, different replica momenta decouple. It is not hard to check that the twist operators are +given by +Tn(u, 0)T −1 +n +(v, 0) = +� +k +Tnk(u, 0)T −1 +nk (v, 0) = +� +k +e−i 2k +n (φk(u)−φk(v)), +(S9) +where n and k label the number of replica copies and replica momentum, respectively, and (u, v) is the interval of the +system we choose to calculate entanglement entropy. Here T −1 denotes the anti-twist operator. +We can relate the twist operator in the free fermion theory to the free boson theory by bosonization [17], +QA,k = +� x0 +0 +dx +� +− 1 +π ∇φk(x) +� += 1 +π (φk(x0) − φk(0)) . +(S10) +The interacting spinless fermion model considered in the main text (1) is +H = −t +� +i +(c† +ici+1 + h.c.) + V +� +i +(ni − 1 +2)(ni+1 − 1 +2), +(S11) +This model is described by the free compact boson theory, the Luttinger liquid theory. A nontrivial V ̸= 0 only +changes the radius of the compact boson field, or equivalently the Luttinger parameter [17], +K = +π +2(π − arccos ∆). +(S12) +The twist operator should accordingly be modified to be [19] +Tnk(u, 0)T −1 +nk (v, 0) = e−i 2k +n +1 +√ +K (φk(u)−φk(v)), +(S13) +which is (8) in the main text. We should note that (S13) only works for the entanglement entropy of a single interval. +For multi disjoint intervals, the compactness of boson field will couple different replica momenta, and more complicated +technique is needed [19]. + +8 +B. +Effective field theory in 1+0D with measurement +Without measurement, the bosonized action of Hamiltonian (1) reads +S[φ] = +1 +2πK +� +dx +� β +0 +dτ +� +˙φ2 + (∇φ)2� +. +(S14) +where th Luttinger parameter is related to the interaction through (2) [17]. For the measurement operator ˆ +M = +e−W � +i(−1)ini = e−W +� +dx cos (2kF x)n(x), where each site is x = nπ/2kF , n ∈ Z. Using the bosonization of fermion +density operator [16] +n(x) = − 1 +π ∇φ(x) + 1 +π cos [2(kF x − φ)], +(S15) +we have ˆ +M = e−W/2π +� +dx cos (2φ), where we keep the leading term and neglect higher-order terms like cos [2(nkF x + φ)] +with n ̸= 0. Here, notice that φ(x) = φ(x, τ = 0). +The density-density correlation for the post-selected state is ⟨n(x)n(0)⟩ = Tr[ ˆ +Mρ ˆ +M †n(x)n(0)]/Tr[ ˆ +Mρ ˆ +M †]. Since +⟨n(x)n(0)⟩ commutes with the measurement operator, the numerator equals Tr[ρ ˆ +M 2n(x)n(0)]. Using path integral +representation, the numerator can be expressed as +Tr[ρ ˆ +M 2n(x)n(0)] = +� +Dφe−Sn(x)n(0), +(S16) +with the action S +S = +� +dxdτ +� +1 +2πK [(∂τφ)2 + (∂xφ)2] − δ(τ)v cos 2φ +� +, +(S17) +and v = W/π. For other case where the operators do not commute with the measurement ˆ +M, we need to be careful +about the order of operators. +With the effective action (S14) we can integrate out the time direction and keep the “boundary” at τ = 0 [38]. Here +we consider a more general case with an additional mass term +1 +2πK +� +dxdτΩ2φ2. To integrate out the time direction, +we first solve the equation of motion ∂2 +τφ + ∂2 +xφ − Ω2φ = 0. In the momentum space, φ(p, τ) = +� +dxφ(p, τ)eipx, we +have ∂2 +τφ(p, τ) = (p2 + Ω2)φ(p, τ), which has the general solution +φ(p, τ) = φ(p, 0)(cosh ωpτ − coth ωpβ sinh ωpτ) + φ(p, β) sinh ωpτ +sinh ωpβ , +(S18) +where ω2 +p = p2 + Ω2. Then plugging it in to the action we have +S = +1 +πK +� +dpωp +2 +� +φ2 +c(p) tanh βωp +2 ++ φ2 +q(p) coth βωp +2 +� +, +(S19) +where φc(p) = [φ(p, β) + φ(p, 0)]/ +√ +2 and φq(p) = [φ(p, β) − φ(p, 0)]/ +√ +2. In the case of gapless fermions at zero +temperature, Ω = 0 and β = ∞, we have ωp = |p| and S = +1 +πK +� +dp |p| +2 +� +φ2 +c(p) + φ2 +q(p) +� += +1 +πK +� +dp|p|φ2(p), where we +apply the periodic boundary condition. With measurement, the final effective theory is +s[φ] = s0[φ] − v +� +dx cos [2φ], +s0[φ] = +1 +πK +� dq +2π |q||φ(q)|2. +(S20) +Also a remark is that the sign of v is unimportant, since a simple translation relates the two cases. +In Ref. [16], authors choose the convention that the field φ is related to the fermion density operator, and the dual +field θ, defined by [∂xφ, θ] = iπδ(x − x′), is related to the phase. Then the corresponding actions are +SG[φ] = +1 +2πK +� +dxdτ +� +(∇φ)2 + ˙φ2� +, +SG[θ] = K +2π +� +dxdτ +� +(∇θ)2 + ˙θ2� +. +(S21a) +After integrating out time direction, the actions are +sG[φ] = +1 +πK +� dq +2π |q||φ(q)|2, +sG[θ] = K +π +� dq +2π |q||θ(q)|2. +(S22a) + +9 +C. +Entanglement entropy transition induced by measurement +1. +Transformation of Different Fields with Strong Interaction +The interactions in dual theories have a strong-weak duality. Therefore, we can use the dual theory at strong +measurement strength or when the measurement is relevant, i.e., v ≫ 1. +In the following, we discuss the dual +transformation in more detail [21]. The initial action of the field φ is +s[φ] = +1 +πK +� dq +2π |q||φ(q)|2 − v +� +dx cos [2mφ], +(S23) +where m ∈ Z+ is an integer. For v ≫ 1, we know that the configuration of φ must be consisted of domain walls. +Therefore, we define h = ∂xφ = �n +i=1 eif(x − xi) with the constraint +� +∞ +−∞ dxf(x − xi) = 2π +2m. And ei = ±1 denotes +the (anti) domain wall. So, the Fourier transform of f(x) gives f(0) = +2π +2m, and a large v means that f(x) can +be approximate by δ-function, i.e., f(x) = +� dq +2πeikxf(q) ≈ +� dq +2πeikxf(0) = +2π +2mδ(x − xi). +Then φ(q) = +1 +iqh(q) = +f(0) +iq +�n +i=1 eie−iqxi. +Plugging the relation in action (S23), we arrive at +s[¯φ] = +1 +πK +� dq +2π |q||f(0)|2 +|q|2 +n +� +ij +eieje−iq(xi−xj) + nSd.w., +(S24) +Using Hubbard–Stratonovich (HS) transformation, we will get +Z = +� +n +� +{ei=±1} +e−nSd.w. +� +Dθe− 4m2K +16π +� +dq +2π |q||θ(q)|2 1 +n! +n +� +i=1 +� +dxiei �n +i=1 ei +� +dq +2π e−iqxiθ(q), +(S25) +where θ(−xi) = +� dq +2πe−iqxiθ(q). Defining γ = 2e−Sd.w. and summing all possible {ei}, we will get the final result +Z = +� +Dθ e− 4m2K +16π +� +dq +2π |q||θ(q)|2 � +n +γn 1 +n! +n +� +i=1 +�� +dxi cos θ(−xi) +� += +� +Dθ e− 4m2K +16π +� +dq +2π |q||θ(q)|2+γ +� +dx cos θ(x). +(S26) +If we only consider m = 1 and define g = e−Sd.w. where Sd.w. is attributed to the action of a single domain wall, we +get the consistent results with Ref. [16]. +Here, we briefly discuss the action of domain wall configuration. In Ref. [16], to introduce a UV-cutoff, the authors +add another term 1 +2 +� +dx(∇φ)2. For large v limit, rescaling x = v−1/2x′ and solving equation of motion for domain +wall configuration, they give +φd.w.(x) = π +2 + arctan[sinh(2v1/2x)]. +(S27) +For the contribution of domain wall configuration to (S23), there are two parts. One comes from the last term in +(S23) and the additional term, which gives ∆s = 4v1/2. Another part is attributed to the first term in (S23). With +the rescaling x = v−1/2x′, and q = v1/2q′, the first term becomes +1 +πK +� dq +2π |q||φd.w.(v−1/2q)|2av, +a = +1 +πK +� dq′ +2π |q′||φd.w.(q′)|2, +(S28) +where φd.w.(q′) is the Fourier transformation of φd.w.(x′) = π +2 + arctan[sinh(2x′)]. Therefore, Sd.w. = av + 4v1/2. +We remark on the commutation relation. +When we apply HS transformation in (S25), the commutation rela- +tion is [ei, θ(xj)] = iδij, which means [∂xφ(x), θ(xj)] = [�n +i=1 eif(x − xi), θ(xj)] = i 2π +2mδ(x − xj). Define ˜θ = mθ, +� +∂xφ(x), ˜θ(x′) +� += iδ(x − x′), ˜θ is the dual field of φ. Using ˜θ to rewrite the action, we will get the action +s[˜θ] = K +4π +� dq +2π |q||˜θ(q)|2 − γ +� +dx cos +˜θ(x) +m . +(S29) +While in the following, we mainly consider m = 1, we remark that the results will not rely on m up to the zero order +with γ = 0. + +10 +2. +Correlation Functions +In the following, we consider three different correlation functions, including +� +ei[θ(x)−θ(0)]� +, +� +ei[φ(x)−φ(0)]� +and +� +TnkT −1 +nk +� += +� +e−i +2k +√ +Kn [φ(x)−φ(0)]� +. +We first consider zero order results. For the phase correlation function, in the free case (v = 0) with the action +(S23), Ref. [16] shows that +� +ei[θ(x0)−θ(0)]� += e− π +4K +� +dq +2π |q||T0,x0(q)|2 ∼ x +− +1 +2K +0 +. +(S30) +Here, with abuse of notation, we define +T0,x0(x) = +� +1, +0 < x < x0 +0, +x < 0, x > x0 +. +(S31) +It should be clear from the context that this is different from the twist operator TA. +In the strong measurement limit (v ≫ 1), we take the action (S29) with γ = 0. In Ref. [16] the authors have shown +that the result should be +� +ei[θ(x0)−θ(0)]� +∼ x +− 1 +K +0 +. +We derive the correlation function above in another way which takes care of the order of operators, and perform +an explicit calculation. We first consider s[φ + πT0,x0]. With similar approximations, we have +φ + πT0,x0 = 1 +iq f(q) +n +� +i=1 +(eieiqxi + e0eiqx0 + en+1eiqxn+1), +(S32) +where e0 = 1, en+1 = −1, x0 = 0, xn+1 = x0. Plugging it in the action (S23) with v → ∞, γ = 0, we have +s0 = π +K +� dq +2π +1 +|q| +n +� +i,j=1 +eieje−iq(xi−xj) ++ π +K +� dq +2π +1 +|q| +� +� +n +� +j=1 +1 · eje−iq(0−xj) + +n +� +j=1 +(−1) · eje−iq(x0−xj) + (−1)e−iq(0−x0) + h.c. + 2 +� +� . +(S33) +It means that the function T0,x0(x) behaves like two domain walls at x = 0, x0 with opposite signs. Then the partition +function will be +Z = +� +n +� +{ei=±1} +e−nSd.w. +� +Dθe− K +4π +� +dq +2π |q||θ(q)|2e +π +2K +� +dq +2π +1 +|q| (2 cos qx0−2) +1 +n! +n +� +i=1 +� +dxiei �n +i=1 ei +� +dq +2π e−iqxiθ(q)− π +2K +� +dq +2π +1 +|q| +�n +i=1 ei(e−iq(0−xi)−e−iq(x0−xi)+h.c.) += +� +Dθe− K +4π +� +dq +2π |q||θ(q)|2+γ +� +dx cos [θ(x)+∆x0(x)]e +π +2K +� +dq +2π +1 +|q| (2 cos qx0−2), +(S34) +where ∆x0(x) = i π +2K +� dq +2π +1 +|q|(eiqx − eiq(x−x0) + h.c.). Then directly we get +� +ei[θ(x0)−θ(0)]� += +� +Dφe− 1 +2 (s[φ]+s[φ+πT0,x0]) +� +Dφe−s[φ] += e +π +2K +� +dq +2π +1 +|q| (2 cos qx0−2) ∼ x +− 1 +K +0 +, +(S35) +which is consistent with Ref. [16]. +Now we consider the correlation function of vertex operators and twist operators. For the free case (v = 0) with +the action (S23), we have +� +ei[φ(x0)−φ(0)]� += eG(x0) ∼ x +− K +2 +0 +, +(S36) +where G(x0) = − πK +2π ln x0. Similarly, we can calculate the correlation function of twist operators, which will give +entanglement entropy. For the free case (v = 0) with the action (S23), we have +� +e−i +2k +√ +Kn [φ(x0)−φ(0)]� += e +4k2 +Kn2 G(x0) ∼ x +− 2k2 +n2 +0 +, +(S37) + +11 +which is consistent with Ref. [23] and gives entanglement entropy S = 1 +3 log L. +In the following, we derive the results of the correlation functions of vertex operators and twist operators for strong +measurement strength by using the similar method of (S34). For the correlation function of vertex operators, we have +� +ei[φ(x0)−φ(0)]� += +� +e− +� +dq +2π qφ(q)T0,x0(−q)� += +� +exp +� +− +� dq +2π +1 +iq +n +� +i=1 +eie−iqxiπqT0,x0(−q) +�� +, +(S38) +where the expectation value is according to the action (S29). Then the numerator is +Z = +� +n +� +{ei=±1} +e−nSd.w. +� +Dθe− K +4π +� +dq +2π |q||θ(q)|2 1 +n! +n +� +i=1 +� +dxiei �n +i=1 ei +� +dq +2π e−iqxiθ(q)+i +� +dq +2π πT0,x0(−q) �n +i=1 eie−iqxi += +� +Dθ e− K +4π +� +dq +2π |q||θ(q)|2+γ +� +dx cos [θ(x)+∆x0(x)], +(S39) +where ∆x0(x) = +� dq +2ππT0,x0(−q)e−iqxi = πT0,x0(x). Therefore, up to the zeroth order γ = 0, the denominator equals +the numerator, which means +� +ei[φ(x0)−φ(0)]� += 1. There is another way to double-check the result above. We can +rewrite the exponent of numerator as follows. +−s′ = − +1 +πK +� dq +2π |q||φ(q)|2 + v +� +dx cos [2φ] − +� dq +2π qφ(q)T0,x0(−q) += − +1 +πK +� dq +2π |q|(φ(q) + πKq +2|q| T0,x0(q))(φ(−q) + πKq +2|q| T0,x0(−q)) + v +� +dx cos [2φ] + +1 +πK +� dq +2π |q|π2K2 +4 +|T0,x0(q)|2, +(S40) +where (− +� dq +2πqφ(q)T0,x0(−q))∗ = − +� dq +2πqφ(−q)T0,x0(q) = +� dq +2πqφ(q)T0,x0(−q). +Then there are two terms con- +tributing to the final result. +One is − 1 +πK +� dq +2π|q||φ(q) + πKq +2|q| T0,x0(q)|2, which is similar to (S35) and gives +e +π +K +� +dq +2π +1 +|q| (2 cos qx0−2)·( Kq +2|q|) +2 +∼ x +− K +2 +0 +. Another term is +1 +πK +� dq +2π|q| π2K2 +4 +|T0,x0(q)|2 which is similar to (S30) and gives +∼ x +K +2 +0 . Therefore, two terms’ contributions will cancel out and lead to +� +ei[φ(x0)−φ(0)]� += 1. +For twist operators, the correlation function is similar to vertex operators case. The only difference in (S39) is that +there is a prefactor in front of ∆x0(x), which will not change the zeroth order results and gives a trivial correlation +function +� +TnkT −1 +nk +� += 1. +Here, we briefly discuss the strong measurement results of several correlation functions. The nontrivial result of +phase correlation function is because of the interaction of two domain walls in T0,x0(x). But for vertex operator, the +additional term is linear with φ, which doesn’t contribute the quadratic terms like phase correlation. So the result is +trivial. +Finally, we consider the correlation functions of density operators. For the free case (v = 0) we have +⟨∇φ(x0)∇φ(0)⟩ = −∇2G(x0) = K +2 x−2 +0 . +(S41) +For the strong measurement case, Ref. [16] shows that ⟨∇φ(x0)∇φ(0)⟩ ∼ γ2x +− 2 +K +0 +. It is worth mentioning that this +density correlation function will rely on different m. +According to the discussion above, for the strong measurement case, we always get vanishing results at the zeroth +order. So now we consider the first order of γ. To simplify the problem, we only consider m = 1 case (For general m +the results will be similar.) In the following, we will only use the method of (S34) and (S39). +3. +Phase Correlation Function +Firstly, we consider phase correlation function. To consider the domain wall contribution, we include another term +to (S23), +s[φ] = 1 +πK +� dq +2π |q||φ(q)|2 − v +� +dx cos [2φ] + 1 +2 +� +dx(∇φ)2 += 1 +πK +� dq +2π |q||φ(q)|2 − v +1 +2 +�� +dx′ cos [2φ] − 1 +2 +� +dx′(∇′φ)2 +� +, +(S42) + +12 +where x′ = v +1 +2 x. Similar to (S35) we have +1 +2(s[φ] + s[φ + πT0,x0]) = +1 +πK +� dq +2π |q||φ(q)|2 + 1 +K +� dq +2π |q|φ(q)T0,x0(−q) + π +2K +� dq +2π |q||T0,x0(q)|2 +− 1 +2v +1 +2 +�� +dx′ cos [2(φ + πT0,x0)] − 1 +2 +� +dx′(∇′(φ + πT0,x0))2 + +� +dx′ cos [2φ] − 1 +2 +� +dx′(∇′φ)2 +� +. +(S43) +Therefore, there are two additional terms in the second line (S43). T0,x0 is unimportant for cos term, because we can +safely ignore the step function. But we need to consider the contribution of it to quadratic terms. For the strong +measurement strength, we need to change the configuration of φ to minimize two cos terms, which means for first +order we take φ as two domain walls (beside two domain walls of T0,x0). Now we have +1 +2(s[φ] + s[φ + πT0,x0]) = +1 +2πK +� dq +2π |q||φ(q)|2 + +1 +2πK +� dq +2π |q||φ(q) + πT0,x0(q)|2 +− v +1 +2 +�� +dx′ cos [2φ] − 1 +2 +� +dx′(∇′φ)2 +� ++ π +2 +� +dx∇φ∇T0,x0 + +˜C +2 , +(S44) +where ˜C = 1 +2 +� +dx(π∇T0,x0)2. Then we consider (S34) with the additional terms +Z =e− ˜ +C +2 � +n +� +{ei=±1} +e−nSd.w. +� +Dθe− K +4π +� +dq +2π |q||θ(q)|2e +π +2K +� +dq +2π +1 +|q| (2 cos qx0−2) 1 +n! +n +� +i=1 +� +dxi +ei �n +i=1 ei +� +dq +2π eiqxiθ(q)− π +2K +� +dq +2π +1 +|q| +�n +i=1 ei(e−iq(0−xi)−e−iq(x0−xi)+h.c.)− π2 +2 +� +dq +2π q2T0,x0(−q)· 1 +iq +�n +i=1 eie−iqxi +=e− ˜ +C +2 +� +Dθe− K +4π +� +dq +2π |q||θ(q)|2+γ +� +dx cos [θ(x)+∆x0(x)+ ˜∆x0(x)]e +π +2K +� +dq +2π +1 +|q| (2 cos qx0−2), +(S45) +where +∆x0(xi) =i π +2K +� dq +2π +1 +|q|(e−iq(0−xi) − e−iq(x0−xi) + h.c.) = i π +K +� dq +2π +1 +|q| [cos qxi − cos q(x0 − xi)] , +(S46) +˜∆x0(xi) =π2 +2 +� dq +2π qT0,x0(−q)e−iqxi = iπ2 +2 ∂xiT0,x0(xi). +(S47) +Then we double-check the results above and give some comments. (i) The zero order term satisfies +e +π +2K +� +dq +2π +1 +|q| (2 cos qx0−2) = e− π +2K +� +dq +2π |q||T0,x0(q)|2 ∼ x +− 2 +K +0 +. +(S48) +(ii) The additional ∆x0(x) is from the crossing term +− +1 +2πK +� dq +2π |q| [φ(q)πT0,x0(−q) + φ(−q)πT0,x0(q)] = − 1 +K +� dq +2π |q|T0,x0(−q) · 1 +iq +n +� +i=1 +eie−iqxif(q) +=i +n +� +i=1 +ei +π +K +� dq +2π +|q| +q e−iqxiT0,x0(−q) = i +n +� +i=1 +ei +iπ +K +� dq +2π +1 +|q|e−iqxi(1 − eiqx) = i +n +� +i=1 +ei∆x0(xi), +(S49) +where f(q) = π. Simplifying ∆x0(x) we have +∆x0(x) = i π +K +� dq +2π +1 +|q| +� +2 sin2 q(xi − x0) +2 +− 2 sin2 qxi +2 +� += i 1 +K ln +���� +xi − x0 +xi +����. +(S50) +(iii) For ˜∆x0(xi) we have +˜∆x0(xi) = π2 +2 +� dq +2π q 1 +−iq (eiq·0 − eiqx0)e−iqxi = iπ2 +2 [δ(xi) − δ(xi − x0)] , +(S51) +which is consistent with (S47). + +13 +We calculate the phase correlation function as follows, +� +ei[θ(x0)−θ(0)]� += e− ˜ +C +2 e +π +2K +� +dq +2π +1 +|q| (2 cos qx0−2) +� +Dθe− K +4π +� +dq +2π |q||θ(q)|2+γ +� +dx cos [θ(x)+∆x0(x)+ ˜∆x0(x)] +� +Dθe− K +4π +� +dq +2π |q||θ(q)|2+γ +� +dx cos [θ(x)] +, +(S52) +where the first two parts above give the zero order result e− ˜ +C +2 x +− 1 +K +0 +. In the following, we evaluate the path integral. +Expanding the cos term gives +num = +� +Dθe− K +4π +� +dq +2π |q||θ(q)|2+γ +� +dx cos [θ(x)+∆x0(x)+ ˜∆x0(x)] += +� +Dθe− K +4π +� +dq +2π |q||θ(q)|2 +∞ +� +n=0 +1 +n! +� +γ +� +dx cos [θ(x) + ∆x0(x) + ˜∆x0(x)] +�n +. +(S53) +For the denominator we can just set ∆x0(x) = ˜∆x0(x) = 0. As we know, the expectation value of “vertex” operators +is nontrivial only when the total charge is zero, which means the first order term is n = 2. With the form of “vertex” +operator we have γ2 term +1 +2γ2 +� +dx1dx2 +� +cos [θ(x1) + ∆x0(x1) + ˜∆x0(x1)] cos [θ(x2) + ∆x0(x2) + ˜∆x0(x2)] +� +=1 +2γ2 × 1 +4 +� +dx1dx2 +� +ei[∆x0(x1)+ ˜∆x0(x1)−∆x0(x2)− ˜∆x0(x2)] � +eiθ(x1)e−iθ(x2)� ++ h.c. +� +, +(S54) +where +� +eiθ(x1)e−iθ(x2)� += e +1 +2 G(x1−x2)+ 1 +2 G(x2−x1) = e− 1 +2π +4π +K ln |x1−x2|. +(S55) +So the γ2 term is +1 +2γ2 × 1 +4 +� +dx1dx2 +� +ei[∆x0(x1)+ ˜∆x0(x1)−∆x0(x2)− ˜∆x0(x2)]− 2 +K ln |x1−x2| + h.c. +� += 1 +2γ2I1. +(S56) +Similarly, γ2 term in denominator is +1 +2γ2 × 1 +4 +� +dx1dx2 +� +e− 2 +K ln |x1−x2| + h.c. +� += 1 +2γ2I2. +(S57) +Therefore, we get the phase correlation function +� +ei[θ(x0)−θ(0)]� +≈ e− ˜ +C +2 x +− 1 +K +0 +(1 + γ2 +2 I1)(1 + γ2 +2 I2)−1 ≈ e− ˜ +C +2 x +− 1 +K +0 +(1 + γ2 +2 (I1 − I2)). +(S58) +Now we evaluate the integral I1 − I2. Plugging ∆x0(x) and ˜∆x0(x) in I1 we have +I1 =2 × 1 +4 +� +dx1dx2e +� +− 1 +K ln +��� +(x1−x0)x2 +x1(x2−x0) +���− π2 +2 (δ(x1)−δ(x1−x0)−δ(x2)+δ(x2−x0))− 2 +K ln |x1−x2| +� +=1 +2 +� +dx1dx2e− π2 +2 (δ(x1)−δ(x1−x0)−δ(x2)+δ(x2−x0)) +����� +(x1 − x0)x2 +x1(x2 − x0) +���� (x1 − x2)2 +�− 1 +K +≈1 +2 +� +dx1dx2 +����� +(x1 − x0)x2 +x1(x2 − x0) +���� (x1 − x2)2 +�− 1 +K +=1 +2 +� +dx1dx2e− 1 +K (ln |x1−x0|+ln |x2−0|−ln |x1−0|−ln |x2−x0|+2 ln |x1−x2|). +(S59) +From the above result we find that the contribution of ˜∆x0(x) is only on some isolated points with the measure that +is dimension zero. If we apply some UV cutoff that require different domain walls cannot be too close to each other +the exponent will always be zero. Besides, there is a comment on (S59). From the last line we can understand the +meaning of the contribution. Because we consider the sub-leading term, we assume there are two domain walls with +opposite signs. And there are also two domain walls because of T0,x0(x). Therefore, we actually have two domain + +14 +walls at x = 0, x1 with positive charge and two domain walls at x = x0, x2 with negative charge. There are C2 +4 = 6 +pairs, where the interaction between x = 0 and x = x0 is zero order result in (S58) and the other five pairs are in +(S59). The pairs with opposite (same) charge will have factor +1(−1). And the factor 1 +2 is because in (S35) numerator +the pair (x1, x2) contributes two actions and other pairs only contribute s[φ + πT0,x0]. +Finally, we have +I1 − I2 =1 +2 +� +dx1dx2 +����� +(x1 − x0)x2 +x1(x2 − x0) +���� +− 1 +K +− 1 +� +|x1 − x2|− 2 +K +=1 +2 +� +d˜x1d˜x2 +����� +(˜x1 − 1)˜x2 +˜x1(˜x2 − 1) +���� +− 1 +K +− 1 +� +|˜x1 − ˜x2|− 2 +K · x +2− 2 +K +0 += ∆I1 · x +2− 2 +K +0 +, +(S60) +where ˜xi = xi/x0. Therefore, we get the phase correlation function is +� +ei[θ(x0)−θ(0)]� +≈ e− ˜ +C +2 x +− 1 +K +0 +(1 + γ2 +2 ∆I1 · x +2− 2 +K +0 +), +(S61) +where we have obtained subleading correction. +4. +Vertex Operator Correlation and Entanglement Entropy +Now we consider the correlation function +� +ei[φ(x)−φ(0)]� +. With (S39) as numerator, we can expand it like (S53), +num = +� +Dθe− K +4π +� +dq +2π |q||θ(q)|2 +∞ +� +n=0 +1 +n! +� +γ +� +dx cos [θ(x) + ∆x0(x)] +�n +, +(S62) +where ∆x0(x) = πT0,x0(x). The first nontrivial term is γ2 term +1 +2γ2 +� +dx1dx2 ⟨cos [θ(x1) + πT0,x0(x1)] cos [θ(x2) + πT0,x0(x2)]⟩ +=γ2 +2 × 1 +4 +� +dx1dx2eiπ[T0,x0(x1)−T0,x0(x2)] � +eiθ(x1)e−iθ(x2)� ++ h.c. = 1 +2γ2I3, +(S63) +where +� +eiθ(x1)e−iθ(x2)� += e− 2 +K ln |x1−x2|. And the denominator is the same as (S57). So, +� +ei[φ(x0)−φ(0)]� +≈ (1 + γ2 +2 I3)(1 + γ2 +2 I2)−1 ≈ (1 + γ2 +2 (I3 − I2)). +(S64) +Here is a remark about the result above. It seems that (S63) is similar to (S56). But actually, their final result is +not the same. For (S56) the function ∆x0(x) represent the interaction of two domain walls which contribute a real +factor. For (S63), here the function ∆x0(x) play the role of phase factor that makes the integrand oscillate. The +key difference is a sign factor |q| = q · sgn(q) in (S49). Comparing (S39) and (S49), we can find that although both +∆x0(x) is from the linear term to φ. But for phase correlation case (S49) the linear term is +� dq +2π|q|φ(q)πT0,x0(−q) +with the factor |q|, but for the vertex correlation (S39) the linear term is +� dq +2πqφ(q)T0,x0(−q) with the factor q. It is +the difference sgn(q) that makes the difference. +We can calculate the integral I3 and I2. +I3 =1 +4 +� +dx1dx2eiπ[T0,x0(x1)−T0,x0(x2)]− 2 +K ln |x1−x2| + h.c +=1 +2 +� +dx1dx2e− 2 +K ln |x1−x2|eiπΘ, +(S65) +where Θ = T0,x0(x1) − T0,x0(x2). Then +∆I3 = I3 − I2 = 2 × 1 +2 +� +F (Θ=1) +dx1dx2e− 2 +K ln |x1−x2| × (−1), +(S66) + +15 +𝑥! +𝑥" +𝑦" +𝑦! +0 +𝑥# +𝑥# +𝑒$% +𝑒$% +𝑒&$% +𝑒&$% +I +I +II +II +(a) +𝑥! +𝑥" +𝑦" +𝑦! +0 +𝑥# +𝑥# +𝑒$%&!" +𝑒$%&!" +𝑒'$%&!" +𝑒'$%&!" +I +I +II +II +(b) +Supplementary Figure S1. (a) Diagram of Integral for integral (S67). (b) Diagram of Integral for integral (S75) +where F(Θ = 1) means the region that (x1, x2) satisfies Θ = 1. In Fig. S1 (a), the region F(Θ = 1) is labeled by e±iπ. +If we change the coordinate (x1, x2) to (y1, y2) = ( x1−x2 +√ +2 , x1+x2 +√ +2 ), it gives +∆I3 = − +� +F (Θ=1) +dy1dy2e− 2 +K ln | +√ +2y1| = −2 +� +I+II +dy1dy2e− 2 +K ln | +√ +2y1|, +(S67) +where the region I and II is also shown in Fig. S1 (a). So, we have +∆I3,I = −2 +� x0/ +√ +2 +0 +dy1 · 4y1e− 2 +K ln +√ +2y1 = −8 +� x0/ +√ +2 +0 +dy1( +√ +2y1)− 2 +K · y1 = +−4 +2 − 2 +K +x +2− 2 +K +0 +, +(S68) +∆I3,II = −2 +� √ +2x0 +− +√ +2x0 +dy2 +� +∞ +x0/ +√ +2 +dy1e− 2 +K ln +√ +2y1 = −4 +√ +2x0 +� +∞ +x0/ +√ +2 +dy1( +√ +2y1)− 2 +K = +4 +1 − 2 +K +x +2− 2 +K +0 +, +(S69) +where we assume K < 1. Therefore, we have +∆I3 = ∆I3,I + ∆I3,II = +� +4 +1 − 2 +K +− +4 +2 − 2 +K +� +x +2− 2 +K +0 +, +(S70) +which means the correlation function is +� +ei[φ(x0)−φ(0)]� +≈ 1 + γ2 +2 · +� +4 +1 − 2 +K +− +4 +2 − 2 +K +� +x +2− 2 +K +0 +. +(S71) +For entanglement entropy, we consider the correlation function +� +e−iak +n[φ(x0)−φ(0)]� +with ak +n = +2k +√ +Kn. Therefore, we +just need to rescale the function T0,x0 with ak +n. The numerator is +num = +� +Dθe− K +4π +� +dq +2π |q||θ(q)|2 +∞ +� +n=0 +1 +n! +� +γ +� +dx cos [θ(x) + ak +nπT0,x0(x)] +�n +. +(S72) +If we define +I4 = 1 +4 +� +dx1dx2eiπak +n[T0,x0(x1)−T0,x0(x2)]− 2 +K ln |x1−x2| + h.c., +(S73) +the correlation function is +� +e−i +2k +√ +Kn [φ(x0)−φ(0)]� +≈ (1 + γ2 +2 I4)(1 + γ2 +2 I2)−1 ≈ (1 + γ2 +2 (I4 − I2)). +(S74) + +16 +Similar to the results above, +∆I4 =1 +4 +� +dx1dx2e− 2 +K ln |x1−x2| � +eiπak +n[T0,x0(x1)−T0,x0(x2)] − 1 +� ++ h.c +=1 +2 +� +I+II +dx1dx2e− 2 +K ln |x1−x2| � +eiπak +n[T0,x0(x1)−T0,x0(x2)] − 1 +� ++ h.c +=[cos (πak +n) − 1] +� +I+II +dx1dx2e− 2 +K ln |x1−x2| = [cos (πak +n) − 1] +� +I+II +dy1dy2e− 2 +K ln +√ +2y1. +(S75) +Using the result above with Fig. S1 (b), we have +∆I4 = [cos (πak +n) − 1] +� +−2 +1 − 2 +K +− +−2 +2 − 2 +K +� +x +2− 2 +K +0 +. +(S76) +Therefore, we have correlation function +� +e−i +2k +√ +Kn [φ(x0)−φ(0)]� +≈ 1 + γ2 +2 [cos (πak +n) − 1] +� +−2 +1 − 2 +K +− +−2 +2 − 2 +K +� +x +2− 2 +K +0 +. +(S77) +Finally, according to Section A 1, we have +ln Zk = ln +� +1 + γ2 +2 [cos (πak +n) − 1] +� +−2 +1 − 2 +K +− +−2 +2 − 2 +K +� +x +2− 2 +K +0 +� +≈ [cos (πak +n) − 1]γ2 +2 +� +−2 +1 − 2 +K +− +−2 +2 − 2 +K +� +x +2− 2 +K +0 +, +(S78) +so the entanglement entropy is +S = lim +n→1 +1 +1 − n +� +k +ln Zk = − lim +n→1 +∂ +∂n +� +k +[cos (πak +n) − 1]γ2 +2 +� +−2 +1 − 2 +K +− +−2 +2 − 2 +K +� +x +2− 2 +K +0 += γ2f(K)x +2− 2 +K +0 +, +(S79) +where f(K) = +� +1 +1− 2 +K − +1 +2− 2 +K +� +limn→1 +∂ +∂n +� +k[cos (πak +n) − 1] = +� +1 +1− 2 +K − +1 +2− 2 +K +� +[ +π +√ +K cot +π +√ +K − 1]. +D. +Entanglement entropy at the critical point +For K = 1 exactly non-interacting case with measurement, we can rotate the x−t plane such that the measurement +effectively act on x = 0 for all time. After the rotation, the system is time translation invariant, so we can transform this +problem to solving the time-independent Schrodinger equation. Now, the system is still a non-interacting free fermion +system, but has a mass term at x = 0. To calculate the entanglement entropy, we need to calculate the correlation +function +� +Q2 +A +� += − ⟨QAQ ¯ +A⟩ [24]. Here ¯A is the complement of A, and the above equation is valid for a pure state. +In the reference, the operator is defined as a time-independent operator (t = 0) that QA = +� +A dDrψ†(r)ψ(r). Now +with rotation we have QA = +� +tA dte−iHtψ†(0)ψ(0)eiHt with fixed location x = 0 but integration of “time interval” tA. +Then we have +⟨QAQ ¯ +A⟩ = +� +A +� +¯ +A +dtdt′Tr +� +ρ(0)eiH(t−t′)ρ(0)e−iH(t−t′)e−βH� += +� +dE +� +dE′ +� +A +dt +� +¯ +A +dt′ ⟨E|ρ(0)|E′⟩ eiE′(t−t′) ⟨E′|ρ(0)|E⟩ e−iE(t−t′)e−βE += +� +dE +� +dE′ +� +A +dt +� +¯ +A +dt′ |⟨E|ρ(0)|E′⟩|2 ei(E′−E)(t−t′)e−βE, +(S80) +where |E⟩ , |E′⟩ are many-body eigenstates and E, E′ are total energy. In the second quantization formalism, ψ(r) = +� +de ⟨r|e⟩ ce = +� +deψe(r)ce, where |e⟩ is the eigenstate of a single particle. At zero temperature β → ∞, the many-body +ground state with energy E0 can be represented by |GS⟩ = ⊗ei<0 |1⟩ ⊗ei>0 |0⟩. Therefore, +⟨E|ρ(r)|E′⟩ = ⟨E| ψ†(r)ψ(r) |E′⟩ = +� +de1e2ψ∗ +e1(r)ψe2(r) ⟨E| ψ†(e1)ψ(e2) |E′⟩ = ψ∗ +e1(r)ψe2(r), +(S81) + +17 +where e1, e2 satisfy that when we annihilate a state |e1⟩ and create a state |e2⟩, |E′⟩ will change to |E⟩. Finally, at +zero temperature we have +⟨QAQ ¯ +A⟩ = +� +dE′ +� +A +dt +� +¯ +A +dt′ |⟨GS|ρ(0)|E′⟩|2 ei(E′−E0)(t−t′) += +� +A +dt +� +¯ +A +dt′ +� +e1<0,e2>0 +de1de2 |ψe1(0)|2 |ψe2(0)|2 ei(e1−e2)(t−t′). +(S82) +We can first consider the free case that |ψe(0)|2 = +1 +2π, then the correlation is +⟨QAQ ¯ +A⟩ = +� +A +dt +� +¯ +A +dt′ +� +d(e1 − e2)ei(e1−e2)(t−t′) +(2π)2 +� +de2 = +� +A +dt +� +¯ +A +dt′ +� d(∆e) +2π +s(∆e)ei∆e(t−t′), +(S83) +where s(∆e) = +� de2 +2π = |∆e| +2π , which can be considered that for a fixed ∆e, e2 can range from e = −∆e to e = 0. +Similar to the Ref. [24], it gives ⟨QAQ ¯ +A⟩ = − 1 +π2 log x0 and entanglement entropy S = 1 +3 log x0. In the following, +we will show that for Luttinger liquid with K = 1 and measurement strength v, the entanglement entropy is just +renormalized by a prefactor which only relies on measurement strength. +As mentioned above, after rotation, we will have an effective Lagrangian +L = ψ†(i∂t + i∂xσz + vδ(x)σx)ψ, +(S84) +where ψ = (ψL ψR)T represents the left and right mover. Then the effective Schrodinger equation reads +i∂tψ = (−i∂xσz + vδ(x)σx)ψ = H′ψ, +(S85) +where ψ = (ψ1 ψ2)T and the sign of v is unimportant. The system has the following symmetries. (i) Denoting the +operator T as reflecting symmetry operator, i.e., x → −x with conjugation of σx, we have [T, H′] = 0. With T 2 = 1, +we have T = ±1 which means Tψ = ±ψ. (ii) Considering x → −x, ψ2 → −ψ2, then we have H′ → −H′ which means +particle-hole symmetry. +Because the δ−function is hard to deal with, we consider the limit that there is a square wall potential m in the +range [− L +2 , L +2 ] and mL = v. Because the Hamiltonian is time-independent, we have ψ(t) = ψe−iEt and H′ψ = Eψ. +In the following, E is single particle energy for state ψ. For x ∈ (−∞, − L +2 ] ∪ [ L +2 , ∞), the wave function is a plane +wave, while for x ∈ [− L +2 , L +2 ], the eigen-equation reads +� +−i∂xψ1 + mψ2 = Eψ1 +(−∂2 +x − E2 + m2)ψ2 = 0 +. +(S86) +Therefore, the corresponding wave function is +ψ(x < −L +2 ) = +� +Aeikx +Be−ikx +� +ψ(−L +2 < x < L +2 ) = +� +� +κ + E +m +Ceiκx + −κ + E +m +De−iκx +Ceiκx + De−iκx +� +� +ψ(x > L +2 ) = +� +A′eikx +B′e−ikx +� +, +(S87) +where k = E and κ2 = E2 − m2. With the boundary condition, we can solve this problem. To simplify the problem, +with the symmetry [T, H] = 0, we consider the wave function that is also the eigenstate of the operator T. For +T = 1, we have Aeik −L +2 += B′e−ik L +2 , so A = B′ and similarly B = A′. Also for the middle region κ+E +m Ceiκ −L +2 + +−κ+E +m +De−iκ −L +2 = Ceiκ L +2 + De−iκ L +2 , so C = −κ+E +m +D, D = κ+E +m C which are consistent with E2−κ2 +m2 += 1. Therefore, the +wave function with T = 1 can be simplified as +ψ(x < −L +2 ) = +� +Aeikx +Be−ikx +� +ψ(−L +2 < x < L +2 ) = +� +Deiκx + Ce−iκx +Ceiκx + De−iκx +� +ψ(x > L +2 ) = +� +Beikx +Ae−ikx +� +. +(S88) +Similarly, for the odd case with T = −1, we have A = −B′, B = −A′, C = − −κ+E +m +D, D = − κ+E +m C, and the wave +function is +ψ(x < −L +2 ) = +� +Aeikx +Be−ikx +� +ψ(−L +2 < x < L +2 ) = +� +−Deiκx − Ce−iκx +Ceiκx + De−iκx +� +ψ(x > L +2 ) = +� +− Beikx +− Ae−ikx +� +. +(S89) + +18 +For both cases, we just need to consider one boundary condition e.g., x = − L +2 . Combining boundary condition and +the relation of factor C and D we can represent all factors with D, +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +A =eik L +2 +� +±e−iκ L +2 + −κ + E +m +eiκ L +2 +� +D +B =e−ik L +2 +� +eiκ L +2 ± −κ + E +m +e−iκ L +2 +� +D +C = ± −κ + E +m +D, +(S90) +where ± is corresponding to T = ±1. Besides, we can notice that A = ±eikLB. +Now, to evaluate the expectation value ⟨QAQ ¯ +A⟩, we only need to calculate the factor D. +It can be fixed by +normalization condition ⟨k′|k⟩ = δ(k − k′). Assuming the total system length is 2R, we have the inner product +⟨E′|E⟩ = +� − L +2 +−R +� +A∗ +k′Akei(k−k′)x + B∗ +k′Bke−i(k−k′)x� +dx ++ +� +L +2 +− L +2 +� +(D∗ +κ′e−˜κ′x + C∗ +κ′e˜κ′x)(Dκe−˜κx + Cκe˜κx) + (D∗ +κ′e˜κ′x + C∗ +κ′e−˜κ′x)(Dκe˜κx + Cκe−˜κx) +� +dx ++ +� R +L +2 +� +B∗ +k′Bkei(k−k′)x + A∗ +k′Ake−i(k−k′)x� +dx += 2 +� +A∗ +k′Ak +ei(k−k′) −L +2 − ei(k−k′)(−R) +i(k − k′) ++ B∗ +k′Bk +e−i(k−k′) −L +2 − e−i(k−k′)(−R) +−i(k − k′) +� ++ 2 +� +(D∗ +κ′Dκ + C∗ +κ′Cκ)e−(˜κ′+˜κ) L +2 − e(˜κ′+˜κ) L +2 +−(˜κ′ + ˜κ) ++ (D∗ +κ′Cκ + C∗ +κ′Dκ)e−(˜κ′−˜κ) L +2 − e(˜κ′−˜κ) L +2 +−(˜κ′ − ˜κ) +� +, +(S91) +where ˜κ2 = m2 − E2. Because we will take the limit of δ−function that m → ∞, L → 0 and mL = v, if we only +consider low-energy states that E << m, κ is a imaginary number but ˜κ is corresponding real number. With the +limit, we can use approximation ˜κ = +√ +m2 − E2 ≈ m − E2 +2m, which means ˜κ · L ≈ v − E2L +2m = v + O( 1 +m2 ). For the first +square brackets of (S91), with the relation of factor Ak, Bk and Dk we have +First term = 2B∗ +k′Bk +i(k − k′) +� +ei(k−k′)R − ei(k−k′)(L−R)� +=2(e−˜κ′ L +2 ± i˜κ′ + E′ +m +e˜κ′ L +2 )(e−˜κ L +2 ± −i˜κ + E +m +e˜κ L +2 )ei(k−k′)(R− L +2 ) − e−i(k−k′)(R− L +2 ) +i(k − k′) +D∗ +κ′Dκ. +(S92) +Plugging the approximation ˜κ · L ≈ v − E2L +2m = v(1 − +k2 +2m2 ), i˜κ+E +m +≈ i(1 − k2 +2m + k +m) in the above equation and only +keeping up to O( 1 +m) term, we get +First term =2 +� +(e−v + ev + (e− v +2 ∓ ie +v +2 )2 vk′2 +4m + (±1 − iev)k′ +m + (e− v +2 ± ie +v +2 )2 vk2 +4m ++(±1 + iev) k +m +� 2 sin (k − k′)(R − L +2 ) +(k − k′) +D∗ +κ′Dκ + O( 1 +m2 ). +(S93) +For the second square brackets of (S91), with the relation of factor Ak, Bk and Dκ we have +Second term =2D∗ +κ′Dκ +� +(1 + i˜κ′ + E′ +m +−i˜κ + E +m +)e−(˜κ′+˜κ) L +2 − e(˜κ′+˜κ) L +2 +−(˜κ′ + ˜κ) +± (i˜κ′ + E′ +m ++ −i˜κ + E +m +)e−(˜κ′−˜κ) L +2 − e(˜κ′−˜κ) L +2 +−(˜κ′ − ˜κ) +� +=2D∗ +κ′Dκ +�2(ev − e−v) +2m ++ rmi(k − k′)(ev − e−v) +4m2 +± v(k + k′) +m2 +� ++ O( 1 +m3 ). +(S94) +If we take the limit m → ∞ and only keep zero order, (S91) is +⟨E′|E⟩ = 4(ev + e−v)sin (k − k′)(R − L +2 ) +(k − k′) +D∗ +κ′Dκ + O( 1 +m) +R→∞ += +4π(ev + e−v)D∗ +κ′Dκδ(k − k′). +(S95) + +19 +Therefore, we can set D = +1 +√ +4π · +1 +(ev+e−v)1/2 , and the corresponding factor B is +B = e−ik L +2 +� +eiκ L +2 ± −κ + E +m +e−iκ L +2 +� +D ≈ (e−v/2 ∓ iev/2)D = +1 +√ +4π +e−v/2 ∓ iev/2 +(ev + e−v)1/2 = +1 +√ +4π e∓iθ, +(S96) +where tan θ = ev. There are some remarks about the results above. (i) For any value of v, we always have |B| = +1 +√ +4π, +which is consistent because when we take the limit L → 0, almost all the contribution for inner product comes from +the free plane wave solution. Because of two components of the wave function, the prefactor must be +1 +√ +2 · +1 +√ +2π = +1 +√ +4π. +(ii) For different v, the factor B has different phases, but the key difference is D which relies on v. With the limit +L → 0, if we set v = 0, then D = +1 +√ +4π +√ +2. Recalling the wave function has the form Ce−˜κx + De˜κx and C ≈ iD, so +the total amplitude is +√ +2D = +1 +√ +4π = B, which means the wave function is continuous. But for v ̸= 0 especially for +large v, we have |D| ≪ |B|, which means it is discontinuous. +Finally, for the entanglement entropy of a general v, with the amplitude of the wave function +|ψe(0)|2 =(D∗ +κ′e−˜κ′x + C∗ +κ′e˜κ′x)(Dκe−˜κx + Cκe˜κx) + (D∗ +κ′e˜κ′x + C∗ +κ′e−˜κ′x)(Dκe˜κx + Cκe−˜κx)|x=0 +=2|C + D|2 ≈ 4|D|2 = +1 +2π cosh v , +(S97) +we arrive at +⟨QAQ ¯ +A⟩ = +� +A +dt +� +¯ +A +dt′ +� +e1<0,e2>0 +de1de2 |ψe1(0)|2 |ψe2(0)|2 ei(e1−e2)(t−t′) = +1 +cosh2 v [⟨QAQ ¯ +A⟩]v=0 , +(S98) +which means the entanglement entropy is S = +1 +cosh2 v · 1 +3 log x0. +E. +Numerical calculation of the entanglement entropy +1. +Free fermion calculation with Julia +Here we consider the Luttinger liquid model with K = 1, which corresponds to a free fermion system with periodic +boundary condition +H = +� +i +(c† +ici+1 − cic† +i+1), +(S99) +where c† +i (ci) denotes the fermion creation (annihilation) operator at site i. To calculate entanglement entropy after +measurement [25], we first diagonalize the Hamiltonian (S99) and calculate its correlation matrix Γ +Γ := +� +Γc†c +Γc†c† +Γcc +Γcc† +� +, +(S100) +where Γc†c +ij += +� +c† +icj +� +and Γc†c† +ij += +� +c† +ic† +j +� +. Then we apply imaginary time evolution with measurement Hamiltonian +Hm, which has the same effect as measurement operator ˆ +M. For different system sizes L and strength of measurement +W that is represented by the imaginary time τ of evolution, we calculate the entanglement entropy of half of the +system. Then we obtain the relation of entanglement entropy and L, W. +We consider the system size L ∈ [2, 200] and the measurement W ∈ [0, 5]. For different measurement strength we fit +the relation between entanglement entropy and system size L with a+b log L, the prefactor b is related to the effective +central charge, i.e. b = ceff +3 . In Fig. S2 (a), we plot the relation between effective central charge and measurement +strength W. +The dot points are the numerical results we get from Julia, and the orange line is the analytical +result +1 +cosh2 1.7W . +Here we take measurement Hamiltonian Hm = Diag(0, 1, 0, 1, ..., 0, −1, 0, −1, ...) and imaginary +time evolution eW Hm. +It is equivalent to the measurement Diag(−1/2, 1/2, −1/2, 1/2, ..., 1/2, −1/2, 1/2, −1/2, ...) +considered in the main text (3) up to an identity operator. At W = 0, we have central charge c = 1 with prefactor +b = 1/3. For W ̸= 0 numerical results and analytical results are consistent with each other very well, leading to the +effective central charge ceff = cosh−2 1.7W. Fig. S2 (c) shows the coefficient of determination R2 when we interpolate +the entanglement entropy S with log L. 1 − R2 ≈ 10−9 means the fitting is accurate. + +20 +0 +1 +2 +3 +4 +5 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +W +ceff +Δ=0 +(a) +0 +1 +2 +3 +4 +5 +0 +2.×10-9 +4.×10-9 +6.×10-9 +8.×10-9 +1.×10-8 +W +1-R2 +(b) +Supplementary Figure S2. +(a) The effective central charge as a function of the measurement strength W. (b) Fitting coefficient +1 − R2 at different measurement strength. +0 +1 +2 +3 +4 +5 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +W +ceff +Δ=0 +(a) +0 +1 +2 +3 +4 +0 +5.×10-7 +1.×10-6 +1.5×10-6 +2.×10-6 +W +1-R2 +(b) +Supplementary Figure S3. +The results for ∆ = 0 from DMRG calculation. (a) The effective central charge as a function of +the measurement strength W. (b) Fitting coefficient 1 − R2 at different measurement strength. +2. +Density matrix renormalization group calculation of the XXZ model +Using Jordan-Wigner transformation, the Hamiltonian (1) becomes (recall that we set t = 1) +H = +� +i +(Sx +i Sx +i+1 + Sy +i Sy +i+1 + ∆Sz +i Sz +i+1), +(S101) +with Sj +α, α = x, y, z spin 1/2 operators at site i. We first calculate the ground state wave function with Matrix +Product State (MPS), then apply the measurement by considering the imaginary time evolution of the ground state +wave function. Here we take Hm as evolution Hamiltonian. We consider the system size L ∈ [10, 200] and the strength +of measurement W ∈ [0, 5]. +At K = 1(∆ = 0) we can plot similar results as free fermion in Fig. S3. The results are consistent with the free +fermion case. Here we define H′ +m = � +i(−1)iSz +i , then the effective central charge is ceff = cosh−2 1.7W. If we take the +(a) +(b) +Supplementary Figure S4. Effective central charge ceff = 3b as a function of ∆ for different measurement strength: (a) ∆ < 0, +(b) ∆ > 0. + +△ +8.0- +a.0- +-0'4 +S.0- +0.0 +0.0 +8.0=W +a.0=w +2.0 +.0=W +cel +S.0=W +0.7 +0.0=W +2.1V +0.0 +S.0 +0'4 +a.0 +8.0 +0.7 +S.0- +8.0=W +0.0 +a.0=w +S.0 +←.0=W +0'4 +ceu +M=0'S +a.0 +0.0=W +8.0 +0.721 +0.5 +1.0 +1.5 +2.0 +-15 +-10 +-5 +0 +W 12 +factor a +Δ=0.8 +(a) +0.5 +1.0 +1.5 +2.0 +-15 +-10 +-5 +0 +W 12 +factor b +Δ=0.8 +(b) +Supplementary Figure S5. Prefactor fitting with quadratic polynomials: (a) prefactor a with fitting results log a ∼ −4.14W + +1.40 +√ +W − 0.19, (b) prefactor b with fitting results log b ∼ −4.23W + 1.95 +√ +W + 0.70. +L=16 +L=32 +L=64 +L=128 +L=200 +-4 +-2 +0 +2 +4 +0.0 +0.5 +1.0 +1.5 +2.0 +(Δ-Δc)logL +Sent(Δ,L)-Sent(Δc,L) +Supplementary Figure S6. Half-chain entanglement entropy as a function of ∆ for different sizes L. The measurement strength +is W = 1.0. +measurement Hamiltonian Hm = � +i Sz +2i+1. The only difference is that the fitting function of effective central charge +is ceff = cosh−2 0.85W. +For K > 1(∆ < 0) we consider two cases and fit the entanglement entropy with S = a+b log L and plot the effective +central charge ceff = 3b with respect to ∆. This is shown in Fig. S4 (a). For W = 0 and −0.9 < ∆ < 0, the central +charge is ceff = 1. For ∆ ∈ (−0.8, −0.2) and W ≤ 0.6, the prefactor b ≈ 1/3 which means the central charge remains +approximately ceff ≈ 1. Because the effective central charge exactly at ∆ = 0 is a function of W, the deviation of the +central charge from ceff = 1 near ∆ = 0 is due to the finite size effect. These results verify that the measurement is +irrelevant for K > 1. +For K < 1(∆ > 0), we similarly plot the fitting effective central charge ceff = 3b with respect to different ∆. In +Fig. S4 (b), at W = 0 we always have b = 1/3, showing the central charge ceff = 1, while for W > 0, we see that +the effective central charge decreases to zero, indicating that the log L behavior of the entanglement entropy breaks +down. To characterize the entanglement behavior, we explore the algebraic correction, and identify the power, as +shown in Fig. 2 in the main text. Besides, we also fit the prefactors a and b in the power law fitting S = a+b/Lc with +respect to measurement strength W. With (S28) and (S79), we know a and b exponentially decay with exponents +which are a quadratic polynomial of +√ +W. In Fig. S5, we plot log a and log b with respect to W 1/2 and fit them with +quadratic polynomials, which show perfect fitting results. Moreover, two quadratic polynomials have coefficients of +the quadratic term ( +√ +W)2. +The distinct behaviors of entanglement entropy for ∆ > 0 and ∆ < 0 clearly reveal an entanglement transition at +∆c = 0. Unlike conventional transition, the scaling dimension of the tuning parameter is zero at the critical point. +This indicates 1/ν = 0. We plot the data collapse to demonstrate this critical exponent. In Fig. S6, we plot half-chain +entanglement entropy as a function of ∆ for different sizes L. All data collapse onto a smooth function when the +argument is chosen to be (∆ − ∆c) log L. + diff --git a/mNFIT4oBgHgl3EQfsyss/content/tmp_files/load_file.txt b/mNFIT4oBgHgl3EQfsyss/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..330211d0a2581db7e342f485f569188065e4f367 --- /dev/null +++ b/mNFIT4oBgHgl3EQfsyss/content/tmp_files/load_file.txt @@ -0,0 +1,1042 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf,len=1041 +page_content='New critical states induced by measurement Xinyu Sun,1 Hong Yao,1, ∗ and Shao-Kai Jian2, † 1Institute for Advanced Study, Tsinghua University, Beijing 100084, China 2Department of Physics and Engineering Physics, Tulane University, New Orleans, Louisiana, 70118, USA (Dated: January 30, 2023) Finding new critical states of matter is an important subject in modern many-body physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Here we study the effect of measurement and postselection on the critical ground state of a Luttinger liquid theory and show that it can lead to qualitatively new critical states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Depending on the Luttinger parameter K, the effect of measurement is irrelevant (relevant) at K > 1 (K < 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' We reveal that this causes an entanglement transition between two phases, one with logarithmic entanglement entropy for a subregion (K > 1), and the other an algebraic entanglement entropy (K < 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' At the critical point K = 1, the measurement is marginal, and we find new critical states whose entanglement entropy exhibits a logarithmic behavior with a continuous effective central charge as a function of measurement strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' We also performed numerical density matrix renormalization group and fermionic Gaussian state simulations to support our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' We believe that our work provides a promising and feasible route to experimentally realize new critical states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='—Critical state underlies various inter- esting physics in phase transition, hydrodynamics, and even quantum gravity according to AdS/CFT correspon- dence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' While conformal field theory (CFT) describes a huge class of critical states, it is of impact to find critical states beyond the description of CFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Recently, stud- ies on quantum trajectory with local measurement re- veal a critical point separating two quantum phases with distinct entanglement structures [1–5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' It is natural to study the effect of measurement in CFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' The effect of local projective measurement in CFT is well described by boundary CFT [6–10], where (a region of) the critical state is projected onto a Cardy state [8] with zero spatial entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Apart from the projective measurement in boundary CFT, less is known about general measure- ment, such as local weak measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' In this paper, we are interested in the effect of weak measurement on the ground state of a Luttinger liquid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Different from the constant monitoring in free fermion systems studied previously [11–15], we consider perform- ing weak measurement to the critical ground state of the Luttinger liquid without time evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' [16] re- ports a transition as a function of Luttinger parameter between two phases with algebraic correlation functions of distinct power laws after weak measurement, neverthe- less, the entanglement property after weak measurement is left unanswered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Since measurement can change entan- glement radically, the scaling of entanglement entropy in these two phases is not immediately obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' We study in depth the entanglement properties of the (spinless) Luttinger liquid theory after weak measure- ment and postselection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' A schematic representation of the model and the phase diagram are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' 1(a) and (b), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Depending on the Luttinger pa- rameter K, we find that for K > 1 the measurement is irrelevant, and the state retains logarithmic entangle- ment entropy with a central charge c = 1 for a subregion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' (a) A schematic plot of the spinless fermion chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' t and V denote the hopping and the nearest-neighbor interac- tion, respectively, in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' (1) W is the measurement strength in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' (b) Phase diagram of the Luttinger liquid after weak measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' K and v ∝ W denote the Luttinger pa- rameter and effective measurement strength, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' For K > 1, the measurement is irrelevant, and the entanglement entropy of a subregion A with length xA satisfies a log-law with central charge c = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' For K < 1, the measurement is relevant, and changes the entanglement entropy from a log- law to an area law with a subleading algebraic correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' At K = 1, there is a continuous critical line, at which the mea- surement is marginal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' The entanglement entropy satisfies a log-law with an effective central charge ceff = 1/ cosh2 v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' While for K < 1, the measurement becomes relevant, and we find that the state exhibits an area law with a (subleading) algebraic entanglement entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' The alge- braic power-law is obtained in the dual theory by tak- ing advantage of a strong-weak duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' We further per- form a large-scale density matrix renormalization group arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='11337v1 [quant-ph] 26 Jan 2023 V t MV 1 SA log xA 3cosh2 V SA = So + xA 2/K+2 logxA 3 0 1/2 1 K2 (DMRG) calculation of an equivalent XXZ model to sup- port our findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' In particular, the (subleading) alge- braic entropy is consistent with our prediction over a wide range of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Furthermore, at the critical point K = 1, the measure- ment is marginal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' We identify a critical line for different measurement strength, on which the entanglement en- tropy exhibits a logarithmic behavior with an effective central charge continuously changing with the measure- ment strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Since at K = 1 the theory is equivalent to a free fermion model, the effective central charge is cal- culated by a spacetime rotation of the low-energy Dirac fermion theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' We also perform a fermionic Gaussian state simulation to calculate the half-chain entanglement entropy, and the effective central charge extracted from the data verifies our prediction for a wide range of mea- surement strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='— We consider spinless fermions in a 1D chain with the Hamiltonian H = −t � i (c† ici+1 + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=') + V � i (ni − 1 2)(ni+1 − 1 2),(1) where c† i (ci) denotes the fermion creation (annihilation) operator at site i, and ni = c† ici is the density operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' t and V denote the hopping and the interaction between nearest neighbor sites, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' We define ∆ = V/t, and set t = 1 without loss of generality (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=', energy is measured in unit of t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' It is well known [17] that for |∆| < 1 the ground state is described by a free compact boson with the Luttinger parameter [see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' (5) below] K = π 2(π − arccos ∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' (2) We now consider a weak measurement to the ground state of such a Luttinger liquid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' For the model we consid- ered, there is one qubit (given by the occupation number of a spinless fermion) at each site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' The measurement at site i is described by the following Kraus operator {e−W Pi, √ 1 − e−2W Pi}, where W ≥ 0 is the measure- ment strength, and P2i−1 = 1 − n2i−1, P2i = n2i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' A physical implementation of this Kraus operator is given in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' After each measurement, the first outcome is post-selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Because measurement operators at dif- ferent sites commute, we arrive at the total measurement operator (up to an unimportant constant) M = e−W � i(−1)ini, (3) and the post-selected density matrix ρm = MρM † Tr[MρM †], (4) where ρ = limβ→∞ e−βH is an unnormalized projection onto the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' We are interested in the entanglement properties of ρm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' For weak measurement strength W ≪ 1, we expect the low energy bosonized theory is still valid, and a nontrivial critical state can be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' To this end, using the path integral representation (see Supplemental Material for the derivation), the post-selected density matrix is proportional to ⟨˜φ(x)|MρM †|˜φ′(x)⟩ = � b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Dφe−S, with the action S = � dτdx � 1 2πK [(∂τφ)2 + (∂xφ)2] + δ(τ)v cos 2φ � ,(5) where φ is the boson field, and the boundary condition φ(x, 0−) = ˜φ(x), φ(x, 0+) = ˜φ′(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Here, |˜φ⟩ is the state with field configuration given by ˜φ, while φ(x, τ) is the compact boson field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Here v ∝ W, and the Delta func- tion takes care of the measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' The last term should be considered as a sum of two terms at an infinitesimal positive and negative τ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' It is worth not- ing that while the model in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' (1) is at half-filling, our main results is still valid for general filling (having Fermi momentum kF ) with the effective measurement operator having a 2kF momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' On the other hand, at strong measurement strength W ≫ 1, the post-selected state is close to a product state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' In the following, we will first study the weak mea- surement case in depth, which leads to qualitatively new critical states, and defer the strong measurement case to the discussion in the end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Measurement induced transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='— To characterize the critical state at weak measurement, we are particularly interested in its entanglement entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' It can be calcu- lated via replica trick as SA = −Tr[ρA log ρA] = lim n→1 1 1 − nTr[ρn A], (6) where ρA = Tr ¯ A[ρm] is the reduced density matrix in the subregion A (here ¯A denotes the complement of A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' This amounts to replicate the theory ci → ci,a, a = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=', n is the replica index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Because the mea- surement operator is bilinear in the fermionic operator, different replica momenta will decouple after we make a Fourier transform w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' the replica index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' More explicitly, the measurement operator in the replicated theory is M = e−W � i,a(−1)ini,a (with abuse of nota- tion, we use the same symbol M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' The replica Fourier transform is defined by ci,k = 1 √n � a ci,aei 2πka n , with k the replica momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' In the replica momentum basis, M = � n−1 2 k=− n−1 2 Mk, and Mk = e−W � i(−1)ini,k can be straightforwardly bosonized to get δ(τ)v cos 2φk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Combining the quadratic term originated from the pre- measured Hamiltonian, we arrive at the following decou- pled action for each replica momentum k (see Supple- mental Material for details), sk = 1 πK � dq 2π |q||φk(q)|2 + v � dx cos 2φk(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' (7) Here we have further integrated over the time direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' 3 q denotes the momentum from Fourier transform φk(q) = � dxφk(x)eiqx, and φk(x) = φk(x, τ = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' The entanglement entropy SA of compact bosons boils down to the expectation value of the twist operator [18, 19] TA = � n−1 2 k=− n−1 2 TA,k with TA,k = e−i k n √ 4 K (φk(xA)−φk(0)), (8) where we assume that the interval A = {x|x ∈ (0, xA)}, and take the replica limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Before we present results of the entanglement entropy, we discuss the renormalization group (RG) flow at low energies to determine the phase diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' (7), the momentum k is a dummy index because different replica momenta decouple;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' therefore, the RG equation for v is the same as that of a single replica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' It is given by [16, 20, 21] dv dl = (1 − K)v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' (9) On the other hand, K is exactly marginal ( dK dl = 0) be- cause the first term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' (7) is non-analytical that does not receive a correction from RG process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Heuristically, the measurement term is only present at τ = 0, so it cannot renormalize K that is present in 1+1D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Combin- ing these two facts, the flow of v is simple: it is relevant (irrelevant) for K < 1 (K > 0), and marginal at K = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' For K > 1, v is irrelevant, and in the lowest energy scale, the entanglement entropy of A after weak measure- ment reduces to that of a free boson with central charge c = 1, SA = 1 3 log xA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' (10) While for K < 1, v is relevant, we will show in the follow- ing that its entanglement property changes qualitatively after weak measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Thus, there is an entanglement transition at K = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' To study the entanglement entropy for K < 1, because v is relevant, it is easier to work with the dual field [17, 21] θ, defined via [∂xφ(x), θ(x′)] = iπδ(x−x′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' We give a de- tailed derivation of the dual field theory in Supplemental Material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' The dual action reads sk = K 4π � dq 2π |q||θk(q)|2 + γ � dx cos θk(x), (11) where γ = 2e−av−4√v, (a is a constant, whose expression is given in Supplemental Material), and k denotes the replica momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' When v flows to a large number, it means γ is small, so we can perform a perturbation cal- culation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Thanks to decoupling between different replica momenta, ⟨TA⟩ = � k⟨TA,k⟩, where ⟨·⟩ = Tr[·ρ⊗n m ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' At the leading order, we arrive at (see Supplemental Mate- rial for detail) ⟨TA,k⟩ = exp � γ2fk(K)x − 2 K +2 A � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Finally, W=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='0 W=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='6 10 20 50 100 200 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='5 L Sent Δ=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='6 (a) W=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='0 50 100 150 200 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='2 L Sent Δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='6 (b) W=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='6 50 100 150 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='275 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='285 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='295 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='305 L Sent Δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='6 (c) W=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='57 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='63 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='69 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='4 K power (d) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' The half-chain entanglement entropy as a function of different sizes at (a) ∆ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='6, (b,c) ∆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' (a) The blue (orange) curve is given by measurement strength W = 0 (W = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' The same slope indicates the effective central charges are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' (b) shows the data at measurement strength W = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' It shows a logarithmic function with central charge c = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' (c) shows the data at measurement strength W = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' It shows an algebraic function with power 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' (d) The algebraic power as a function of different K for K < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' The measurement strength is W = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' The black dots show the power fitted by numerical data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' The orange curve is our prediction 2/K − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' The numerical calculation is obtained with bond dimension χ = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' taking the replica limit, the entanglement entropy for K < 1 reads SA = γ2 � S0 + f(K)x − 2 K +2 A � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' (12) Here, the expressions for fk(K) and f(K) are given in Supplemental Material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' The first term is a non-universal constant piece that accounts for the leading area-law contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Therefore, the entanglement entropy for K < 1 shows an area law with a subleading power-law behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' With Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' (10) at K > 1 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' (12) at K < 1, we can conclude a measurement-induced entanglement tran- sition between two phases with a logarithmic entangle- ment and a (subleading) algebraic entanglement, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' To further support our conclusion, we perform DMRG calculation [22] of entanglement entropy for ρm in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Details of the simulation can be found in Supplemental Material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' For K > 1, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' 2(a) shows the entanglement entropy for W = 0 (blue) and W = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='6 (orange).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' The same slope 1/3 indicates that the central charge of both cases is c = 1, and the measurement is irrelevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' For K < 1, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' 2 shows that the entangle- ment entropy w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='t system sizes is a logarithmic function for W = 0 (b), an algebraic function for W = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='6 (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' It 4 demonstrates that the measurement is relevant for K < 1 and changes the entanglement entropy from a logarithm law without measurement to an area law with sublead- ing algebraic correction with measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' 2 (d), the black dots represent the powers of the entanglement entropy fitted from numerical data for W > 0 and the or- ange curve is our prediction −2/K + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' We can see that they are consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' We further perform data collapse to demonstrate this measurement-induced transition in the Supplemental Material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Critical point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='— We now discuss the entanglement en- tropy at the critical point K = 1, where the measurement parameter v is marginal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' In this case, the interaction V vanishes, and the model reduces to a free fermion the- ory with measurement (3) at τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' For a free fermion theory, the twist operator is simply given by [23, 24] TA = � k ei 2πk n QA, QA = � x∈A dxψ†(x)ψ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' (13) As we are interested in the long-wavelength behavior of entanglement entropy, we have taken a continuum limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Here ψ(x) = ψ(x, 0) denotes the fermion operator in the continuum limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Because different replica momenta de- couple in free fermion theory, we omit the dummy index k in QA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Performing the product over replica momentum, the entanglement entropy is SA = π2 3 ⟨Q2 A⟩ [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Now consider the effect of measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' The measure- ment induces a scattering between left and right movers, and it is not hard to deduce the low-energy theory L = ¯ψ(/∂ + vδ(τ))ψ, (14) where ¯ψ = ψ†γ0, /∂ ≡ ∂µγµ is the Dirac operator, with γ0 = σx, γ1 = −σy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' For simplicity, we scale the coor- dinate to set Fermi velocity to one, and v ∝ W is de- termined by the measurement strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Since the Dirac operator has a rotational symmetry in the Euclidean sig- nature, we can make a space-time rotation such that the measurement operator becomes a defect located at x = 0 [16, 20], with the corresponding Hamiltonian H′ = −i∂xσz + vδ(x)σx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' (15) Moreover, the entanglement entropy becomes the corre- lation function in the time direction SA = π2 3 � t,t′⟨ψ†(0, t)ψ(0, t)ψ†(0, t′)ψ(0, t′)⟩ (16) = π2 3 � t,t′ � dµ|⟨ΨE|ψ†(0)ψ(0)|GS⟩|2e−iE(t−t′), where � t,t′ = � xA 0 dt � xA 0 dt′, |GS⟩ denotes the ground state whose energy is set to be zero without loss of gen- erality, and |ΨE⟩ is a complete eigenbasis with energy E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' The integral over � dµ denotes the integral over this basis that includes energy E and parity r, as we will discuss W=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='0 5 10 50 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='25 L Sent Δ=0 (a) 0 1 2 3 4 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='0 W ceff Δ=0 (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' (a) The half-chain entanglement entropy at the crit- ical point as a function of different system sizes L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' The pa- rameter is chosen to be t = 1, V = 0, W = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' The black dots represent the numerical data, and the orange line is a fitting with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='04 log L + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='0435.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' (b) The effective central charge as a function of different measurement strength W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' The black dots represent effective central charge that is extracted from fitting the numerical data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' The orange curve is our prediction ceff = 1/ cosh2(αW), where α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='7 is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' In the second step, we used the Heisen- berg evolution ψ(0, t) = eiH′tψ(0)e−iH′t, and inserted a complete basis � dµ|ΨE⟩⟨ΨE| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' It is worth noting a reflecting symmetry for Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' (15), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=', x → −x with conjugation of σx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' In the single- particle eigenstate basis of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' (15), ψE,r(x), that satis- fies H′ψE,r(x) = EψE,r(x), r = ±1, and σxψE,±(−x) = ±ψE,±(x), we have ψ(x) = � r � dEψE,r(x)cE,r, (17) where cE,r denotes the annihilation operator for this eigenstate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' In particular, at the symmetric point, x = 0, the eigenstate has the form ψE,±(0) = φE,± √ 2 (1, ±1)T , (18) where φE,± are two scalars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Using the eigenstate expan- sion, we can obtain |⟨ΨE|ψ†(0)ψ(0)|GS⟩|2 = (19) � r � dE1dE2|φ∗ E2,rφE1,r|2|⟨ΨE|c† E2,rcE1,r|GS⟩|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' To derive the above expression, it should be noted that |ΨE⟩ is uniquely determined by a particle-hole excita- tion above the ground state c† E2,rcE1,r|GS⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Given an energy E, we have a constraint E = E2 + E1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=', |⟨ΨE|c† E2,rcE1,r|GS⟩|2 = δ(E − E1 − E2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Thus, the en- tanglement entropy can be simplified as SA = π2 3 � t,t′ � r=± � ∞ 0 dE � E 0 dE′|φ∗ E′,rφE−E′,r|2e−iE(t−t′), which depends only on φE,r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' To calculate of single-particle eigenstate, we regularize the Delta potential by a square potential barrier, vδ(x) = 5 limL→0 UL(x) with UL(x) = v LΘ(L/2 − |x|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Θ(x) is the step function [Θ(x > 0) = 1, Θ(x < 0) = 0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' At low energy, we obtain φE,± as (see Supplemental Material for a derivation) φE,± = 1 √ 2π cosh v eiπ/4, (20) which is independent of the energy and the parity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Plug- ging this into the entanglement entropy, it gives SA = � t,t′ � ∞ 0 dE Ee−iE(t−t′) 6 cosh2 v = log xA 3 cosh2 v , (21) and this shows that the effective central charge after mea- surement is given by ceff = 1 cosh2 v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' At v = 0, it cor- rectly reduces to the free fermion case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' For nontrivial v, we see that it corresponds to a critical line separating two phases, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' 1(b), where the effective central charge is a continuous function of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Note that this is entirely different from the conventional Berezin- skii–Kosterlitz–Thouless transition and the previous en- tanglement transition in monitoring free fermion dynam- ics [5, 11, 13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' To support our analytical calculation, we perform a Gaussian state simulation [25] to calculate half-chain en- tanglement entropy for different size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' 3(a) shows the half-chain entanglement entropy at critical point, t = 1, V = 0, W = 1, for different size L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' It exhibits a logarithmic behavior with an effective central charge that deviates from one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' 3(b), we show the effec- tive central charge extracted from fitting the numerical data in black dots, and our prediction in an orange curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Note that the effective parameter v and the microscopic parameter W are related by a constant factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Discussion and outlook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='— We have performed thor- ough analysis of the resulting state after weak measure- ment W ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' When the measurement is strong, W ≫ 1, it is expected to approach a projective measurement at W → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' In this case, naively the Luttinger liquid theory is not an appropriate starting point, since measurement can insert high energy into the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Remarkably, for K < 1, in the strong-weak duality, the strong measure- ment strength indicates γ → 0, our formula Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' (12) predicts vanishing entanglement entropy, which is con- sistent with a product state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' At K = 1, our result for the critical point works well for large W, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' We leave a detailed study of strong measurement to the future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' In the end, we mention a few open questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' At ∆ = 1, the theory has a larger SU(2) symmetry, though the measurement explicitly breaks it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' It would be an interesting future question to investigate the effect of measurement at this special point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Criticality under measurement in higher dimensions is a natural exten- sion [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Moreover, it would be interesting to investigate the resulting state after measurement without postselec- tion [16, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Finally, measurement effect is currently under investigation in the context of holographic dual- ity [28–33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' It would also be interesting to develop a holo- graphic description for general measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' We leave such an investigation to future works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Acknowledgement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='— This work is supported in part by the MOSTC under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' 2021YFA1400100 and by NSFC under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' 11825404 (X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='- K.' metadata={'source': 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MATERIAL A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Twist operator in 1+1D free fermion theory and compact boson theory In this section, we discuss the twist operator in 1+1D free fermion theory and compact boson theory, which is used to calculate the entanglement entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Recall that the entanglement entropy can be evaluated by replica trick, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=', the entanglement entropy of subregion A reads SA = −Tr[ρA log ρA] = lim n→1 1 1 − nTr[ρn A], (S1) where ρA = Tr ¯ A[ρ] is the reduced density matrix of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' ¯A denotes the complement of subregion A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' In the path integral approach, we make n replicas of the original systems, a = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=', n, and insert twist operators in the subregion A to change the boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Then we can evaluate the path integral for n after which, a continuation of n to a real number is taken followed by the replica limit n → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' In the following two subsections, we discuss the twist operators in free fermion theory and compact boson theory, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Free fermion theory Let ci,a, c† i,a be the spinless fermion operator at site i and replica a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' It satisfies {ci,a, c† j,b} = δijδab The twist operator for a subregion A is defined as T † Aci,aTA = � � � � � ci,a i ∈ ¯A ci,a+1 i ∈ A, a < n (−1)n+1ci,1 i ∈ A, a = n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' (S2) The sign (−1)n+1 is due to the anticommutation of fermion operators [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' This sign can be accounted by making the transformation ci,a → (−1)aci,a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Then the twist operator TA is similar to a translation in the replica space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Therefore, we can make a Fourier transform in the replica space ci,k = 1 √n n � a=1 ci,ae−i 2πka n , (S3) with k denotes the replica momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' In the replica momentum basis, the action of twist operator is diagonal, namely, T † Aci,kTA = � ei 2πk n ci,k i ∈ A ci,k i ∈ ¯A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' (S4) 7 It is then not hard to deduce the twist operator: TA = (n−1)/2 � k=−(n−1)/2 TA,k, TA,k = ei 2πk n QA,k, QA,k = � i∈A c† i,kci,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' (S5) In free fermion theory, different replica momenta decouple, and they are described by a same theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' In this case, we can omit the replica momentum index in QA,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Then taking the continuum limit, we arrive at QA,k = � x∈A dxψ† k(x)ψk(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' (S6) If we omit the dummy replica momentum index in free fermion theory, we arrive at (13) in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Free compact boson theory In 1+1D boson field theory, the entanglement entropy of an interval is related to the two-point correlation function of branch-point twist operator [19, 36, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' In the following, we simply call the branch-point twist operator, denoted as Tn(x, τ), the twist operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Similar to (S3), we make a Fourier transformation in replica space which diagonalizes the twist operator, φk(x) = 1 √n n � a=1 φa(x)e−i 2πka n , (S7) where φa(x) is the boson operator at replica a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Denote the twist operator at replica momentum k as Tnk, which satisfies Tnk(u, 0)φk(v) = � ei 2πk n φk(v), u < v φk(v), u > v , (S8) the twist operator is Tn = � k Tnk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Note that we use a tilde to denote the branch-point twist operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' It is a local operator, which should be contrasted with the twist operator TA defined in (S4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' But they are related by TA = Tnk(0, 0)Tnk(x0, 0) with A = {x|x ∈ (0, x0)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Notice that in the Supplement Material, we use x0 instead of xA to denote the length of subregion A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' We will see this relation more explicitly in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' In the free boson theory, different replica momenta decouple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' It is not hard to check that the twist operators are given by Tn(u, 0)T −1 n (v, 0) = � k Tnk(u, 0)T −1 nk (v, 0) = � k e−i 2k n (φk(u)−φk(v)), (S9) where n and k label the number of replica copies and replica momentum, respectively, and (u, v) is the interval of the system we choose to calculate entanglement entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Here T −1 denotes the anti-twist operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' We can relate the twist operator in the free fermion theory to the free boson theory by bosonization [17], QA,k = � x0 0 dx � − 1 π ∇φk(x) � = 1 π (φk(x0) − φk(0)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' (S10) The interacting spinless fermion model considered in the main text (1) is H = −t � i (c† ici+1 + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=') + V � i (ni − 1 2)(ni+1 − 1 2), (S11) This model is described by the free compact boson theory, the Luttinger liquid theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' A nontrivial V ̸= 0 only changes the radius of the compact boson field, or equivalently the Luttinger parameter [17], K = π 2(π − arccos ∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' (S12) The twist operator should accordingly be modified to be [19] Tnk(u, 0)T −1 nk (v, 0) = e−i 2k n 1 √ K (φk(u)−φk(v)), (S13) which is (8) in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' We should note that (S13) only works for the entanglement entropy of a single interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' For multi disjoint intervals, the compactness of boson field will couple different replica momenta, and more complicated technique is needed [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' 8 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Effective field theory in 1+0D with measurement Without measurement, the bosonized action of Hamiltonian (1) reads S[φ] = 1 2πK � dx � β 0 dτ � ˙φ2 + (∇φ)2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' (S14) where th Luttinger parameter is related to the interaction through (2) [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' For the measurement operator ˆ M = e−W � i(−1)ini = e−W � dx cos (2kF x)n(x), where each site is x = nπ/2kF , n ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Using the bosonization of fermion density operator [16] n(x) = − 1 π ∇φ(x) + 1 π cos [2(kF x − φ)], (S15) we have ˆ M = e−W/2π � dx cos (2φ), where we keep the leading term and neglect higher-order terms like cos [2(nkF x + φ)] with n ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Here, notice that φ(x) = φ(x, τ = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' The density-density correlation for the post-selected state is ⟨n(x)n(0)⟩ = Tr[ ˆ Mρ ˆ M †n(x)n(0)]/Tr[ ˆ Mρ ˆ M †].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Since ⟨n(x)n(0)⟩ commutes with the measurement operator, the numerator equals Tr[ρ ˆ M 2n(x)n(0)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Using path integral representation, the numerator can be expressed as Tr[ρ ˆ M 2n(x)n(0)] = � Dφe−Sn(x)n(0), (S16) with the action S S = � dxdτ � 1 2πK [(∂τφ)2 + (∂xφ)2] − δ(τ)v cos 2φ � , (S17) and v = W/π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' For other case where the operators do not commute with the measurement ˆ M, we need to be careful about the order of operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' With the effective action (S14) we can integrate out the time direction and keep the “boundary” at τ = 0 [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Here we consider a more general case with an additional mass term 1 2πK � dxdτΩ2φ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' To integrate out the time direction, we first solve the equation of motion ∂2 τφ + ∂2 xφ − Ω2φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' In the momentum space, φ(p, τ) = � dxφ(p, τ)eipx, we have ∂2 τφ(p, τ) = (p2 + Ω2)φ(p, τ), which has the general solution φ(p, τ) = φ(p, 0)(cosh ωpτ − coth ωpβ sinh ωpτ) + φ(p, β) sinh ωpτ sinh ωpβ , (S18) where ω2 p = p2 + Ω2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Then plugging it in to the action we have S = 1 πK � dpωp 2 � φ2 c(p) tanh βωp 2 + φ2 q(p) coth βωp 2 � , (S19) where φc(p) = [φ(p, β) + φ(p, 0)]/ √ 2 and φq(p) = [φ(p, β) − φ(p, 0)]/ √ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' In the case of gapless fermions at zero temperature, Ω = 0 and β = ∞, we have ωp = |p| and S = 1 πK � dp |p| 2 � φ2 c(p) + φ2 q(p) � = 1 πK � dp|p|φ2(p), where we apply the periodic boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' With measurement, the final effective theory is s[φ] = s0[φ] − v � dx cos [2φ], s0[φ] = 1 πK � dq 2π |q||φ(q)|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' (S20) Also a remark is that the sign of v is unimportant, since a simple translation relates the two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' [16], authors choose the convention that the field φ is related to the fermion density operator, and the dual field θ, defined by [∂xφ, θ] = iπδ(x − x′), is related to the phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Then the corresponding actions are SG[φ] = 1 2πK � dxdτ � (∇φ)2 + ˙φ2� , SG[θ] = K 2π � dxdτ � (∇θ)2 + ˙θ2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' (S21a) After integrating out time direction, the actions are sG[φ] = 1 πK � dq 2π |q||φ(q)|2, sG[θ] = K π � dq 2π |q||θ(q)|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' (S22a) 9 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Entanglement entropy transition induced by measurement 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Transformation of Different Fields with Strong Interaction The interactions in dual theories have a strong-weak duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Therefore, we can use the dual theory at strong measurement strength or when the measurement is relevant, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=', v ≫ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' In the following, we discuss the dual transformation in more detail [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' The initial action of the field φ is s[φ] = 1 πK � dq 2π |q||φ(q)|2 − v � dx cos [2mφ], (S23) where m ∈ Z+ is an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' For v ≫ 1, we know that the configuration of φ must be consisted of domain walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Therefore, we define h = ∂xφ = �n i=1 eif(x − xi) with the constraint � +∞ −∞ dxf(x − xi) = 2π 2m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' And ei = ±1 denotes the (anti) domain wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' So, the Fourier transform of f(x) gives f(0) = 2π 2m, and a large v means that f(x) can be approximate by δ-function, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=', f(x) = � dq 2πeikxf(q) ≈ � dq 2πeikxf(0) = 2π 2mδ(x − xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Then φ(q) = 1 iqh(q) = f(0) iq �n i=1 eie−iqxi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Plugging the relation in action (S23), we arrive at s[¯φ] = 1 πK � dq 2π |q||f(0)|2 |q|2 n � ij eieje−iq(xi−xj) + nSd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=', (S24) Using Hubbard–Stratonovich (HS) transformation, we will get Z = � n � {ei=±1} e−nSd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' � Dθe− 4m2K 16π � dq 2π |q||θ(q)|2 1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' n � i=1 � dxiei �n i=1 ei � dq 2π e−iqxiθ(q), (S25) where θ(−xi) = � dq 2πe−iqxiθ(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Defining γ = 2e−Sd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' and summing all possible {ei}, we will get the final result Z = � Dθ e− 4m2K 16π � dq 2π |q||θ(q)|2 � n γn 1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' n � i=1 �� dxi cos θ(−xi) � = � Dθ e− 4m2K 16π � dq 2π |q||θ(q)|2+γ � dx cos θ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' (S26) If we only consider m = 1 and define g = e−Sd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' where Sd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' is attributed to the action of a single domain wall, we get the consistent results with Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Here, we briefly discuss the action of domain wall configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' [16], to introduce a UV-cutoff, the authors add another term 1 2 � dx(∇φ)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' For large v limit, rescaling x = v−1/2x′ and solving equation of motion for domain wall configuration, they give φd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' (x) = π 2 + arctan[sinh(2v1/2x)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' (S27) For the contribution of domain wall configuration to (S23), there are two parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' One comes from the last term in (S23) and the additional term, which gives ∆s = 4v1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Another part is attributed to the first term in (S23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' With the rescaling x = v−1/2x′, and q = v1/2q′, the first term becomes 1 πK � dq 2π |q||φd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' (v−1/2q)|2av, a = 1 πK � dq′ 2π |q′||φd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' (q′)|2, (S28) where φd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' (q′) is the Fourier transformation of φd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' (x′) = π 2 + arctan[sinh(2x′)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Therefore, Sd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' = av + 4v1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' We remark on the commutation relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' When we apply HS transformation in (S25), the commutation rela- tion is [ei, θ(xj)] = iδij, which means [∂xφ(x), θ(xj)] = [�n i=1 eif(x − xi), θ(xj)] = i 2π 2mδ(x − xj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Define ˜θ = mθ, � ∂xφ(x), ˜θ(x′) � = iδ(x − x′), ˜θ is the dual field of φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Using ˜θ to rewrite the action, we will get the action s[˜θ] = K 4π � dq 2π |q||˜θ(q)|2 − γ � dx cos ˜θ(x) m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' (S29) While in the following, we mainly consider m = 1, we remark that the results will not rely on m up to the zero order with γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' 10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Correlation Functions In the following, we consider three different correlation functions, including � ei[θ(x)−θ(0)]� , � ei[φ(x)−φ(0)]� and � TnkT −1 nk � = � e−i 2k √ Kn [φ(x)−φ(0)]� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' We first consider zero order results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' For the phase correlation function, in the free case (v = 0) with the action (S23), Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' [16] shows that � ei[θ(x0)−θ(0)]� = e− π 4K � dq 2π |q||T0,x0(q)|2 ∼ x − 1 2K 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' (S30) Here, with abuse of notation, we define T0,x0(x) = � 1, 0 < x < x0 0, x < 0, x > x0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' (S31) It should be clear from the context that this is different from the twist operator TA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' In the strong measurement limit (v ≫ 1), we take the action (S29) with γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' [16] the authors have shown that the result should be � ei[θ(x0)−θ(0)]� ∼ x − 1 K 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' We derive the correlation function above in another way which takes care of the order of operators, and perform an explicit calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' We first consider s[φ + πT0,x0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' With similar approximations, we have φ + πT0,x0 = 1 iq f(q) n � i=1 (eieiqxi + e0eiqx0 + en+1eiqxn+1), (S32) where e0 = 1, en+1 = −1, x0 = 0, xn+1 = x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Plugging it in the action (S23) with v → ∞, γ = 0, we have s0 = π K � dq 2π 1 |q| n � i,j=1 eieje−iq(xi−xj) + π K � dq 2π 1 |q| � � n � j=1 1 · eje−iq(0−xj) + n � j=1 (−1) · eje−iq(x0−xj) + (−1)e−iq(0−x0) + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' + 2 � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' (S33) It means that the function T0,x0(x) behaves like two domain walls at x = 0, x0 with opposite signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Then the partition function will be Z = � n � {ei=±1} e−nSd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' � Dθe− K 4π � dq 2π |q||θ(q)|2e π 2K � dq 2π 1 |q| (2 cos qx0−2) 1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' n � i=1 � dxiei �n i=1 ei � dq 2π e−iqxiθ(q)− π 2K � dq 2π 1 |q| �n i=1 ei(e−iq(0−xi)−e−iq(x0−xi)+h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=') = � Dθe− K 4π � dq 2π |q||θ(q)|2+γ � dx cos [θ(x)+∆x0(x)]e π 2K � dq 2π 1 |q| (2 cos qx0−2), (S34) where ∆x0(x) = i π 2K � dq 2π 1 |q|(eiqx − eiq(x−x0) + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Then directly we get � ei[θ(x0)−θ(0)]� = � Dφe− 1 2 (s[φ]+s[φ+πT0,x0]) � Dφe−s[φ] = e π 2K � dq 2π 1 |q| (2 cos qx0−2) ∼ x − 1 K 0 , (S35) which is consistent with Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Now we consider the correlation function of vertex operators and twist operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' For the free case (v = 0) with the action (S23), we have � ei[φ(x0)−φ(0)]� = eG(x0) ∼ x − K 2 0 , (S36) where G(x0) = − πK 2π ln x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Similarly, we can calculate the correlation function of twist operators, which will give entanglement entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' For the free case (v = 0) with the action (S23), we have � e−i 2k √ Kn [φ(x0)−φ(0)]� = e 4k2 Kn2 G(x0) ∼ x − 2k2 n2 0 , (S37) 11 which is consistent with Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' [23] and gives entanglement entropy S = 1 3 log L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' In the following, we derive the results of the correlation functions of vertex operators and twist operators for strong measurement strength by using the similar method of (S34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' For the correlation function of vertex operators, we have � ei[φ(x0)−φ(0)]� = � e− � dq 2π qφ(q)T0,x0(−q)� = � exp � − � dq 2π 1 iq n � i=1 eie−iqxiπqT0,x0(−q) �� , (S38) where the expectation value is according to the action (S29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Then the numerator is Z = � n � {ei=±1} e−nSd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' � Dθe− K 4π � dq 2π |q||θ(q)|2 1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' n � i=1 � dxiei �n i=1 ei � dq 2π e−iqxiθ(q)+i � dq 2π πT0,x0(−q) �n i=1 eie−iqxi = � Dθ e− K 4π � dq 2π |q||θ(q)|2+γ � dx cos [θ(x)+∆x0(x)], (S39) where ∆x0(x) = � dq 2ππT0,x0(−q)e−iqxi = πT0,x0(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Therefore, up to the zeroth order γ = 0, the denominator equals the numerator, which means � ei[φ(x0)−φ(0)]� = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' There is another way to double-check the result above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' We can rewrite the exponent of numerator as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' −s′ = − 1 πK � dq 2π |q||φ(q)|2 + v � dx cos [2φ] − � dq 2π qφ(q)T0,x0(−q) = − 1 πK � dq 2π |q|(φ(q) + πKq 2|q| T0,x0(q))(φ(−q) + πKq 2|q| T0,x0(−q)) + v � dx cos [2φ] + 1 πK � dq 2π |q|π2K2 4 |T0,x0(q)|2, (S40) where (− � dq 2πqφ(q)T0,x0(−q))∗ = − � dq 2πqφ(−q)T0,x0(q) = � dq 2πqφ(q)T0,x0(−q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Then there are two terms con- tributing to the final result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' One is − 1 πK � dq 2π|q||φ(q) + πKq 2|q| T0,x0(q)|2, which is similar to (S35) and gives e π K � dq 2π 1 |q| (2 cos qx0−2)·( Kq 2|q|) 2 ∼ x − K 2 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Another term is 1 πK � dq 2π|q| π2K2 4 |T0,x0(q)|2 which is similar to (S30) and gives ∼ x K 2 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Therefore, two terms’ contributions will cancel out and lead to � ei[φ(x0)−φ(0)]� = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' For twist operators, the correlation function is similar to vertex operators case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' The only difference in (S39) is that there is a prefactor in front of ∆x0(x), which will not change the zeroth order results and gives a trivial correlation function � TnkT −1 nk � = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Here, we briefly discuss the strong measurement results of several correlation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' The nontrivial result of phase correlation function is because of the interaction of two domain walls in T0,x0(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' But for vertex operator, the additional term is linear with φ, which doesn’t contribute the quadratic terms like phase correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' So the result is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Finally, we consider the correlation functions of density operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' For the free case (v = 0) we have ⟨∇φ(x0)∇φ(0)⟩ = −∇2G(x0) = K 2 x−2 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' (S41) For the strong measurement case, Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' [16] shows that ⟨∇φ(x0)∇φ(0)⟩ ∼ γ2x − 2 K 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' It is worth mentioning that this density correlation function will rely on different m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' According to the discussion above, for the strong measurement case, we always get vanishing results at the zeroth order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' So now we consider the first order of γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' To simplify the problem, we only consider m = 1 case (For general m the results will be similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=') In the following, we will only use the method of (S34) and (S39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Phase Correlation Function Firstly, we consider phase correlation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' To consider the domain wall contribution, we include another term to (S23), s[φ] = 1 πK � dq 2π |q||φ(q)|2 − v � dx cos [2φ] + 1 2 � dx(∇φ)2 = 1 πK � dq 2π |q||φ(q)|2 − v 1 2 �� dx′ cos [2φ] − 1 2 � dx′(∇′φ)2 � , (S42) 12 where x′ = v 1 2 x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Similar to (S35) we have 1 2(s[φ] + s[φ + πT0,x0]) = 1 πK � dq 2π |q||φ(q)|2 + 1 K � dq 2π |q|φ(q)T0,x0(−q) + π 2K � dq 2π |q||T0,x0(q)|2 − 1 2v 1 2 �� dx′ cos [2(φ + πT0,x0)] − 1 2 � dx′(∇′(φ + πT0,x0))2 + � dx′ cos [2φ] − 1 2 � dx′(∇′φ)2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' (S43) Therefore, there are two additional terms in the second line (S43).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' T0,x0 is unimportant for cos term, because we can safely ignore the step function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' But we need to consider the contribution of it to quadratic terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' For the strong measurement strength, we need to change the configuration of φ to minimize two cos terms, which means for first order we take φ as two domain walls (beside two domain walls of T0,x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Now we have 1 2(s[φ] + s[φ + πT0,x0]) = 1 2πK � dq 2π |q||φ(q)|2 + 1 2πK � dq 2π |q||φ(q) + πT0,x0(q)|2 − v 1 2 �� dx′ cos [2φ] − 1 2 � dx′(∇′φ)2 � + π 2 � dx∇φ∇T0,x0 + ˜C 2 , (S44) where ˜C = 1 2 � dx(π∇T0,x0)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Then we consider (S34) with the additional terms Z =e− ˜ C 2 � n � {ei=±1} e−nSd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' � Dθe− K 4π � dq 2π |q||θ(q)|2e π 2K � dq 2π 1 |q| (2 cos qx0−2) 1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' n � i=1 � dxi ei �n i=1 ei � dq 2π eiqxiθ(q)− π 2K � dq 2π 1 |q| �n i=1 ei(e−iq(0−xi)−e−iq(x0−xi)+h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' )− π2 2 � dq 2π q2T0,x0(−q)· 1 iq �n i=1 eie−iqxi =e− ˜ C 2 � Dθe− K 4π � dq 2π |q||θ(q)|2+γ � dx cos [θ(x)+∆x0(x)+ ˜∆x0(x)]e π 2K � dq 2π 1 |q| (2 cos qx0−2), (S45) where ∆x0(xi) =i π 2K � dq 2π 1 |q|(e−iq(0−xi) − e−iq(x0−xi) + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=') = i π K � dq 2π 1 |q| [cos qxi − cos q(x0 − xi)] , (S46) ˜∆x0(xi) =π2 2 � dq 2π qT0,x0(−q)e−iqxi = iπ2 2 ∂xiT0,x0(xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' (S47) Then we double-check the results above and give some comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' (i) The zero order term satisfies e π 2K � dq 2π 1 |q| (2 cos qx0−2) = e− π 2K � dq 2π |q||T0,x0(q)|2 ∼ x − 2 K 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' (S48) (ii) The additional ∆x0(x) is from the crossing term − 1 2πK � dq 2π |q| [φ(q)πT0,x0(−q) + φ(−q)πT0,x0(q)] = − 1 K � dq 2π |q|T0,x0(−q) · 1 iq n � i=1 eie−iqxif(q) =i n � i=1 ei π K � dq 2π |q| q e−iqxiT0,x0(−q) = i n � i=1 ei iπ K � dq 2π 1 |q|e−iqxi(1 − eiqx) = i n � i=1 ei∆x0(xi), (S49) where f(q) = π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Simplifying ∆x0(x) we have ∆x0(x) = i π K � dq 2π 1 |q| � 2 sin2 q(xi − x0) 2 − 2 sin2 qxi 2 � = i 1 K ln ���� xi − x0 xi ����.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' (S50) (iii) For ˜∆x0(xi) we have ˜∆x0(xi) = π2 2 � dq 2π q 1 −iq (eiq·0 − eiqx0)e−iqxi = iπ2 2 [δ(xi) − δ(xi − x0)] , (S51) which is consistent with (S47).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' 13 We calculate the phase correlation function as follows, � ei[θ(x0)−θ(0)]� = e− ˜ C 2 e π 2K � dq 2π 1 |q| (2 cos qx0−2) � Dθe− K 4π � dq 2π |q||θ(q)|2+γ � dx cos [θ(x)+∆x0(x)+ ˜∆x0(x)] � Dθe− K 4π � dq 2π |q||θ(q)|2+γ � dx cos [θ(x)] , (S52) where the first two parts above give the zero order result e− ˜ C 2 x − 1 K 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' In the following, we evaluate the path integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Expanding the cos term gives num = � Dθe− K 4π � dq 2π |q||θ(q)|2+γ � dx cos [θ(x)+∆x0(x)+ ˜∆x0(x)] = � Dθe− K 4π � dq 2π |q||θ(q)|2 ∞ � n=0 1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' � γ � dx cos [θ(x) + ∆x0(x) + ˜∆x0(x)] �n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' (S53) For the denominator we can just set ∆x0(x) = ˜∆x0(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' As we know, the expectation value of “vertex” operators is nontrivial only when the total charge is zero, which means the first order term is n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' With the form of “vertex” operator we have γ2 term 1 2γ2 � dx1dx2 � cos [θ(x1) + ∆x0(x1) + ˜∆x0(x1)] cos [θ(x2) + ∆x0(x2) + ˜∆x0(x2)] � =1 2γ2 × 1 4 � dx1dx2 � ei[∆x0(x1)+ ˜∆x0(x1)−∆x0(x2)− ˜∆x0(x2)] � eiθ(x1)e−iθ(x2)� + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' � , (S54) where � eiθ(x1)e−iθ(x2)� = e 1 2 G(x1−x2)+ 1 2 G(x2−x1) = e− 1 2π 4π K ln |x1−x2|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' (S55) So the γ2 term is 1 2γ2 × 1 4 � dx1dx2 � ei[∆x0(x1)+ ˜∆x0(x1)−∆x0(x2)− ˜∆x0(x2)]− 2 K ln |x1−x2| + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' � = 1 2γ2I1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' (S56) Similarly, γ2 term in denominator is 1 2γ2 × 1 4 � dx1dx2 � e− 2 K ln |x1−x2| + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' � = 1 2γ2I2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' (S57) Therefore, we get the phase correlation function � ei[θ(x0)−θ(0)]� ≈ e− ˜ C 2 x − 1 K 0 (1 + γ2 2 I1)(1 + γ2 2 I2)−1 ≈ e− ˜ C 2 x − 1 K 0 (1 + γ2 2 (I1 − I2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' (S58) Now we evaluate the integral I1 − I2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Plugging ∆x0(x) and ˜∆x0(x) in I1 we have I1 =2 × 1 4 � dx1dx2e � − 1 K ln ��� (x1−x0)x2 x1(x2−x0) ���− π2 2 (δ(x1)−δ(x1−x0)−δ(x2)+δ(x2−x0))− 2 K ln |x1−x2| � =1 2 � dx1dx2e− π2 2 (δ(x1)−δ(x1−x0)−δ(x2)+δ(x2−x0)) ����� (x1 − x0)x2 x1(x2 − x0) ���� (x1 − x2)2 �− 1 K ≈1 2 � dx1dx2 ����� (x1 − x0)x2 x1(x2 − x0) ���� (x1 − x2)2 �− 1 K =1 2 � dx1dx2e− 1 K (ln |x1−x0|+ln |x2−0|−ln |x1−0|−ln |x2−x0|+2 ln |x1−x2|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' (S59) From the above result we find that the contribution of ˜∆x0(x) is only on some isolated points with the measure that is dimension zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' If we apply some UV cutoff that require different domain walls cannot be too close to each other the exponent will always be zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Besides, there is a comment on (S59).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' From the last line we can understand the meaning of the contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Because we consider the sub-leading term, we assume there are two domain walls with opposite signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' And there are also two domain walls because of T0,x0(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Therefore, we actually have two domain 14 walls at x = 0, x1 with positive charge and two domain walls at x = x0, x2 with negative charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' There are C2 4 = 6 pairs, where the interaction between x = 0 and x = x0 is zero order result in (S58) and the other five pairs are in (S59).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' The pairs with opposite (same) charge will have factor +1(−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' And the factor 1 2 is because in (S35) numerator the pair (x1, x2) contributes two actions and other pairs only contribute s[φ + πT0,x0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Finally, we have I1 − I2 =1 2 � dx1dx2 ����� (x1 − x0)x2 x1(x2 − x0) ���� − 1 K − 1 � |x1 − x2|− 2 K =1 2 � d˜x1d˜x2 ����� (˜x1 − 1)˜x2 ˜x1(˜x2 − 1) ���� − 1 K − 1 � |˜x1 − ˜x2|− 2 K · x 2− 2 K 0 = ∆I1 · x 2− 2 K 0 , (S60) where ˜xi = xi/x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Therefore, we get the phase correlation function is � ei[θ(x0)−θ(0)]� ≈ e− ˜ C 2 x − 1 K 0 (1 + γ2 2 ∆I1 · x 2− 2 K 0 ), (S61) where we have obtained subleading correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Vertex Operator Correlation and Entanglement Entropy Now we consider the correlation function � ei[φ(x)−φ(0)]� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' With (S39) as numerator, we can expand it like (S53), num = � Dθe− K 4π � dq 2π |q||θ(q)|2 ∞ � n=0 1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' � γ � dx cos [θ(x) + ∆x0(x)] �n , (S62) where ∆x0(x) = πT0,x0(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' The first nontrivial term is γ2 term 1 2γ2 � dx1dx2 ⟨cos [θ(x1) + πT0,x0(x1)] cos [θ(x2) + πT0,x0(x2)]⟩ =γ2 2 × 1 4 � dx1dx2eiπ[T0,x0(x1)−T0,x0(x2)] � eiθ(x1)e−iθ(x2)� + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' = 1 2γ2I3, (S63) where � eiθ(x1)e−iθ(x2)� = e− 2 K ln |x1−x2|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' And the denominator is the same as (S57).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' So, � ei[φ(x0)−φ(0)]� ≈ (1 + γ2 2 I3)(1 + γ2 2 I2)−1 ≈ (1 + γ2 2 (I3 − I2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' (S64) Here is a remark about the result above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' It seems that (S63) is similar to (S56).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' But actually, their final result is not the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' For (S56) the function ∆x0(x) represent the interaction of two domain walls which contribute a real factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' For (S63), here the function ∆x0(x) play the role of phase factor that makes the integrand oscillate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' The key difference is a sign factor |q| = q · sgn(q) in (S49).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Comparing (S39) and (S49), we can find that although both ∆x0(x) is from the linear term to φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' But for phase correlation case (S49) the linear term is � dq 2π|q|φ(q)πT0,x0(−q) with the factor |q|, but for the vertex correlation (S39) the linear term is � dq 2πqφ(q)T0,x0(−q) with the factor q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' It is the difference sgn(q) that makes the difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' We can calculate the integral I3 and I2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' I3 =1 4 � dx1dx2eiπ[T0,x0(x1)−T0,x0(x2)]− 2 K ln |x1−x2| + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='c =1 2 � dx1dx2e− 2 K ln |x1−x2|eiπΘ, (S65) where Θ = T0,x0(x1) − T0,x0(x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Then ∆I3 = I3 − I2 = 2 × 1 2 � F (Θ=1) dx1dx2e− 2 K ln |x1−x2| × (−1), (S66) 15 𝑥!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' 𝑥" 𝑦" 𝑦!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' 0 𝑥# 𝑥# 𝑒$% 𝑒$% 𝑒&$% 𝑒&$% I I II II (a) 𝑥!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' 𝑥" 𝑦" 𝑦!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' 0 𝑥# 𝑥# 𝑒$%&!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='" 𝑒$%&!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='" 𝑒\'$%&!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='" 𝑒\'$%&!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='" I I II II (b) Supplementary Figure S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' (a) Diagram of Integral for integral (S67).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' (b) Diagram of Integral for integral (S75) where F(Θ = 1) means the region that (x1, x2) satisfies Θ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' S1 (a), the region F(Θ = 1) is labeled by e±iπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' If we change the coordinate (x1, x2) to (y1, y2) = ( x1−x2 √ 2 , x1+x2 √ 2 ), it gives ∆I3 = − � F (Θ=1) dy1dy2e− 2 K ln | √ 2y1| = −2 � I+II dy1dy2e− 2 K ln | √ 2y1|, (S67) where the region I and II is also shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' S1 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' So, we have ∆I3,I = −2 � x0/ √ 2 0 dy1 · 4y1e− 2 K ln √ 2y1 = −8 � x0/ √ 2 0 dy1( √ 2y1)− 2 K · y1 = −4 2 − 2 K x 2− 2 K 0 , (S68) ∆I3,II = −2 � √ 2x0 − √ 2x0 dy2 � +∞ x0/ √ 2 dy1e− 2 K ln √ 2y1 = −4 √ 2x0 � +∞ x0/ √ 2 dy1( √ 2y1)− 2 K = 4 1 − 2 K x 2− 2 K 0 , (S69) where we assume K < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Therefore, we have ∆I3 = ∆I3,I + ∆I3,II = � 4 1 − 2 K − 4 2 − 2 K � x 2− 2 K 0 , (S70) which means the correlation function is � ei[φ(x0)−φ(0)]� ≈ 1 + γ2 2 · � 4 1 − 2 K − 4 2 − 2 K � x 2− 2 K 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' (S71) For entanglement entropy, we consider the correlation function � e−iak n[φ(x0)−φ(0)]� with ak n = 2k √ Kn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Therefore, we just need to rescale the function T0,x0 with ak n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' The numerator is num = � Dθe− K 4π � dq 2π |q||θ(q)|2 ∞ � n=0 1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' � γ � dx cos [θ(x) + ak nπT0,x0(x)] �n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' (S72) If we define I4 = 1 4 � dx1dx2eiπak n[T0,x0(x1)−T0,x0(x2)]− 2 K ln |x1−x2| + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=', (S73) the correlation function is � e−i 2k √ Kn [φ(x0)−φ(0)]� ≈ (1 + γ2 2 I4)(1 + γ2 2 I2)−1 ≈ (1 + γ2 2 (I4 − I2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' (S74) 16 Similar to the results above, ∆I4 =1 4 � dx1dx2e− 2 K ln |x1−x2| � eiπak n[T0,x0(x1)−T0,x0(x2)] − 1 � + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='c =1 2 � I+II dx1dx2e− 2 K ln |x1−x2| � eiπak n[T0,x0(x1)−T0,x0(x2)] − 1 � + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='c =[cos (πak n) − 1] � I+II dx1dx2e− 2 K ln |x1−x2| = [cos (πak n) − 1] � I+II dy1dy2e− 2 K ln √ 2y1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' (S75) Using the result above with Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' S1 (b), we have ∆I4 = [cos (πak n) − 1] � −2 1 − 2 K − −2 2 − 2 K � x 2− 2 K 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' (S76) Therefore, we have correlation function � e−i 2k √ Kn [φ(x0)−φ(0)]� ≈ 1 + γ2 2 [cos (πak n) − 1] � −2 1 − 2 K − −2 2 − 2 K � x 2− 2 K 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' (S77) Finally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' according to Section A 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' we have ln Zk = ln � 1 + γ2 2 [cos (πak n) − 1] � −2 1 − 2 K − −2 2 − 2 K � x 2− 2 K 0 � ≈ [cos (πak n) − 1]γ2 2 � −2 1 − 2 K − −2 2 − 2 K � x 2− 2 K 0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' (S78) so the entanglement entropy is S = lim n→1 1 1 − n � k ln Zk = − lim n→1 ∂ ∂n � k [cos (πak n) − 1]γ2 2 � −2 1 − 2 K − −2 2 − 2 K � x 2− 2 K 0 = γ2f(K)x 2− 2 K 0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' (S79) where f(K) = � 1 1− 2 K − 1 2− 2 K � limn→1 ∂ ∂n � k[cos (πak n) − 1] = � 1 1− 2 K − 1 2− 2 K � [ π √ K cot π √ K − 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Entanglement entropy at the critical point For K = 1 exactly non-interacting case with measurement, we can rotate the x−t plane such that the measurement effectively act on x = 0 for all time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' After the rotation, the system is time translation invariant, so we can transform this problem to solving the time-independent Schrodinger equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Now, the system is still a non-interacting free fermion system, but has a mass term at x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' To calculate the entanglement entropy, we need to calculate the correlation function � Q2 A � = − ⟨QAQ ¯ A⟩ [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Here ¯A is the complement of A, and the above equation is valid for a pure state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' In the reference, the operator is defined as a time-independent operator (t = 0) that QA = � A dDrψ†(r)ψ(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Now with rotation we have QA = � tA dte−iHtψ†(0)ψ(0)eiHt with fixed location x = 0 but integration of “time interval” tA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Then we have ⟨QAQ ¯ A⟩ = � A � ¯ A dtdt′Tr � ρ(0)eiH(t−t′)ρ(0)e−iH(t−t′)e−βH� = � dE � dE′ � A dt � ¯ A dt′ ⟨E|ρ(0)|E′⟩ eiE′(t−t′) ⟨E′|ρ(0)|E⟩ e−iE(t−t′)e−βE = � dE � dE′ � A dt � ¯ A dt′ |⟨E|ρ(0)|E′⟩|2 ei(E′−E)(t−t′)e−βE, (S80) where |E⟩ , |E′⟩ are many-body eigenstates and E, E′ are total energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' In the second quantization formalism, ψ(r) = � de ⟨r|e⟩ ce = � deψe(r)ce, where |e⟩ is the eigenstate of a single particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' At zero temperature β → ∞, the many-body ground state with energy E0 can be represented by |GS⟩ = ⊗ei<0 |1⟩ ⊗ei>0 |0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Therefore, ⟨E|ρ(r)|E′⟩ = ⟨E| ψ†(r)ψ(r) |E′⟩ = � de1e2ψ∗ e1(r)ψe2(r) ⟨E| ψ†(e1)ψ(e2) |E′⟩ = ψ∗ e1(r)ψe2(r), (S81) 17 where e1, e2 satisfy that when we annihilate a state |e1⟩ and create a state |e2⟩, |E′⟩ will change to |E⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Finally, at zero temperature we have ⟨QAQ ¯ A⟩ = � dE′ � A dt � ¯ A dt′ |⟨GS|ρ(0)|E′⟩|2 ei(E′−E0)(t−t′) = � A dt � ¯ A dt′ � e1<0,e2>0 de1de2 |ψe1(0)|2 |ψe2(0)|2 ei(e1−e2)(t−t′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' (S82) We can first consider the free case that |ψe(0)|2 = 1 2π, then the correlation is ⟨QAQ ¯ A⟩ = � A dt � ¯ A dt′ � d(e1 − e2)ei(e1−e2)(t−t′) (2π)2 � de2 = � A dt � ¯ A dt′ � d(∆e) 2π s(∆e)ei∆e(t−t′), (S83) where s(∆e) = � de2 2π = |∆e| 2π , which can be considered that for a fixed ∆e, e2 can range from e = −∆e to e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Similar to the Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' [24], it gives ⟨QAQ ¯ A⟩ = − 1 π2 log x0 and entanglement entropy S = 1 3 log x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' In the following, we will show that for Luttinger liquid with K = 1 and measurement strength v, the entanglement entropy is just renormalized by a prefactor which only relies on measurement strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' As mentioned above, after rotation, we will have an effective Lagrangian L = ψ†(i∂t + i∂xσz + vδ(x)σx)ψ, (S84) where ψ = (ψL ψR)T represents the left and right mover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Then the effective Schrodinger equation reads i∂tψ = (−i∂xσz + vδ(x)σx)ψ = H′ψ, (S85) where ψ = (ψ1 ψ2)T and the sign of v is unimportant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' The system has the following symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' (i) Denoting the operator T as reflecting symmetry operator, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=', x → −x with conjugation of σx, we have [T, H′] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' With T 2 = 1, we have T = ±1 which means Tψ = ±ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' (ii) Considering x → −x, ψ2 → −ψ2, then we have H′ → −H′ which means particle-hole symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Because the δ−function is hard to deal with, we consider the limit that there is a square wall potential m in the range [− L 2 , L 2 ] and mL = v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Because the Hamiltonian is time-independent, we have ψ(t) = ψe−iEt and H′ψ = Eψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' In the following, E is single particle energy for state ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' For x ∈ (−∞, − L 2 ] ∪ [ L 2 , ∞), the wave function is a plane wave, while for x ∈ [− L 2 , L 2 ], the eigen-equation reads � −i∂xψ1 + mψ2 = Eψ1 (−∂2 x − E2 + m2)ψ2 = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' (S86) Therefore, the corresponding wave function is ψ(x < −L 2 ) = � Aeikx Be−ikx � ψ(−L 2 < x < L 2 ) = � � κ + E m Ceiκx + −κ + E m De−iκx Ceiκx + De−iκx � � ψ(x > L 2 ) = � A′eikx B′e−ikx � , (S87) where k = E and κ2 = E2 − m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' With the boundary condition, we can solve this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' To simplify the problem, with the symmetry [T, H] = 0, we consider the wave function that is also the eigenstate of the operator T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' For T = 1, we have Aeik −L 2 = B′e−ik L 2 , so A = B′ and similarly B = A′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Also for the middle region κ+E m Ceiκ −L 2 + −κ+E m De−iκ −L 2 = Ceiκ L 2 + De−iκ L 2 , so C = −κ+E m D, D = κ+E m C which are consistent with E2−κ2 m2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Therefore, the wave function with T = 1 can be simplified as ψ(x < −L 2 ) = � Aeikx Be−ikx � ψ(−L 2 < x < L 2 ) = � Deiκx + Ce−iκx Ceiκx + De−iκx � ψ(x > L 2 ) = � Beikx Ae−ikx � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' (S88) Similarly, for the odd case with T = −1, we have A = −B′, B = −A′, C = − −κ+E m D, D = − κ+E m C, and the wave function is ψ(x < −L 2 ) = � Aeikx Be−ikx � ψ(−L 2 < x < L 2 ) = � −Deiκx − Ce−iκx Ceiκx + De−iκx � ψ(x > L 2 ) = � − Beikx − Ae−ikx � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' (S89) 18 For both cases, we just need to consider one boundary condition e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=', x = − L 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Combining boundary condition and the relation of factor C and D we can represent all factors with D, � � � � � � � � � � � � � � � A =eik L 2 � ±e−iκ L 2 + −κ + E m eiκ L 2 � D B =e−ik L 2 � eiκ L 2 ± −κ + E m e−iκ L 2 � D C = ± −κ + E m D, (S90) where ± is corresponding to T = ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Besides, we can notice that A = ±eikLB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Now, to evaluate the expectation value ⟨QAQ ¯ A⟩, we only need to calculate the factor D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' It can be fixed by normalization condition ⟨k′|k⟩ = δ(k − k′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Assuming the total system length is 2R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' we have the inner product ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='⟨E′|E⟩ = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='� − L ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='−R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='A∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='k′Akei(k−k′)x + B∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='k′Bke−i(k−k′)x� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='dx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='L ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='− L ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='(D∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='κ′e−˜κ′x + C∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='κ′e˜κ′x)(Dκe−˜κx + Cκe˜κx) + (D∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='κ′e˜κ′x + C∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='κ′e−˜κ′x)(Dκe˜κx + Cκe−˜κx) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='dx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='� R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='L ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='B∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='k′Bkei(k−k′)x + A∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='k′Ake−i(k−k′)x� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='dx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='= 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='A∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='k′Ak ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='ei(k−k′) −L ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='2 − ei(k−k′)(−R) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='i(k − k′) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='+ B∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='k′Bk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='e−i(k−k′) −L ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='2 − e−i(k−k′)(−R) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='−i(k − k′) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='+ 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='(D∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='κ′Dκ + C∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='κ′Cκ)e−(˜κ′+˜κ) L ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='2 − e(˜κ′+˜κ) L ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='−(˜κ′ + ˜κ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='+ (D∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='κ′Cκ + C∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='κ′Dκ)e−(˜κ′−˜κ) L ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='2 − e(˜κ′−˜κ) L ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='−(˜κ′ − ˜κ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' (S91) where ˜κ2 = m2 − E2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Because we will take the limit of δ−function that m → ∞, L → 0 and mL = v, if we only consider low-energy states that E << m, κ is a imaginary number but ˜κ is corresponding real number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' With the limit, we can use approximation ˜κ = √ m2 − E2 ≈ m − E2 2m, which means ˜κ · L ≈ v − E2L 2m = v + O( 1 m2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' For the first square brackets of (S91), with the relation of factor Ak, Bk and Dk we have First term = 2B∗ k′Bk i(k − k′) � ei(k−k′)R − ei(k−k′)(L−R)� =2(e−˜κ′ L 2 ± i˜κ′ + E′ m e˜κ′ L 2 )(e−˜κ L 2 ± −i˜κ + E m e˜κ L 2 )ei(k−k′)(R− L 2 ) − e−i(k−k′)(R− L 2 ) i(k − k′) D∗ κ′Dκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' (S92) Plugging the approximation ˜κ · L ≈ v − E2L 2m = v(1 − k2 2m2 ), i˜κ+E m ≈ i(1 − k2 2m + k m) in the above equation and only keeping up to O( 1 m) term, we get First term =2 � (e−v + ev + (e− v 2 ∓ ie v 2 )2 vk′2 4m + (±1 − iev)k′ m + (e− v 2 ± ie v 2 )2 vk2 4m +(±1 + iev) k m � 2 sin (k − k′)(R − L 2 ) (k − k′) D∗ κ′Dκ + O( 1 m2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' (S93) For the second square brackets of (S91), with the relation of factor Ak, Bk and Dκ we have Second term =2D∗ κ′Dκ � (1 + i˜κ′ + E′ m −i˜κ + E m )e−(˜κ′+˜κ) L 2 − e(˜κ′+˜κ) L 2 −(˜κ′ + ˜κ) ± (i˜κ′ + E′ m + −i˜κ + E m )e−(˜κ′−˜κ) L 2 − e(˜κ′−˜κ) L 2 −(˜κ′ − ˜κ) � =2D∗ κ′Dκ �2(ev − e−v) 2m + rmi(k − k′)(ev − e−v) 4m2 ± v(k + k′) m2 � + O( 1 m3 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' (S94) If we take the limit m → ∞ and only keep zero order, (S91) is ⟨E′|E⟩ = 4(ev + e−v)sin (k − k′)(R − L 2 ) (k − k′) D∗ κ′Dκ + O( 1 m) R→∞ = 4π(ev + e−v)D∗ κ′Dκδ(k − k′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' (S95) 19 Therefore, we can set D = 1 √ 4π · 1 (ev+e−v)1/2 , and the corresponding factor B is B = e−ik L 2 � eiκ L 2 ± −κ + E m e−iκ L 2 � D ≈ (e−v/2 ∓ iev/2)D = 1 √ 4π e−v/2 ∓ iev/2 (ev + e−v)1/2 = 1 √ 4π e∓iθ, (S96) where tan θ = ev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' There are some remarks about the results above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' (i) For any value of v, we always have |B| = 1 √ 4π, which is consistent because when we take the limit L → 0, almost all the contribution for inner product comes from the free plane wave solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Because of two components of the wave function, the prefactor must be 1 √ 2 · 1 √ 2π = 1 √ 4π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' (ii) For different v, the factor B has different phases, but the key difference is D which relies on v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' With the limit L → 0, if we set v = 0, then D = 1 √ 4π √ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Recalling the wave function has the form Ce−˜κx + De˜κx and C ≈ iD, so the total amplitude is √ 2D = 1 √ 4π = B, which means the wave function is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' But for v ̸= 0 especially for large v, we have |D| ≪ |B|, which means it is discontinuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Finally, for the entanglement entropy of a general v, with the amplitude of the wave function |ψe(0)|2 =(D∗ κ′e−˜κ′x + C∗ κ′e˜κ′x)(Dκe−˜κx + Cκe˜κx) + (D∗ κ′e˜κ′x + C∗ κ′e−˜κ′x)(Dκe˜κx + Cκe−˜κx)|x=0 =2|C + D|2 ≈ 4|D|2 = 1 2π cosh v , (S97) we arrive at ⟨QAQ ¯ A⟩ = � A dt � ¯ A dt′ � e1<0,e2>0 de1de2 |ψe1(0)|2 |ψe2(0)|2 ei(e1−e2)(t−t′) = 1 cosh2 v [⟨QAQ ¯ A⟩]v=0 , (S98) which means the entanglement entropy is S = 1 cosh2 v · 1 3 log x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Numerical calculation of the entanglement entropy 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Free fermion calculation with Julia Here we consider the Luttinger liquid model with K = 1, which corresponds to a free fermion system with periodic boundary condition H = � i (c† ici+1 − cic† i+1), (S99) where c† i (ci) denotes the fermion creation (annihilation) operator at site i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' To calculate entanglement entropy after measurement [25], we first diagonalize the Hamiltonian (S99) and calculate its correlation matrix Γ Γ := � Γc†c Γc†c† Γcc Γcc† � , (S100) where Γc†c ij = � c† icj � and Γc†c† ij = � c† ic† j � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Then we apply imaginary time evolution with measurement Hamiltonian Hm, which has the same effect as measurement operator ˆ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' For different system sizes L and strength of measurement W that is represented by the imaginary time τ of evolution, we calculate the entanglement entropy of half of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Then we obtain the relation of entanglement entropy and L, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' We consider the system size L ∈ [2, 200] and the measurement W ∈ [0, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' For different measurement strength we fit the relation between entanglement entropy and system size L with a+b log L, the prefactor b is related to the effective central charge, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' b = ceff 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' S2 (a), we plot the relation between effective central charge and measurement strength W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' The dot points are the numerical results we get from Julia, and the orange line is the analytical result 1 cosh2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='7W .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Here we take measurement Hamiltonian Hm = Diag(0, 1, 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=', 0, −1, 0, −1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=') and imaginary time evolution eW Hm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' It is equivalent to the measurement Diag(−1/2, 1/2, −1/2, 1/2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=', 1/2, −1/2, 1/2, −1/2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=') considered in the main text (3) up to an identity operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' At W = 0, we have central charge c = 1 with prefactor b = 1/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' For W ̸= 0 numerical results and analytical results are consistent with each other very well, leading to the effective central charge ceff = cosh−2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='7W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' S2 (c) shows the coefficient of determination R2 when we interpolate the entanglement entropy S with log L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' 1 − R2 ≈ 10−9 means the fitting is accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' 20 0 1 2 3 4 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='0 W ceff Δ=0 (a) 0 1 2 3 4 5 0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='×10-9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='×10-9 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='×10-9 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='×10-9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='×10-8 W 1-R2 (b) Supplementary Figure S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' (a) The effective central charge as a function of the measurement strength W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' (b) Fitting coefficient 1 − R2 at different measurement strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' 0 1 2 3 4 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='0 W ceff Δ=0 (a) 0 1 2 3 4 0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='×10-7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='×10-6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='5×10-6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='×10-6 W 1-R2 (b) Supplementary Figure S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' The results for ∆ = 0 from DMRG calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' (a) The effective central charge as a function of the measurement strength W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' (b) Fitting coefficient 1 − R2 at different measurement strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Density matrix renormalization group calculation of the XXZ model Using Jordan-Wigner transformation, the Hamiltonian (1) becomes (recall that we set t = 1) H = � i (Sx i Sx i+1 + Sy i Sy i+1 + ∆Sz i Sz i+1), (S101) with Sj α, α = x, y, z spin 1/2 operators at site i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' We first calculate the ground state wave function with Matrix Product State (MPS), then apply the measurement by considering the imaginary time evolution of the ground state wave function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Here we take Hm as evolution Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' We consider the system size L ∈ [10, 200] and the strength of measurement W ∈ [0, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' At K = 1(∆ = 0) we can plot similar results as free fermion in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' The results are consistent with the free fermion case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Here we define H′ m = � i(−1)iSz i , then the effective central charge is ceff = cosh−2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='7W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' If we take the (a) (b) Supplementary Figure S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Effective central charge ceff = 3b as a function of ∆ for different measurement strength: (a) ∆ < 0, (b) ∆ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' △ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='0- a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content="0- 0'4 S." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='0- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='0=W a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='0=w 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='0=W cel S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='0=W 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='0=W 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='1V 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='0 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content="0 0'4 a." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='7 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='0- 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='0=W 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='0 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='0=w S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='0 ←.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content="0=W 0'4 ceu M=0'S a." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='0=W 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='721 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='0 15 10 5 0 W 1\uf00c2 factor a Δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='8 (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='0 15 10 5 0 W 1\uf00c2 factor b Δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='8 (b) Supplementary Figure S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Prefactor fitting with quadratic polynomials: (a) prefactor a with fitting results log a ∼ −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='14W + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='40 √ W − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='19, (b) prefactor b with fitting results log b ∼ −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='23W + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='95 √ W + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' L=16 L=32 L=64 L=128 L=200 4 2 0 2 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='0 (Δ-Δc)logL Sent(Δ,L)-Sent(Δc,L) Supplementary Figure S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Half-chain entanglement entropy as a function of ∆ for different sizes L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' The measurement strength is W = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' measurement Hamiltonian Hm = � i Sz 2i+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' The only difference is that the fitting function of effective central charge is ceff = cosh−2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='85W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' For K > 1(∆ < 0) we consider two cases and fit the entanglement entropy with S = a+b log L and plot the effective central charge ceff = 3b with respect to ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' This is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' S4 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' For W = 0 and −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='9 < ∆ < 0, the central charge is ceff = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' For ∆ ∈ (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='8, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='2) and W ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content='6, the prefactor b ≈ 1/3 which means the central charge remains approximately ceff ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Because the effective central charge exactly at ∆ = 0 is a function of W, the deviation of the central charge from ceff = 1 near ∆ = 0 is due to the finite size effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' These results verify that the measurement is irrelevant for K > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' For K < 1(∆ > 0), we similarly plot the fitting effective central charge ceff = 3b with respect to different ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' S4 (b), at W = 0 we always have b = 1/3, showing the central charge ceff = 1, while for W > 0, we see that the effective central charge decreases to zero, indicating that the log L behavior of the entanglement entropy breaks down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' To characterize the entanglement behavior, we explore the algebraic correction, and identify the power, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' 2 in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Besides, we also fit the prefactors a and b in the power law fitting S = a+b/Lc with respect to measurement strength W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' With (S28) and (S79), we know a and b exponentially decay with exponents which are a quadratic polynomial of √ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' S5, we plot log a and log b with respect to W 1/2 and fit them with quadratic polynomials, which show perfect fitting results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Moreover, two quadratic polynomials have coefficients of the quadratic term ( √ W)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' The distinct behaviors of entanglement entropy for ∆ > 0 and ∆ < 0 clearly reveal an entanglement transition at ∆c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' Unlike conventional transition, the scaling dimension of the tuning parameter is zero at the critical point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' This indicates 1/ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' We plot the data collapse to demonstrate this critical exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' S6, we plot half-chain entanglement entropy as a function of ∆ for different sizes L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} +page_content=' All data collapse onto a smooth function when the argument is chosen to be (∆ − ∆c) log L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFIT4oBgHgl3EQfsyss/content/2301.11337v1.pdf'} diff --git a/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf b/ndFKT4oBgHgl3EQfFS0k/content/2301.11719v1.pdf new file 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Concurrent operation with stroke minimization +R. Pourcelot1, E. H. Por2, M. N’Diaye1, H. Benard3, G. Brady2, L. Canas3, M. Carbillet1, +K. Dohlen4, I. Laginja5, J. Lugten2, J. Noss2, M. D. Perrin2, P. Petrone6, L. Pueyo2, +S. F. Redmond7, A. Sahoo2, A. Vigan4, S. D. Will8, and R. Soummer2 +1 Université Côte d’Azur, Observatoire de la Côte d’Azur, CNRS, Laboratoire Lagrange, France +e-mail: raphael.pourcelot@oca.eu +2 Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA +3 Thales Alenia Space, 5 Allée des Gabians - B.P. 99 - 06156 Cannes la Bocca Cedex – France | +4 Aix Marseille Université, CNRS, CNES, LAM (Laboratoire d’Astrophysique de Marseille) UMR 7326, 13388 Marseille, France +5 LESIA, Observatoire de Paris, Université PSL, Sorbonne Université, Université Paris Cité, CNRS, 5 place Jules Janssen, 92195 +Meudon, France +6 Hexagon Federal, Chantilly, VA 20151, USA +7 Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08540, USA +8 NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA +January 10, 2023 +ABSTRACT +Context. Wavefront sensing and control (WFSC) will play a key role in improving the stability of future large segmented space +telescopes while relaxing the thermo-mechanical constraints on the observatory structure. Coupled with a coronagraph to reject the +light of an observed bright star, WFSC enables the generation and stabilisation of a dark hole (DH) in the star image to perform planet +observations. +Aims. While WFSC traditionally relies on a single wavefront sensor (WFS) input to measure wavefront errors, the next generation +of instruments will require several WFSs to address aberrations with different sets of spatial and temporal frequency contents. The +multiple measurements produced in such a way will then have to be combined and converted to commands for deformable mirrors +(DMs) to modify the wavefront subsequently. +Methods. We asynchronously operate a loop controlling the high-order modes digging a DH and a control loop that uses the re- +jected light by a Lyot coronagraph with a Zernike wavefront sensor to stabilize the low-order aberrations. Using the HiCAT testbed +with a segmented telescope aperture, we implement concurrent operations and quantify the expected cross-talk between the two con- +trollers. We then present experiments that alternate high-order and low-order control loops to identify and estimate their respective +contributions. +Results. We show an efficient combination of the high-order and low-order control loops, keeping a DH contrast better than 5 × 10−8 +over a 30 min experiment and stability improvement by a factor of 1.5. In particular, we show a contrast gain of 1.5 at separations +close to the DH inner working angle, thanks to the low-order controller contribution. +Conclusions. Concurrently digging a DH and using the light rejected by a Lyot coronagraph to stabilize the wavefront is a promising +path towards exoplanet imaging and spectroscopy with future large space observatories. +Key words. instrumentation: high angular resolution, methods: data analysis, telescopes +1. Introduction +High-contrast imaging and spectroscopy is one of the path- +ways envisioned by the Astro2020 Decadal Survey (National +Academies of Sciences, Engineering, and Medicine 2021) to +identify Earth-like worlds in other planetary systems and search +for the biochemical signatures of life. To this aim, a telescope +and its instruments need to fulfill at least the following three +main requirements: sensitivity to observe objects with magni- +tude larger than 30, resolution to separate a planet from its host +star at angular distances shorter than 50 mas, and contrast to +disentangle the planet photons from the star glow to observe +exo-Earths around Sun-like stars with a contrast (i.e., flux ratio) +around 10-10 in visible light. +A space telescope with a primary mirror larger than 4 m +(Gaudi et al. 2020; The LUVOIR Team 2019) will have the res- +olution and sensitivity to enable the observation of a large sam- +ple of Earth-like planets (Stark et al. 2019). To save weight and +room in a spacecraft and rocket fairing, a promising option for +the telescope is to be folded thanks to the use of a segmented pri- +mary mirror, such as the recently launched James Webb Space +Telescope (Lightsey et al. 2012). To overcome the flux ratio be- +tween the host star and the planet, one of the promising options +is the use of a coronagraph. This instrument rejects the on-axis +stellar light while retaining the planet light. In the case of geo- +metrically unfriendly apertures of telescopes with pupil features +such as primary mirror segmentation, secondary mirror central +Article number, page 1 of 13 +arXiv:2301.03242v1 [astro-ph.IM] 9 Jan 2023 + +A&A proofs: manuscript no. paper +obstructions, and spider struts, the Lyot-style coronagraph (Lyot +1939) is one of the leading devices for starlight diffraction sup- +pression with arbitrary apertures (e.g. Soummer 2005; Soummer +et al. 2011; N’Diaye et al. 2016). Including the combination of a +focal-plane mask and a Lyot stop, this concept has already been +implemented in several ground-based high-contrast instruments +(Hinkley et al. 2011; Macintosh et al. 2014; Beuzit et al. 2019), +allowing the detection of several planetary companions and disks +so far (e.g., Macintosh et al. 2015; Chauvin et al. 2017; Keppler +et al. 2018) and large-scale surveys of nearby stars (e.g., Nielsen +et al. 2019; Vigan et al. 2021). +However, on top of the complicated diffraction pattern due to +telescope segmentation (Yaitskova et al. 2003), the contrast goal +of 10-10 translates into tight requirements in terms of observatory +stability, with wavefront error budgets on the order of a few hun- +dreds of picometers during the observations (Coyle et al. 2019). +To alleviate the constraints on the observatory structure, a solu- +tion is to make use of wavefront sensing and control (WFSC) +strategies. These methods consist in measuring the wavefront +errors with wavefront sensors (WFSs) and correcting for them +to obtain the desired shape with one or more deformable mir- +rors (DMs). In this context, it is possible to distinguish differ- +ent WFSC applications. One of the main goals is to dig a dark +hole (DH, Malbet et al. 1995; Bordé & Traub 2006), an opti- +mized high-contrast region in the image of an observed star to +enable exoplanet observations. This approach can also be used +to correct for perturbations originating from observatory vibra- +tions and mechanical or thermal drifts, in particular those of the +primary mirror segment alignment. These effects lead to a con- +trast degradation in the coronagraphic image of an observed star, +hindering the detection of planets. The wavefront errors cover a +wide range of spatial frequencies, from low-order, up to 4-5 cy- +cles per pupil (c/p), to high-order aberrations, at a few tens of +c/p. The wavefront errors also cover a scale of temporal frequen- +cies, from low to high range going from the mHz to the kHz +regime. Optimal correction requires the combination of simul- +taneous control loops (Pueyo et al. 2019; Guyon et al. 2020a,b; +Currie et al. 2022) to address these errors at different parts of the +spatial or temporal frequency ranges. +In the case of a Lyot coronagraph, the focal-plane mask +aims to reject the central part of the stellar point-spread func- +tion, which corresponds to the fraction of the beam that contains +the low-order content of the aberrations. We therefore explore +the use of this rejected light to feed a low-order wavefront con- +trol (LOWFC) loop that stabilizes the low-order modes of the +wavefront. To analyse the low-order information in this rejected +light, we use a Zernike Wavefront Sensor (ZWFS, Zernike 1934; +Bloemhof & Wallace 2003; Dohlen 2004; Wallace et al. 2011; +N’Diaye et al. 2013b) to take advantage of its high sensitivity +(Guyon 2005; Chambouleyron et al. 2021), the simplicity of its +wavefront reconstructor, and the ease of its hardware implemen- +tation. In Pourcelot et al. (2022) hereafter Paper I, we demon- +strated the wavefront stabilisation with a standalone closed loop, +stabilizing both the wavefront, and the DH contrast at an aver- +age and standard deviation better than 10-7 and 10-8 respectively, +under artificially introduced perturbations. While a similar ap- +proach has also been demonstrated for the Roman Space Tele- +scope (RST) coronagraphic instrument (CGI, Shi et al. 2016, +2018; Mennesson et al. 2018; Kasdin et al. 2020), enabling +deeper contrasts will require the study of the simultaneous use +of the ZWFS-based control loop combined with a high-order +wavefront control (HOWFC) loop. The algorithms used for the +HOWFC in this paper are pair-wise (PW) probing (Bordé & +Traub 2006; Give’on et al. 2007; Potier et al. 2020) for the elec- +tric field estimation in the focal plane and the stroke minimiza- +tion (SM, Pueyo et al. 2009) algorithm to compute the desired +DM commands. +In this work, we investigate a concurrent control loop ap- +proach on the High-contrast imager for Complex Aperture Tele- +scopes testbed (HiCAT, N’Diaye et al. 2013a, 2014, 2015; +Leboulleux et al. 2016; Soummer et al. 2018; Moriarty et al. +2018; Soummer et al. 2019, 2022) in monochromatic light at the +Space Telescope Science Institute (STScI) in Baltimore, USA. In +particular, we present an estimation of the interactions between +the HOWFC and LOWFC loops, and run it through various con- +figurations. These setups include switching both loops on and +off, and two variants of PW probing to estimate the impact of +LOWFC on the focal-plane wavefront estimation. In an addi- +tional experiment, we limit the maximum speed of the HOWFC +loop to allow the LOWFC loop to address more turbulence, and +emphasize its positive impact on the DH contrast. +2. Experimental setup +2.1. Optical setup +We recall the main details of the optical configuration for our +study, further details can be found in Paper I. We work on sta- +bilizing the contrast with a classical Lyot coronagraph, working +with a segmented aperture. The classical Lyot coronagraph is +composed of two masks. The first component is an amplitude +focal-plane mask that rejects the on-axis light. In our studies, it +is implemented as a pinhole in a mirror that lets the rejected light +through. The second component, the Lyot Stop, is a diaphragm in +the re-imaged pupil plane downstream of the focal-plane mask. +Usually implemented with a circular diaphragm, the Lyot stop is +slightly undersized with respect to the entrance pupil. The wave- +front corrections are performed by using two DMs upstream of +the classical Lyot coronagraph, DM1 in a pupil plane and DM2 +outside of a pupil plane, for both phase and amplitude control. +To address the low-order aberrations, we use a ZWFS in the light +rejected by the focal-plane mask to measure them and DM1 to +correct for them. Concerning the HOWFC loop, the PW esti- +mation uses several coronagraphic images taken with probes ap- +plied on DM1 to estimate the focal-plane electric field, and SM +computes commands for both DMs to generate a DH. +2.2. Implementation on HiCAT +To test and validate our approach, we use HiCAT in a configura- +tion identical to the one presented in Fig. 3 of Paper I and repro- +duced in Fig. 1, with a monochromatic laser source. The relevant +values describing the experiment implementation are summa- +rized in Table 1. The testbed is composed of several parts. First, +there is a telescope simulator that mimics the light from a star +through a segmented-aperture telescope. The following arm with +the coronagraph includes the continuous DMs, DM1 in a pupil +plane and DM2 out of pupil plane, the focal-plane mask, from +the Lyot project (Oppenheimer et al. 2004), and the Lyot Stop. +The light reflected by the focal-plane mask is sent to the imaging +camera, where some light can be picked up with a beamsplitter +to re-image the pupil plane. The light rejected by the focal-plane +mask is sent to the low-order wavefront sensing (LOWFS) arm +with three different detectors: a ZWFS, a target acquisition cam- +era, and a phase retrieval camera. In this work, we focus on the +ZWFS as the LOWFS. To isolate it from external perturbations, +HiCAT is protected by an enclosure. A flux of dried air at a con- +Article number, page 2 of 13 + +R. Pourcelot et al.: Low-order wavefront control using a Zernike sensor through Lyot coronagraphs for exoplanet imaging II +Table 1: Physical parameter values of the HiCAT testbed used +for the experiments. +Parameter +Variable +Value +Wavelength +λ +640 nm +Entrance pupil diameter +Dpup +19.55 mm +Number of segments +- +37 +DM 1&2 actuator count +- +952 +Total actuator count +q +1904 +focal-plane mask diameter +dFPM +455 µm +8.52 λ/Dpup +ZWFS mask diameter +dZ +54.3 µm +1.02 λ/Dpup +ZWFS mask depth +- +280 nm +Lyot Stop diameter +DLS +15 mm +ZWFS camera model +- +ZWO ASI178 +ZWFS camera pixel pitch +- +2.4 µm +ZWFS Region of Interest size +- +800 pix +Region of Interest number of pixels +n +64000 +ZWFS camera max framerate +- +80 Hz +Imaging camera model +- +ZWO ASI178 +stant temperature is continuously injected to keep the conditions +stable and safe for the DM operations. +2.3. ZWFS control loop +Figure 2 represents the block diagram of HiCAT for our ex- +periments and in which the ZWFS control loop is represented +in orange. The overall principle remains the same, except that +the software implementation of HiCAT has been upgraded since +Paper I (Soummer et al. 2022; Por in prep.). With the use +of a service-oriented architecture using shared memory for +low-latency inter-process communication, it is now possible to +run different testbed operations independently. The calibration +method has slightly evolved as well. Instead of poking only 12 +Zernike modes, we now consider the first m Zernike modes (ex- +cluding piston) following the Noll convention (Noll 1976), with +m = 20. The Jacobian matrix JZ of dimensions m × n between +DM1 and the ZWFS is built using 1000 draws of random combi- +nations of the first m Zernike modes. These draws are poked on +DM1, with an expected wavefront error of 10 nm Root-Mean- +Square (RMS) per poke. The control matrix C, of dimensions +n × m is then computed by inverting JZ using Tikhonov regular- +ization, and a scalar relative regularization parameter of 0.03. +The current speed limit imposed on the testbed operations is +coming from the maximum frame rate of cameras of 80 Hz. With +this control, the temporal standard deviation for each Zernike +mode coefficient is reduced to a few hundreds of picometers +RMS. The LOWFC controller is a pure integrator with a gain +of 0.01, at a loop speed of 80 Hz, limited by the camera readout +speed. +2.4. Dark hole digging with stroke minimization +DH digging usually relies on two steps: the measurement of the +electric field in the DH region in the focal plane and the com- +putation of the DM commands that will efficiently remove the +light in the DH. HiCAT currently relies on PW for the determi- +nation of the electric field in the focal plane. PW probing applies +a set of DM commands, called probes, that modulate the energy +distribution on the imaging camera, and especially in the DH +area. By using probes with enough diversity, a testbed model +and by solving a linear problem between the DM commands and +the focal-plane electric field, an estimation of both the ampli- +tude and the phase of the electric field is obtained in the focal +plane. The choice of the probes is made by considering several +parameters, such as the area in the science image we want to +control. Two sets of probes are currently used on HiCAT for PW +performance studies. The most basic probes are made by poking +single DM actuators, hereafter single-actuator probes. The sec- +ond set called “HiCAT probes” can be computed by solving an +optimization problem to find the DM command that produces a +specific modulation of the electric field in the DH (Will et al. +2021). +By using the PW estimation and a DM-to-focal-plane rela- +tion such as a Jacobian matrix built between the DMs and the +focal-plane electric field, it is possible to perform an optimiza- +tion of the DM commands to minimize the energy in the DH. +Several algorithms exist, among which SM (Pueyo et al. 2009; +Mazoyer et al. 2018b,a) that is implemented on HiCAT. An al- +ternative algorithm is electric field conjugation (EFC, Give’On +et al. 2007) that aims at solving a similar problem with a dif- +ferent energy minimization strategy. If the electric field conju- +gation algorithm requires less computation time and therefore +runs faster than SM, it usually provides larger actuator strokes +on the DMs. Alternative strategies have successfully been im- +plemented and tested on HiCAT (e.g., Will et al. 2021) but they +are not considered here. Fig. 3 shows an example of DHs gen- +erated with SM during HiCAT experiments with an outer work- +ing angle of 12.8 λ/Dpup and two different inner working angles +(IWA). While the contrast of 2 × 10−8 could only be reached +with a large IWA of 7.6 λ/Dpup in Paper I, the new, fast soft- +ware architecture allows for a loop frequency gain of a factor of +10 and therefore a better compensation of the turbulence in the +HiCAT environment. This enables the digging of a DH with the +same contrast like previously but with a reduced IWA down to +4.6 λ/Dpup. This is the value set for the IWA of the DH used in +the rest of the paper. +3. Interactions between the loops +When running two wavefront control loops in parallel, we want +to avoid any cross-talk between the processes. As the experiment +goes on, SM is going to refine the DM commands to reduce the +intensity in the DH. From the ZWFS point of view, it means the +application of a new DM offset at each iteration. The sensitivity +loss of the ZWFS due to this offset change has been addressed +in Paper I in the case of a large IWA of 7.6 λ/Dpup, and a sim- +ilar behavior is observed with an IWA at 4.6 λ/Dpup. Overall, +this offset change does not prevent the ZWFS control loop to +efficiently measure the low-order aberrations. +Conversely, we want to avoid the ZWFS control loop to cor- +rect for SM updates or for the probes introduced by PW. The +success of this will depend on the response measurements of +these commands by the ZWFS. In our configuration with con- +trast levels of 10-8, we estimate the projection of the PW probes +or SM commands on the modes controlled by the LOWFC loop +using the following procedure. For a given command v control- +ling both DMs, of dimensions 1 × q, we project v on the Zernike +polynomial basis that is used to compute JZ, leading to the term +v// of dimensions 1 × m. As JZ is a model of our system, we +can therefore estimate the response on the ZWFS detector bimg +Article number, page 3 of 13 + +A&A proofs: manuscript no. paper +Fig. 1: Simplified HiCAT layout in a semi-transmissive representation. The IrisAO segmented DM creates the segemented aperture. +The apodizer, in the top-right corner is currently replaced by a flat mirror. The cameras used in this paper are the Zernike WFS +camera, in the light transmitted by the focal plane mask (FPM), and the imaging camera in the light reflected by the FPM. +of size n by computing bimg = JZv//. Finally, by using the con- +trol matrix C, the pseudo-inverse of the Jacobian matrix J, we +can assess the command ˆv sent by the ZWFS control loop to +DM1 to correct for the perturbation induced by v before apply- +ing the control gain. This estimation can also be used to avoid +the correction of SM commands and PW probes by the ZWFS, +as showed later in Sec. 4.3. +These estimations are given in Fig. 4 for a pure Zernike +mode, a single SM update after convergence of the DH contrast, +and a correction for a single perturbation measured by the ZWFS +while in closed loop. For all the estimated commands, we use +the same colorbar and we display the amplification factor ap- +plied to each command in the plot to match the colorbar scale. +In this configuration, the SM update triggers a faint response of +the LOWFC loop. This result is expected as the SM command +contains small DM strokes and it is computed to modulate the +DH intensity. Since the DH reaches in to the edge of the focal- +plane mask, its corresponding spatial frequencies differ from the +spatial frequencies that are seen by the ZWFS. In comparison +with the SM commands, the command sent by the LOWFC loop +to correct for turbulence is 10 times fainter in peak-to-valley but +triggers a response with a peak-to-valley value which is 20 times +larger than the SM command. In practice, we expect the com- +mand triggered by the ZWFS for a SM update to be even smaller +because we use two DMs on HiCAT to correct for phase and +amplitude. In the case of a classical Lyot coronagraph with a +segmented aperture, the amplitude correction dominates over the +phase correction due to the segment gaps in our pupil. DM2 will +be used to correct for these amplitude errors, introducing both +the desired amplitude correction and undesired phase correction +which is compensated by DM1 (Mazoyer et al. 2018a). +The other conflicting commands with the ZWFS arise from +the probes introduced by PW. Following the same principle as +for Fig. 4, Fig. 5 shows examples of commands triggered by the +ZWFS loop when single-actuator and HiCAT probes are sent to +DM1. To match the plots with the same color bars, the DM re- +sponse have been amplified by a factor of 100. Overall, these re- +sponses are also small, with the same order of magnitude as the +single ZWFS turbulence correction. Similarly to the SM com- +mands, the HiCAT probes are less seen by the ZWFS than the +single-actuator probes: they are optimized to modulate the DH +intensity, which means they are targeting wavefront error spatial +frequencies beyond the ZWFS range. +Overall, the commands introduced by SM and PW present a +limited impact on a single ZWFS control-loop command. Fur- +thermore, on HiCAT, the PW estimation runs at 80 Hz which is +close to the frequency of the LOWFC loop, and SM at 0.4 Hz +with an iteration every 2.5 s. This translates into approximately +200 LOWFC loop iterations for one SM command. With a con- +trol gain of 0.01 and considering the ZWFS amplification factors +in Figs. 4 and 5, the ZWFS control loop is therefore unlikely to +disturb the probes and the SM updates by more than 0.01% and +0.1%, respectively. As a result, we choose to disregard these per- +turbations for the experiments shown in Sec. 4.2 with a 10-8 con- +trast regime. Further analysis of the PW estimation degradation +due to LOWFC would be required for deeper contrasts but this +is beyond the scope of this paper and will be addressed in future +work. +Article number, page 4 of 13 + +Light + Sources +Segmented Telescope Simulator +Apodizer + Laser diode +: Monochromatic +Iris-AO Segmented DM +Beam +Pupil Mask +Supercontinuum +Filter +ND +Launcher +laser source +Wheel +Wheel +Broadband +Zernike WFS and Tip/Tilt Sensing +Coronagraph and Wavefront Control +FPM +Flip Mirror +DM2 +DM1 +Zernike +Phase Retrieval +Target Acq +Mask +Camera +Camera +"Science Instrument" +Lyot Stop +Low-order +Zernike WFS +phase retrieval +Camera +Pupil +Camera +Camera +Imaging +Linear +Camera +PolarizerR. Pourcelot et al.: Low-order wavefront control using a Zernike sensor through Lyot coronagraphs for exoplanet imaging II +Fig. 2: Simplified block diagram of HiCAT with the control loops and their interactions for our experiment. The IrisAO segmented +DM creates the segmented aperture. The LOWFC sends commands to DM1 only, while the HOWFC sends commands to both DMs. +Loops can be closed or opened during the operations. +10 +0 +10 +Separation ( /Dpup) +10 +0 +10 +Separation ( /Dpup) +IWA: 7.6 /Dpup +10 +0 +10 +Separation ( /Dpup) +10 +0 +10 +IWA: 4.6 /Dpup +10 +8 +10 +7 +10 +6 +Contrast +Fig. 3: DH examples with an outer working angle of +12.8 λ/Dpup, with an IWA of 7.6 λ/Dpup (left) and 4.6 λ/Dpup +(right). The blue circles define the edges of the controlled DH. +The black dashed circle delimits the focal-plane mask area. The +residual DH speckles are removed with a faster HOWFC loop +thanks to the new HiCAT software architecture. +4. Parallel operations of low-order and high-order +wavefront control loops +4.1. Control loops architecture +In this section, we present the results of simultaneous operations +of the DH digging process with PW and SM, and ZWFS con- +trol. Considering the conclusions of Sec. 3, we run the loops in +a completely asynchronous manner. The principle of these op- +erations is detailed in Fig. 6, where both loops run in a separate +process and send commands to a virtual DM channel. Another +process then sums the commands sent to the channels and ap- +plies the final command to the DMs. Two configurations are ex- +plored for PW: using single-actuator or HiCAT probes. For both +of these experiments, the protocol is the same: we pre-compute +a DH solution using a fast electric field conjugation algorithm +that manages to provide DM solutions yielding an averaged con- +trast smaller than 5 × 10−8 in the DH. The experiment then goes +through 5 parts: (0) SM runs alone; (1) SM runs in parallel with +ZWFS; (2) SM runs alone again; (3) SM runs in parallel with +ZWFS again; (4) SM is stopped while ZWFS is running. During +(4), for each contrast measurement, another measurement is per- +formed with the ZWFS loop opened and DM command reset to +the first command of (4). All along the experiments, a measure- +ment of the direct flux without the focal-plane mask is performed +every 10 iterations to get an accurate contrast normalization. +Recent studies of the testbed stability have identified a warm +flip-mount motor generating the air turbulence that was charac- +terized in Paper I. After moving this component, the beam is +now more stable without any noticeable contrast drift during ex- +periments running longer than 1 h, even without control and with +contrasts around 3 × 10−8. To study the behavior of the ZWFS +control loop with DH digging in a less favorable environment, +these two experiments are performed with a 10 × 50 cm2 open- +ing in the HiCAT enclosure, creating extra turbulence within the +testbed. In addition, to keep the DMs safe during the operations, +the dry air flow injected in the bench has been increased to com- +pensate the influx of humid air through the opening in the en- +closure and stabilize the environment humidity and temperature +in the turbulent airflow on testbed. Examples of power spectral +density of the turbulence for the tip and tilt modes is presented +in Fig. 7. They show a turbulent motion in the low frequency +range below 1 Hz that we associate to turbulence. Perturbations +at higher frequencies around 10 Hz are also present and are as- +sociated to vibrations of the testbed. They are not specifically +linked to the generated turbulence. The behavior is very similar +with the other Zernike modes but almost two orders of magni- +tude smaller. +Article number, page 5 of 13 + +TTTIT +LOWFC +Segmented +Science +DM +camera +DM1 +HOWFC +Zernike +Wavefront +LS +sensor +o +DM2 +Focal plane +maskA&A proofs: manuscript no. paper +Input perturbation +on DM1 +Jacobian +Zernike poke +20 +10 +0 +10 +20 +SM update +0.2 +0.0 +0.2 +ZWFS update +0.1 +0.0 +0.1 +Expected OPD +(nm RMS) +Projected response +from LOWFC +x 1.0 +20 +10 +0 +10 +20 +x 5268 +20 +10 +0 +10 +20 +x 339 +20 +10 +0 +10 +20 +Expected cmd (nm) +Fig. 4: Examples of different perturbations introduced on DM1 (top) and their corresponding LOWFC loop responses on DM1 +(bottom), obtained by projecting the command on the controlled basis of the LOWFC loop and using the control matrix to estimate +the feedback command. From left to right: the Zernike mode Z4 used for the interaction matrix calibration, a single update from +SM after convergence of the DH digging, and a command update from the LOWFC loop to correct for residual turbulence in closed +loop. In the bottom row, all the displayed commands are scaled to fit the same color bar. From left to right, the commands have been +multiplied by 1, 5268 and 339 respectively to be displayed with the same color bar, and the respectively applied amplification factor +is displayed on top of each command. The commands shown here expand over the whole controllable area of DM1. The hexagons +represent the projection of the segmented aperture onto DM1. +On DM1 +Single act #1 +Single act #2 HiCAT probe 1 HiCAT probe 2 +10 +0 +10 +Expected DM WFE +(nm) +Projected response +from LOWFC +10 +0 +10 +Projected cmd (nm) +x 100 +Fig. 5: Examples of different PW probes introduced on DM1 (top) and their corresponding LOWFC loop responses on DM1 +(bottom) obtained by projecting the command on the controlled basis of the LOWFC loop and using the control matrix to estimate +the feedback command. From left to right: two examples of single-actuator probes poked by 10 nm in wavefront error; two of the +4 HiCAT probes designed to specifically modulate the electric field in the DH. To match the top and bottom row color bars, the +bottom commands have been scaled by a factor of 100. The commands shown here are over the whole controllable area of the DM1. +The hexagons represent the projection of the segmented aperture onto DM1. +Article number, page 6 of 13 + +R. Pourcelot et al.: Low-order wavefront control using a Zernike sensor through Lyot coronagraphs for exoplanet imaging II +Fig. 6: Vertical timeline of the parallel operations on HiCAT +with three processes. The DMs are operated by a process that +reads from all the different channels (LOWFC, PW and HOWFC +here) and apply the shape corresponding to the sum of the differ- +ent contributions. The LOWFC process starts by taking an initial +reference, and then reads images from the camera and computes +corrections sent to the LOWFC DM channel. The HOWFC reads +runs PW probing, using a dedicated DM channel, and computes +DM corrections with SM for the HOWFC channel. It can also +interrupt the LOWFC by clearing the channel for specific acqui- +sitions without LOWFC. Each process runs independently from +the other. Data is transferred from cameras or to DMs through +shared memory. +10 +2 +10 +1 +100 +101 +10 +7 +10 +5 +10 +3 +10 +1 +101 +PSD (nm2/Hz) +Tip +Closed enclosure +Open enclosure +10 +2 +10 +1 +100 +101 +Frequency (Hz) +10 +7 +10 +5 +10 +3 +10 +1 +101 +PSD (nm2/Hz) +Tilt +Fig. 7: Examples of power spectral density (PSD) of the tip (top) +and tilt (bottom) perturbations at the level of the ZWFS on Hi- +CAT. For both modes, a measurement was performed with all the +panels closed (in blue) and with an open panel of the enclosure +and a high dry air flow (orange). +4.2. Results +The mean DH contrast is presented as a function of the iterations +in Figs. 8 and 9 for the single-actuator and HiCAT probe scenar- +ios. Examples of DH images extracted from the two experiments +are given in Fig. 10, with the image yielding the best DH con- +trast for intervals (0) to (3) and the respective last images for +intervals (4). The corresponding contrast statistics are detailed +in Table 2. For both probe types, the HOWFC loop manages to +correct for most of the wavefront errors, yielding a fairly sta- +ble contrast in the intervals (0) to (3), especially in the intervals +(0) and (2) where no LOWFC loop is running. Conversely, large +drifts are visible in (4), where we observe a contrast loss of an +order of magnitude compared to sections (0)-(3). Consistently +with the previous blind DH stabilization (Paper I), the LOWFC +loop slows down the contrast drift. From the difference between +intervals (0) and (2), and intervals (1) and (3), we draw several +conclusions. +First, we show a HOWFC and LOWFC loop combination +that manages to keep the DH stable at a contrast of ∼ 5 × 10−8 +under large natural drifts. Concerning the standard deviation σt, +it is typically maintained at an order of magnitude lower than the +averaged contrast, and with σt = 5 × 10−9 over 25 min of an ex- +periment. This asynchronous implementation does not introduce +conflicts between the two controls. At slightly better contrasts, +the “Très Haute Dynamique 2” testbed is routinely operating a +fast low-order correction loop (Galicher et al. 2020). However, +their approach uses the light reflected by the Lyot stop and it +addresses only the control of tip and tilt on a separate steering +mirror. Contrasts deeper than 10-8 have also been obtained by +RST/CGI (Zhou et al. 2019, 2020) with a HOWFC loop includ- +ing a LOWFC loop (Shi et al. 2016, 2017, 2018), but our ap- +proach differs in several aspects. (i) While RST relies on a mono- +lithic primary mirror, HiCAT implements a segmented aperture +Article number, page 7 of 13 + +Initial ref +PW cmd +Acquistion +New cmd +Probe 1 +cmd +Acquistion +New cmd +pW cmd +cmd +Acquistion +Probe 2 +New cmd +Acquistion +pW cmd +New cmd +... +clear PW +Solving +electric field +Stroke +minimization +cmd +Coron. +image +With LOWFC +Clear LO +Coron. +image +No LOWFC +Restore LO +Iteration n+1A&A proofs: manuscript no. paper +0 +100 +200 +300 +400 +500 +600 +700 +Iteration # +2 × 10 +8 +4 × 10 +8 +7 × 10 +8 +10 +7 +2 × 10 +7 +Mean contrast in the DH +0 +1 +2 +3 +4 +SM off +SM only +No control +SM + ZWFS +ZWFS only +Fig. 8: Mean DH contrast while PW with single-actuator probes and SM are running, without LOWFC in blue in section (0) and +(2), and with it in red in sections (1) and (3). SM is turned off in section (4). In (4), the mean contrast is measured while LOWFC +is running. Once per contrast measurement, the LOWFC is interrupted and reset to its initial command in (4) to get an open-loop +measurement, plotted in black. When on, the HOWFC loop runs at 0.4 Hz and the LOWFC loop at 80 Hz. +0 +100 +200 +300 +400 +500 +600 +700 +Iteration # +2 × 10 +8 +4 × 10 +8 +7 × 10 +8 +10 +7 +2 × 10 +7 +Mean contrast in the DH +0 +1 +2 +3 +4 +SM off +SM only +No control +SM + ZWFS +ZWFS only +Fig. 9: Same as Fig. 8 but using PW with HiCAT probes instead of single-actuator probes. +with an IrisAO deformable mirror. (ii) Although the tests on RST +have been performed with a fast LOWFC loop at frequencies up +to 1 kHz, this loop only addresses tip/tilt (Z2, Z3) at this speed +with a steering mirror to control them. The other modes are +controlled at much slower rates, around 0.2 Hz for defocus (Z4) +with their DM2 and 5 mHz for other modes (Cady et al. 2017; +Shi et al. 2017; Seo et al. 2017, 2018) with their DM1. At the +same time, the HOWFC loop on RST is running on both DMs +at around 0.1 Hz, limiting the possible cross-talks with LOWFC. +The RST tests were also performed on a testbed located in a vac- +uum, where it is only perturbed by known artificial drifts with a +few Zernike modes while we operate our testbed in air and under +strong turbulence with an opened enclosure. (iii) The RST/CGI +observation mode is designed to dig a DH on a bright star and +Article number, page 8 of 13 + +R. Pourcelot et al.: Low-order wavefront control using a Zernike sensor through Lyot coronagraphs for exoplanet imaging II +Table 2: Mean DH contrast statistics over the different intervals for the single-actuator probes experiment and HiCAT probes +experiment. +Situation +Stat +0 +1 +2 +3 +4 with ZWFS +4 no ZWFS +ZWFS +off +on +off +on +on +off +SM +on +on +on +on +off +off +Single-actuator probes +Mean +4.5e-08 +3.6e-08 +4.0e-08 +3.4e-08 +5.4e-08 +1.4e-07 +σt +6.0e-09 +3.6e-09 +4.7e-09 +3.3e-09 +2.1e-08 +1.3e-07 +HiCAT probes +Mean +4.5e-08 +4.7e-08 +5.3e-08 +4.8e-08 +7.0e-08 +1.3e-07 +σt +4.7e-09 +3.6e-09 +6.7e-09 +5.9e-09 +1.5e-08 +7.3e-08 +then slew to the science target, using the LOWFC loop only to +correct for perturbations. In our case, we aim at demonstrating +the continuous use of both loops while observing. +Second, the contrast gain in both temporal average and stan- +dard deviation from Table 2 are moderate but noticeable, espe- +cially in the single-actuator probe experiment, reaching a mean +contrast of 3.4 × 10−8 and a value of σt of 3.3 × 10−9 over the in- +terval (3). While we expect the contrast with LOWFC to be bet- +ter because of the turbulence that introduces a lot of low-order +aberrations, the new implementation of the HOWFC control loop +runs fast enough to handle a large part of the low-order cor- +rections. From our experience with electric field conjugation on +HiCAT, digging a DH with faster iterations allows even deeper +contrasts in the presence of turbulence. Therefore, the contrast +we observe is very likely to be driven by the internal turbulence +that evolves faster than what the HOWFC loop can handle in +the high-order modes. Further experiments are required to fully +understand the limitations in these conditions. +Third, even if the temporal contrast averages are moderate, +the ratio of contrast between the cases without and with LOWFC +as a function of angular separation in the focal plane emphasizes +the action of the loop. The corresponding curves are displayed +in Fig. 11. For each probe, the plots show the ratio between the +azimuthally averaged contrast over sections (0) and (2), and over +sections (1) and (3). The maximum contrast improvement by a +factor of up to 1.5 is located at separations around 4.6 λ/Dpup, as +expected - this corresponds to the edge of the focal-plane mask, +and therefore the theoretical sensing limit of the ZWFS through +the Lyot mask. Since the LOWFC loop controls a limited number +of 20 Zernike modes here, its impact is negligible at larger sepa- +rations. The same plot is also displayed for section (4) in Fig. 11 +bottom plot, showing the same behavior, but with a much larger +gain up, to a factor of 5. This is consistent with the first point +above: in this experiment, both loops control overlapping spatial +frequencies of the wavefront error without cross-talk. The gains +at larger separations are negligible here as well. +Finally, the experiments have slightly better results with +single-actuator probes, even though their interaction with the +LOWFC loop was supposed to be greater with the HiCAT +probes. However, these single-shot experiments do not allow us +to conclude on any significant difference between the two types +of probes, the different behaviors in Fig. 8 and 9 being possibly +due to external factors such as turbulence. The real limitation +due to loop interactions will probably appear when working at +deeper contrasts. We are still investigating the exact limitations +on HiCAT of the electric field estimation by PW. These inter- +actions could degrade wavefront estimation that would result in +reaching a contrast floor at some point. Therefore, using probes +as invisible as possible to the ZWFS could prove of interest in +the future. +4.3. Experiment with slower high-order wavefront control +loop +The gain provided by the ZWFS depends not only on the in- +put turbulence, but also on the amount of low-order aberrations +the HOWFC loop corrects for. By degrading the correction per- +formed by the high-order controller we can emphasize the im- +provement due to the parallel use of the ZWFS. This also de- +grades the contrast performance by an order of magnitude, and +is not representative of the current operations of HiCAT. +For this goal, we use HiCAT in a different setup, with +a HOWFC loop running 8 times slower, at 0.05 Hz and the +LOWFC loop around 2.5 times slower at 30 Hz. In this exper- +iment, the HiCAT probes are used for PW. With the slower +HOWFC and similar air turbulence, the contrast in the DH is +stuck above 10-7. The speed difference between the two con- +trollers now being much larger, we expect the cross-talk between +them to be more challenging. For this experiment, we have im- +plemented a communication between the loops to offload the +HOWFC loop commands to the LOWFC reference: each time +PW or SM sends a command on the DMs, the command is sent to +the LOWFC loop as well. The command is then projected on the +controlled modes of the LOWFC loop and multiplied by the Ja- +cobian matrix JZ to estimate the corresponding variations on the +ZWFS signal. While there are many ways for improvement here, +our code modifications for synchronisation add overheads in the +computation and slow down the LOWFC loop from 80 Hz to +30 Hz. As a result we increase its gain from 0.01 to 0.15, a value +that empirically proved to be efficient. Considering the loop fre- +quencies, the LOWFC loop produces around 600 corrections in +a single HOWFC loop iteration, while with the previous setup it +was only around 200. +The contrast curve is presented in Fig. 12. In this configura- +tion, the LOWFC loop proves very efficient. While in interval (0) +the contrast reaches a floor at 1.9 × 10−7, turning on the LOWFC +loop in (1) allows for an almost immediate contrast improvement +by a factor of 1.5. This behavior is similar with the intervals (2) +and (3). During interval (4), when the HOWFC loop is turned off, +the drift of the uncontrolled DH is immediate, while the LOWFC +loop manages to slow it down, limited by the reduced number of +controllable modes. Overall, the contrast standard deviation over +the intervals proves also to be greatly improved, from around +1.5 × 10−7 to 1.5 × 10−8 without and with LOWFC. The az- +imuthally averaged gain is represented in Fig. 13, showing a gain +of 2.5 and 6 with and without SM at short separations as in the +previous experiments. But it also shows a gain larger than 1.5 at +all separations, whether SM is on or off. +Under these different experimental conditions, we also show +a successful concurrent operation. Similarly to the results in +Sec. 4.2, most of the improvement by the LOWFC is located at +the inner radius of the DH, improving the contrast at short sep- +arations. By comparing the two setups, it is possible to extract +Article number, page 9 of 13 + +A&A proofs: manuscript no. paper +SA probes +Best (0) +HiCAT probes +Best (1) +Best (2) +Best (3) +Last (4) +10 +8 +10 +7 +10 +6 +10 +5 +Contrast +Fig. 10: Examples of focal-plane DH images for the experiment with single-actuator probes (top) and HiCAT probes (bottom). For +intervals (0) to (3), this corresponds to the image yielding the best mean spatial contrast in the respective operational sections in +Figs. 8 and 9. The image for interval (4) is the last one from the open-loop data, shown in black in both figures. +0.0 +0.5 +1.0 +1.5 +2.0 +Gain (No ZWFS / with ZWFS) +IWA +OWA +With SM +Single act probes +HiCAT probes +4 +6 +8 +10 +12 +14 +Separation /D +2.5 +0.0 +2.5 +5.0 +7.5 +10.0 +Gain (No ZWFS / with ZWFS) +No SM +Fig. 11: Contrast gain due to ZWFS as a function of angular +separation in the DH, for experiments with single-actuator and +HiCAT probes. Solid lines represent the ratio of the contrast az- +imutal average between the cases without and with LOWFC. The +filling colors show the standard deviation of the measurement. +The dashed horizontal lines are drawn at a gain of 1 in both +plots. The top plot represents the values of the intervals (0) and +(2) over the values of the intervals (1) and (3). The bottom plot +corresponds to the values of the interval (4). The gain lower than +1 at the OWA is due to edge effects in the numerical computation +of the contrast. +properties of the control loops. In both setups, when the control +loops are off, the contrast drift is immediate, emphasizing the +presence of the perturbation which is partially corrected when +the LOWFC loop is on. +The remaining contrast drift is due to the higher order pertur- +bation. This perturbation is a main driver of the averaged contrast +in the DH, and is out of reach for the LOWFS. This is particularly +visible when looking at the intervals (0) in both setups, where +the contrast levels only depend on the HOWFC speed. In similar +conditions, the faster setup reaches a contrast level deeper than +the slower setup by almost an order of magnitude, showing the +impact of loop speed with the air turbulence on HiCAT. +With a faster speed than the HOWFC loop, the LOWFC loop +can therefore correct for the low-order turbulence that evolves +too fast for the high-order controller. As the leftover is more im- +portant in the slower setup, the contrast gain due to the ZWFS is +greater and more obvious. On the contrary, with the faster setup, +since the HOWFC is more efficient at correcting low-order aber- +rations, the gains are more moderate. Overall, there is a gain in +using the LOWFS in both setups, showing that the loop combi- +nation does not generate cross-talks at the contrast levels in our +experiments. +5. Conclusions +Wavefront error stability is a key parameter in the success of fu- +ture space missions with exoplanet imaging capabilities. To im- +prove the stellar light rejection by the coronagraph, WFSC ap- +pears essential to alleviate the requirements on the observatory +stability for high-contrast observations. In particular, WFSC has +been proven efficient to correct for different kinds of aberrations +such as low-order drifts, or improve the image plane DH by opti- +mizing the DM shapes. However, this was only done sequentially +so far. Following the demonstration of DH stabilization with a +ZWFS-based LOWFC loop in the rejected light of a Lyot coron- +agraph in Paper I, we have here studied its concurrent operation +with a HOWFC loop using different WFSs but the same DM. +Using the HiCAT testbed for our experiments, we first study +the impact of PW probes and SM commands on the LOWFC +loop by projecting these commands on its controlled modes. We +Article number, page 10 of 13 + +O0R. Pourcelot et al.: Low-order wavefront control using a Zernike sensor through Lyot coronagraphs for exoplanet imaging II +0 +200 +400 +600 +800 +1000 +Iteration # +10 +7 +3 × 10 +7 +6 × 10 +7 +10 +6 +Mean DH contrast +0 +1 +2 +3 +4 +SM off +SM only +No control +SM + ZWFS +ZWFS only +Fig. 12: Evolution of the mean contrast in the DH during HOWFS (using SM) as a function of iterations during different intervals: +without LOWFC in blue in section (0) and (2), and with it in red in sections (1) and (3). SM is turned off in section (4). In (4), the +mean contrast is measured while LOWFC loop is running (green). Once per contrast measurement, the LOWFC loop is interrupted +and reset to its initial command in (4) to get an open-loop measurement, in black. When on, the HOWFC loop runs at 0.05 Hz and +the LOWFC loop at 30 Hz. +4 +6 +8 +10 +12 +14 +Separation /D +0 +2 +4 +6 +8 +Gain (No ZWFS / with ZWFS) +IWA +OWA +With SM +No SM +Fig. 13: Contrast gain as a function of angular separation in the +DH when the HOWFC loop is slowed down to 0.05 Hz on pur- +pose. Solid lines represent the ratio of the azimuthally averaged +contrast between the cases without and with LOWFC. The fill- +ing colors show the standard deviation of the measurement. The +dashed horizontal line is drawn at a gain of 1. The blue curve +represents the gain between the values of the intervals (0) and +(2) over the values of the intervals (1) and (3). The yellow curve +corresponds to intervals (4) values. +find a limited response, and considering the gain of the LOWFC +loop of 0.01 as well as the different loop frequencies on HiCAT, +the loop response to a single command is unlikely to modify the +original command by more than 0.1%. We find similar results +for two kinds of probes for PW, whether single actuators pok- +ing or HiCAT probes. Then we run the HOWFC and LOWFC +loops to dig or stabilize a focal-plane DH under the conditions +of increased perturbation on the testbed. While the first loop is +on, we maintain the contrast levels around 5 × 10−8, showing no +cross-talk between the two control loops. Even if the main DH +contrast driver is due to high-order modes that evolve too fast for +the HOWFC loop, it is possible to observe a contrast improve- +ment with the LOWFC loop, with a factor of up to 1.5, and a +factor of 5 when the HOWFC loop is on and off. To prove the ef- +ficiency of the LOWFC loop, we compare the performances with +a reduced speed of the HOWFC loop. With the same IWA of +4.6 λ/Dpup this greatly degrades the achievable contrast, but em- +phasizes the capability of both loops to work together. A classi- +cal Lyot coronagraph with a smaller focal-plane mask and there- +fore a smaller IWA will be more sensitive to low-order aberra- +tions, leading to an increased need for the LOWFC loop. Further +investigations are required to find the best functioning point be- +tween the HOWFC and LOWFC loops for this setup. Prelimi- +nary tests have already been successfully performed on HiCAT +with a knife-edge coronagraph, showing promising contrast sta- +bility results at shorter separations which will be detailed in a +forthcoming paper (Por in prep.). +We validated our approach in a specific environment, in +which the PW sensing runs at temporal frequencies very close +to the LOWFC loop at 80 Hz with a loop gain of 0.01 and a SM +iteration every 2.5 s. On average, a single LOWFC loop correc- +tion is applied at each PW probe. However, when we reduce the +HOWFC loop speed, we manage to achieve a contrast of 10−7 +with a very simple loop synchronization to avoid cross-talk. The +asynchronous operations tend to converge toward DH solutions +that are dependent on the LOWFC loop contribution, while we +would prefer the two control loops to be as independent as pos- +Article number, page 11 of 13 + +A&A proofs: manuscript no. paper +sible. We currently operate HiCAT around contrasts of 5 × 10−8. +These loop interactions could create a bias in the wavefront es- +timation, thus representing a possible limitation when pushing +the contrast to lower values. Regarding these points, it might be +worth re-evaluating the asynchronous operations and consider- +ing an improved communication. This will enable adapting the +LOWFC loop reference depending on the commands sent by the +HOWFC loop, in a similar fashion as Guyon et al. (2020b). In +the scope of combining more loops, the asynchronous approach +will remain applicable as long as there is no cross-talk between +the loops. Further investigations on the loop communication will +otherwise be required as the number of communications for of- +floading grows quadratically with the number of loops. +We conducted our experiments in idealized conditions. First +the tests are done in monochromatic light. While the ZWFS is +expected to work relatively well with broadband light (N’Diaye +et al. 2013b), the DM DH commands will be different from those +in monochromatic light. In further studies, we will investigate +the stability of the parallel loop operations in broadband light, +pending the delivery of a new broadband source. Second, our +work is made with enough photons to correct as much turbu- +lence as possible, the main limitation being the camera max- +imum frame rate. Assuming a JWST-like primary mirror, the +equivalent magnitude of the monochromatic source on HiCAT +would be about -6 in the visible. This is clearly an ideal case and +the impact of signal-to-noise ratio on ZWFS measurements re- +mains to be explored. We will investigate these aspects further, +following Sahoo et al. (2022) who recently developed a novel ap- +proach to determine the optimal wavefront sensor exposure time, +for a given contrast requirement at a given stellar magnitude. +A flux limitation will also likely increase the relevance of +the LOWFS. In a photon-limited regime, a reduced photon flux +requires longer exposure times on the science camera to keep +the signal-to-noise ratio constant, leading to a possible reduction +of the HOWFC efficiency. Since the ZWFS uses the light from +the core of the source image, it collects more photons than the +science camera. Its associated loop will therefore be able to run +faster and correct for the aberrations beyond the HOWFC tempo- +ral bandpass. As emphasized by the configuration in Sec. 4.3, the +low-order controller can then be complementary with the high- +order controller. +Finally, these results have been obtained by degrading the +HiCAT environment to generate larger drifts in an uncontrolled +DH. In this context, we do not reach the optimal performance +of HiCAT that is way more stable in nominal conditions. In +particular, with the injected turbulence, here we are limited in +contrast by the speed of the HOWFC loop that runs at 0.4 Hz. +Using LOWFC, the contrast improvement is limited to separa- +tions close to DH IWA, where the control regions of both loops +is smoothly overlapping. It would be relevant to study how far +out in the focal plane the LOWFC loop could effectively control +by considering more than the current 20 Zernike modes. This +could allow a gain in the temporal bandpass of correction as the +LOWFC loop runs orders of magnitude faster than the HOWFC +loop and alleviates the latter. +At the moment, our demonstration of the concurrent use of +LOWFC and HOWFC loops at the 5× 10-8 contrast point repre- +sent a first milestone towards concurrent operations with a 10-10 +contrast goal. Several aspects in our current experiment such as +the selected control modes, the accuracy of the DM behavior +modeling, or the sensor sensitivity, will be further explored in +the regime of wavefront error fluctuations down to the picomet- +ric level to further advance exo-Earth imaging with concurrent +loops. +This combination of several control loops is a necessary step +toward a system-level demonstration of a future high-contrast in- +strument for exoplanet imaging with a future large space obser- +vatory. To stabilize the whole range of aberrations that can dis- +turb the observations, these control loops have to be associated to +others. These include a loop dedicated to tip/tilt correction, with +a dedicated steering mirror already implemented on HiCAT but +currently unused, another to primary mirror segment alignment +or one dedicated to vibration rejections. Fully operating and un- +derstanding this setup will help develop future instrumentation +for exoplanet and more particularly exo-Earth imaging. +Acknowledgements. R.P. acknowledges PhD scholarship funding from Région +Provence-Alpes-Côte d’Azur and Thales Alenia Space. The authors are espe- +cially thankful to the extended HiCAT team (over 50 people) who have worked +over the past several years to develop this testbed. 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D., et al. 2019, in Society of Photo-Optical Instru- +mentation Engineers (SPIE) Conference Series, Vol. 11117, Society of Photo- +Optical Instrumentation Engineers (SPIE) Conference Series, 111170H +Article number, page 13 of 13 + diff --git a/ntE1T4oBgHgl3EQfhgSw/content/tmp_files/load_file.txt b/ntE1T4oBgHgl3EQfhgSw/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d3e38d842778a7298964b5f6dfaa64ec4d398d19 --- /dev/null +++ b/ntE1T4oBgHgl3EQfhgSw/content/tmp_files/load_file.txt @@ -0,0 +1,1111 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf,len=1110 +page_content='Astronomy & Astrophysics manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' paper ©ESO 2023 January 10, 2023 Low-order wavefront control using a Zernike sensor through Lyot coronagraphs for exoplanet imaging: II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Concurrent operation with stroke minimization R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Pourcelot1, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Por2, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' N’Diaye1, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Benard3, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Brady2, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Canas3, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Carbillet1, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Dohlen4, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Laginja5, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Lugten2, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Noss2, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Perrin2, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Petrone6, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Pueyo2, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Redmond7, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Sahoo2, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Vigan4, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Will8, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Soummer2 1 Université Côte d’Azur, Observatoire de la Côte d’Azur, CNRS, Laboratoire Lagrange, France e-mail: raphael.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='pourcelot@oca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='eu 2 Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA 3 Thales Alenia Space, 5 Allée des Gabians - B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 99 - 06156 Cannes la Bocca Cedex – France | 4 Aix Marseille Université,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' CNRS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' CNES,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' LAM (Laboratoire d’Astrophysique de Marseille) UMR 7326,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 13388 Marseille,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' France 5 LESIA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Observatoire de Paris,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Université PSL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Sorbonne Université,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Université Paris Cité,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' CNRS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 5 place Jules Janssen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 92195 Meudon,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' France 6 Hexagon Federal,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Chantilly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' VA 20151,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' USA 7 Department of Mechanical and Aerospace Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Princeton University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Princeton,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' NJ 08540,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' USA 8 NASA Goddard Space Flight Center,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Greenbelt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' MD 20771,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' USA January 10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 2023 ABSTRACT Context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Wavefront sensing and control (WFSC) will play a key role in improving the stability of future large segmented space telescopes while relaxing the thermo-mechanical constraints on the observatory structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Coupled with a coronagraph to reject the light of an observed bright star, WFSC enables the generation and stabilisation of a dark hole (DH) in the star image to perform planet observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Aims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' While WFSC traditionally relies on a single wavefront sensor (WFS) input to measure wavefront errors, the next generation of instruments will require several WFSs to address aberrations with different sets of spatial and temporal frequency contents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' The multiple measurements produced in such a way will then have to be combined and converted to commands for deformable mirrors (DMs) to modify the wavefront subsequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' We asynchronously operate a loop controlling the high-order modes digging a DH and a control loop that uses the re- jected light by a Lyot coronagraph with a Zernike wavefront sensor to stabilize the low-order aberrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Using the HiCAT testbed with a segmented telescope aperture, we implement concurrent operations and quantify the expected cross-talk between the two con- trollers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' We then present experiments that alternate high-order and low-order control loops to identify and estimate their respective contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' We show an efficient combination of the high-order and low-order control loops, keeping a DH contrast better than 5 × 10−8 over a 30 min experiment and stability improvement by a factor of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' In particular, we show a contrast gain of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='5 at separations close to the DH inner working angle, thanks to the low-order controller contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Concurrently digging a DH and using the light rejected by a Lyot coronagraph to stabilize the wavefront is a promising path towards exoplanet imaging and spectroscopy with future large space observatories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Key words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' instrumentation: high angular resolution, methods: data analysis, telescopes 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Introduction High-contrast imaging and spectroscopy is one of the path- ways envisioned by the Astro2020 Decadal Survey (National Academies of Sciences, Engineering, and Medicine 2021) to identify Earth-like worlds in other planetary systems and search for the biochemical signatures of life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' To this aim, a telescope and its instruments need to fulfill at least the following three main requirements: sensitivity to observe objects with magni- tude larger than 30, resolution to separate a planet from its host star at angular distances shorter than 50 mas, and contrast to disentangle the planet photons from the star glow to observe exo-Earths around Sun-like stars with a contrast (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=', flux ratio) around 10-10 in visible light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' A space telescope with a primary mirror larger than 4 m (Gaudi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' The LUVOIR Team 2019) will have the res- olution and sensitivity to enable the observation of a large sam- ple of Earth-like planets (Stark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' To save weight and room in a spacecraft and rocket fairing, a promising option for the telescope is to be folded thanks to the use of a segmented pri- mary mirror, such as the recently launched James Webb Space Telescope (Lightsey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' To overcome the flux ratio be- tween the host star and the planet, one of the promising options is the use of a coronagraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' This instrument rejects the on-axis stellar light while retaining the planet light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' In the case of geo- metrically unfriendly apertures of telescopes with pupil features such as primary mirror segmentation, secondary mirror central Article number, page 1 of 13 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='03242v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='IM] 9 Jan 2023 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' paper obstructions, and spider struts, the Lyot-style coronagraph (Lyot 1939) is one of the leading devices for starlight diffraction sup- pression with arbitrary apertures (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Soummer 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Soummer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' N’Diaye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Including the combination of a focal-plane mask and a Lyot stop, this concept has already been implemented in several ground-based high-contrast instruments (Hinkley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Macintosh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Beuzit et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 2019), allowing the detection of several planetary companions and disks so far (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=', Macintosh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Chauvin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Keppler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 2018) and large-scale surveys of nearby stars (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=', Nielsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Vigan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' However, on top of the complicated diffraction pattern due to telescope segmentation (Yaitskova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 2003), the contrast goal of 10-10 translates into tight requirements in terms of observatory stability, with wavefront error budgets on the order of a few hun- dreds of picometers during the observations (Coyle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' To alleviate the constraints on the observatory structure, a solu- tion is to make use of wavefront sensing and control (WFSC) strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' These methods consist in measuring the wavefront errors with wavefront sensors (WFSs) and correcting for them to obtain the desired shape with one or more deformable mir- rors (DMs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' In this context, it is possible to distinguish differ- ent WFSC applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' One of the main goals is to dig a dark hole (DH, Malbet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Bordé & Traub 2006), an opti- mized high-contrast region in the image of an observed star to enable exoplanet observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' This approach can also be used to correct for perturbations originating from observatory vibra- tions and mechanical or thermal drifts, in particular those of the primary mirror segment alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' These effects lead to a con- trast degradation in the coronagraphic image of an observed star, hindering the detection of planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' The wavefront errors cover a wide range of spatial frequencies, from low-order, up to 4-5 cy- cles per pupil (c/p), to high-order aberrations, at a few tens of c/p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' The wavefront errors also cover a scale of temporal frequen- cies, from low to high range going from the mHz to the kHz regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Optimal correction requires the combination of simul- taneous control loops (Pueyo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Guyon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 2020a,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Currie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 2022) to address these errors at different parts of the spatial or temporal frequency ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' In the case of a Lyot coronagraph, the focal-plane mask aims to reject the central part of the stellar point-spread func- tion, which corresponds to the fraction of the beam that contains the low-order content of the aberrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' We therefore explore the use of this rejected light to feed a low-order wavefront con- trol (LOWFC) loop that stabilizes the low-order modes of the wavefront.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' To analyse the low-order information in this rejected light, we use a Zernike Wavefront Sensor (ZWFS, Zernike 1934;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Bloemhof & Wallace 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Dohlen 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Wallace et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' N’Diaye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 2013b) to take advantage of its high sensitivity (Guyon 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Chambouleyron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 2021), the simplicity of its wavefront reconstructor, and the ease of its hardware implemen- tation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' In Pourcelot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' (2022) hereafter Paper I, we demon- strated the wavefront stabilisation with a standalone closed loop, stabilizing both the wavefront, and the DH contrast at an aver- age and standard deviation better than 10-7 and 10-8 respectively, under artificially introduced perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' While a similar ap- proach has also been demonstrated for the Roman Space Tele- scope (RST) coronagraphic instrument (CGI, Shi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 2016, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Mennesson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Kasdin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 2020), enabling deeper contrasts will require the study of the simultaneous use of the ZWFS-based control loop combined with a high-order wavefront control (HOWFC) loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' The algorithms used for the HOWFC in this paper are pair-wise (PW) probing (Bordé & Traub 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Give’on et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Potier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 2020) for the elec- tric field estimation in the focal plane and the stroke minimiza- tion (SM, Pueyo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 2009) algorithm to compute the desired DM commands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' In this work, we investigate a concurrent control loop ap- proach on the High-contrast imager for Complex Aperture Tele- scopes testbed (HiCAT, N’Diaye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 2013a, 2014, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Leboulleux et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Soummer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Moriarty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Soummer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 2019, 2022) in monochromatic light at the Space Telescope Science Institute (STScI) in Baltimore, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' In particular, we present an estimation of the interactions between the HOWFC and LOWFC loops, and run it through various con- figurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' These setups include switching both loops on and off, and two variants of PW probing to estimate the impact of LOWFC on the focal-plane wavefront estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' In an addi- tional experiment, we limit the maximum speed of the HOWFC loop to allow the LOWFC loop to address more turbulence, and emphasize its positive impact on the DH contrast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Experimental setup 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Optical setup We recall the main details of the optical configuration for our study, further details can be found in Paper I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' We work on sta- bilizing the contrast with a classical Lyot coronagraph, working with a segmented aperture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' The classical Lyot coronagraph is composed of two masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' The first component is an amplitude focal-plane mask that rejects the on-axis light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' In our studies, it is implemented as a pinhole in a mirror that lets the rejected light through.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' The second component, the Lyot Stop, is a diaphragm in the re-imaged pupil plane downstream of the focal-plane mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Usually implemented with a circular diaphragm, the Lyot stop is slightly undersized with respect to the entrance pupil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' The wave- front corrections are performed by using two DMs upstream of the classical Lyot coronagraph, DM1 in a pupil plane and DM2 outside of a pupil plane, for both phase and amplitude control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' To address the low-order aberrations, we use a ZWFS in the light rejected by the focal-plane mask to measure them and DM1 to correct for them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Concerning the HOWFC loop, the PW esti- mation uses several coronagraphic images taken with probes ap- plied on DM1 to estimate the focal-plane electric field, and SM computes commands for both DMs to generate a DH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Implementation on HiCAT To test and validate our approach, we use HiCAT in a configura- tion identical to the one presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 3 of Paper I and repro- duced in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 1, with a monochromatic laser source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' The relevant values describing the experiment implementation are summa- rized in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' The testbed is composed of several parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' First, there is a telescope simulator that mimics the light from a star through a segmented-aperture telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' The following arm with the coronagraph includes the continuous DMs, DM1 in a pupil plane and DM2 out of pupil plane, the focal-plane mask, from the Lyot project (Oppenheimer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 2004), and the Lyot Stop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' The light reflected by the focal-plane mask is sent to the imaging camera, where some light can be picked up with a beamsplitter to re-image the pupil plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' The light rejected by the focal-plane mask is sent to the low-order wavefront sensing (LOWFS) arm with three different detectors: a ZWFS, a target acquisition cam- era, and a phase retrieval camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' In this work, we focus on the ZWFS as the LOWFS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' To isolate it from external perturbations, HiCAT is protected by an enclosure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' A flux of dried air at a con- Article number, page 2 of 13 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Pourcelot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' : Low-order wavefront control using a Zernike sensor through Lyot coronagraphs for exoplanet imaging II Table 1: Physical parameter values of the HiCAT testbed used for the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Parameter Variable Value Wavelength λ 640 nm Entrance pupil diameter Dpup 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='55 mm Number of segments 37 DM 1&2 actuator count 952 Total actuator count q 1904 focal-plane mask diameter dFPM 455 µm 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='52 λ/Dpup ZWFS mask diameter dZ 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='3 µm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='02 λ/Dpup ZWFS mask depth 280 nm Lyot Stop diameter DLS 15 mm ZWFS camera model ZWO ASI178 ZWFS camera pixel pitch 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='4 µm ZWFS Region of Interest size 800 pix Region of Interest number of pixels n 64000 ZWFS camera max framerate 80 Hz Imaging camera model ZWO ASI178 stant temperature is continuously injected to keep the conditions stable and safe for the DM operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' ZWFS control loop Figure 2 represents the block diagram of HiCAT for our ex- periments and in which the ZWFS control loop is represented in orange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' The overall principle remains the same, except that the software implementation of HiCAT has been upgraded since Paper I (Soummer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Por in prep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' With the use of a service-oriented architecture using shared memory for low-latency inter-process communication, it is now possible to run different testbed operations independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' The calibration method has slightly evolved as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Instead of poking only 12 Zernike modes, we now consider the first m Zernike modes (ex- cluding piston) following the Noll convention (Noll 1976), with m = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' The Jacobian matrix JZ of dimensions m × n between DM1 and the ZWFS is built using 1000 draws of random combi- nations of the first m Zernike modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' These draws are poked on DM1, with an expected wavefront error of 10 nm Root-Mean- Square (RMS) per poke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' The control matrix C, of dimensions n × m is then computed by inverting JZ using Tikhonov regular- ization, and a scalar relative regularization parameter of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' The current speed limit imposed on the testbed operations is coming from the maximum frame rate of cameras of 80 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' With this control, the temporal standard deviation for each Zernike mode coefficient is reduced to a few hundreds of picometers RMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' The LOWFC controller is a pure integrator with a gain of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='01, at a loop speed of 80 Hz, limited by the camera readout speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Dark hole digging with stroke minimization DH digging usually relies on two steps: the measurement of the electric field in the DH region in the focal plane and the com- putation of the DM commands that will efficiently remove the light in the DH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' HiCAT currently relies on PW for the determi- nation of the electric field in the focal plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' PW probing applies a set of DM commands, called probes, that modulate the energy distribution on the imaging camera, and especially in the DH area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' By using probes with enough diversity, a testbed model and by solving a linear problem between the DM commands and the focal-plane electric field, an estimation of both the ampli- tude and the phase of the electric field is obtained in the focal plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' The choice of the probes is made by considering several parameters, such as the area in the science image we want to control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Two sets of probes are currently used on HiCAT for PW performance studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' The most basic probes are made by poking single DM actuators, hereafter single-actuator probes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' The sec- ond set called “HiCAT probes” can be computed by solving an optimization problem to find the DM command that produces a specific modulation of the electric field in the DH (Will et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' By using the PW estimation and a DM-to-focal-plane rela- tion such as a Jacobian matrix built between the DMs and the focal-plane electric field, it is possible to perform an optimiza- tion of the DM commands to minimize the energy in the DH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Several algorithms exist, among which SM (Pueyo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Mazoyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 2018b,a) that is implemented on HiCAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' An al- ternative algorithm is electric field conjugation (EFC, Give’On et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 2007) that aims at solving a similar problem with a dif- ferent energy minimization strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' If the electric field conju- gation algorithm requires less computation time and therefore runs faster than SM, it usually provides larger actuator strokes on the DMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Alternative strategies have successfully been im- plemented and tested on HiCAT (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=', Will et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 2021) but they are not considered here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 3 shows an example of DHs gen- erated with SM during HiCAT experiments with an outer work- ing angle of 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='8 λ/Dpup and two different inner working angles (IWA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' While the contrast of 2 × 10−8 could only be reached with a large IWA of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='6 λ/Dpup in Paper I, the new, fast soft- ware architecture allows for a loop frequency gain of a factor of 10 and therefore a better compensation of the turbulence in the HiCAT environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' This enables the digging of a DH with the same contrast like previously but with a reduced IWA down to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='6 λ/Dpup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' This is the value set for the IWA of the DH used in the rest of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Interactions between the loops When running two wavefront control loops in parallel, we want to avoid any cross-talk between the processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' As the experiment goes on, SM is going to refine the DM commands to reduce the intensity in the DH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' From the ZWFS point of view, it means the application of a new DM offset at each iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' The sensitivity loss of the ZWFS due to this offset change has been addressed in Paper I in the case of a large IWA of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='6 λ/Dpup, and a sim- ilar behavior is observed with an IWA at 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='6 λ/Dpup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Overall, this offset change does not prevent the ZWFS control loop to efficiently measure the low-order aberrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Conversely, we want to avoid the ZWFS control loop to cor- rect for SM updates or for the probes introduced by PW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' The success of this will depend on the response measurements of these commands by the ZWFS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' In our configuration with con- trast levels of 10-8, we estimate the projection of the PW probes or SM commands on the modes controlled by the LOWFC loop using the following procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' For a given command v control- ling both DMs, of dimensions 1 × q, we project v on the Zernike polynomial basis that is used to compute JZ, leading to the term v// of dimensions 1 × m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' As JZ is a model of our system, we can therefore estimate the response on the ZWFS detector bimg Article number, page 3 of 13 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' paper Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 1: Simplified HiCAT layout in a semi-transmissive representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' The IrisAO segmented DM creates the segemented aperture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' The apodizer, in the top-right corner is currently replaced by a flat mirror.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' The cameras used in this paper are the Zernike WFS camera, in the light transmitted by the focal plane mask (FPM), and the imaging camera in the light reflected by the FPM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' of size n by computing bimg = JZv//.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Finally, by using the con- trol matrix C, the pseudo-inverse of the Jacobian matrix J, we can assess the command ˆv sent by the ZWFS control loop to DM1 to correct for the perturbation induced by v before apply- ing the control gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' This estimation can also be used to avoid the correction of SM commands and PW probes by the ZWFS, as showed later in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' These estimations are given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 4 for a pure Zernike mode, a single SM update after convergence of the DH contrast, and a correction for a single perturbation measured by the ZWFS while in closed loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' For all the estimated commands, we use the same colorbar and we display the amplification factor ap- plied to each command in the plot to match the colorbar scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' In this configuration, the SM update triggers a faint response of the LOWFC loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' This result is expected as the SM command contains small DM strokes and it is computed to modulate the DH intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Since the DH reaches in to the edge of the focal- plane mask, its corresponding spatial frequencies differ from the spatial frequencies that are seen by the ZWFS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' In comparison with the SM commands, the command sent by the LOWFC loop to correct for turbulence is 10 times fainter in peak-to-valley but triggers a response with a peak-to-valley value which is 20 times larger than the SM command.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' In practice, we expect the com- mand triggered by the ZWFS for a SM update to be even smaller because we use two DMs on HiCAT to correct for phase and amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' In the case of a classical Lyot coronagraph with a segmented aperture, the amplitude correction dominates over the phase correction due to the segment gaps in our pupil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' DM2 will be used to correct for these amplitude errors, introducing both the desired amplitude correction and undesired phase correction which is compensated by DM1 (Mazoyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 2018a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' The other conflicting commands with the ZWFS arise from the probes introduced by PW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Following the same principle as for Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 4, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 5 shows examples of commands triggered by the ZWFS loop when single-actuator and HiCAT probes are sent to DM1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' To match the plots with the same color bars, the DM re- sponse have been amplified by a factor of 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Overall, these re- sponses are also small, with the same order of magnitude as the single ZWFS turbulence correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Similarly to the SM com- mands, the HiCAT probes are less seen by the ZWFS than the single-actuator probes: they are optimized to modulate the DH intensity, which means they are targeting wavefront error spatial frequencies beyond the ZWFS range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Overall, the commands introduced by SM and PW present a limited impact on a single ZWFS control-loop command.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Fur- thermore, on HiCAT, the PW estimation runs at 80 Hz which is close to the frequency of the LOWFC loop, and SM at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='4 Hz with an iteration every 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='5 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' This translates into approximately 200 LOWFC loop iterations for one SM command.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' With a con- trol gain of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='01 and considering the ZWFS amplification factors in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 4 and 5, the ZWFS control loop is therefore unlikely to disturb the probes and the SM updates by more than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='01% and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='1%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' As a result, we choose to disregard these per- turbations for the experiments shown in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='2 with a 10-8 con- trast regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Further analysis of the PW estimation degradation due to LOWFC would be required for deeper contrasts but this is beyond the scope of this paper and will be addressed in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Article number,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' page 4 of 13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='Light ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='Sources ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='Segmented Telescope Simulator ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='Apodizer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='Laser diode ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=': Monochromatic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='Iris-AO Segmented DM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='Beam ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='Pupil Mask ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='Supercontinuum ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='Filter ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='ND ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='Launcher ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='laser source ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='Wheel ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='Wheel ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='Broadband ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='Zernike WFS and Tip/Tilt Sensing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='Coronagraph and Wavefront Control ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='FPM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='Flip Mirror ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='DM2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='DM1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='Zernike ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='Phase Retrieval ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='Target Acq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='Mask ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='Camera ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='Camera "' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='Science Instrument" ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='Lyot Stop ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='Low-order ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='Zernike WFS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='phase retrieval ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='Camera ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='Pupil ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='Camera ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='Camera ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='Imaging ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='Linear ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='Camera ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='PolarizerR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Pourcelot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' : Low-order wavefront control using a Zernike sensor through Lyot coronagraphs for exoplanet imaging II Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 2: Simplified block diagram of HiCAT with the control loops and their interactions for our experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' The IrisAO segmented DM creates the segmented aperture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' The LOWFC sends commands to DM1 only, while the HOWFC sends commands to both DMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Loops can be closed or opened during the operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 10 0 10 Separation ( /Dpup) 10 0 10 Separation ( /Dpup) IWA: 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='6 /Dpup 10 0 10 Separation ( /Dpup) 10 0 10 IWA: 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='6 /Dpup 10 8 10 7 10 6 Contrast Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 3: DH examples with an outer working angle of 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='8 λ/Dpup, with an IWA of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='6 λ/Dpup (left) and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='6 λ/Dpup (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' The blue circles define the edges of the controlled DH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' The black dashed circle delimits the focal-plane mask area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' The residual DH speckles are removed with a faster HOWFC loop thanks to the new HiCAT software architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Parallel operations of low-order and high-order wavefront control loops 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Control loops architecture In this section, we present the results of simultaneous operations of the DH digging process with PW and SM, and ZWFS con- trol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Considering the conclusions of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 3, we run the loops in a completely asynchronous manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' The principle of these op- erations is detailed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 6, where both loops run in a separate process and send commands to a virtual DM channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Another process then sums the commands sent to the channels and ap- plies the final command to the DMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Two configurations are ex- plored for PW: using single-actuator or HiCAT probes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' For both of these experiments, the protocol is the same: we pre-compute a DH solution using a fast electric field conjugation algorithm that manages to provide DM solutions yielding an averaged con- trast smaller than 5 × 10−8 in the DH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' The experiment then goes through 5 parts: (0) SM runs alone;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' (1) SM runs in parallel with ZWFS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' (2) SM runs alone again;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' (3) SM runs in parallel with ZWFS again;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' (4) SM is stopped while ZWFS is running.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' During (4), for each contrast measurement, another measurement is per- formed with the ZWFS loop opened and DM command reset to the first command of (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' All along the experiments, a measure- ment of the direct flux without the focal-plane mask is performed every 10 iterations to get an accurate contrast normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Recent studies of the testbed stability have identified a warm flip-mount motor generating the air turbulence that was charac- terized in Paper I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' After moving this component, the beam is now more stable without any noticeable contrast drift during ex- periments running longer than 1 h, even without control and with contrasts around 3 × 10−8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' To study the behavior of the ZWFS control loop with DH digging in a less favorable environment, these two experiments are performed with a 10 × 50 cm2 open- ing in the HiCAT enclosure, creating extra turbulence within the testbed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' In addition, to keep the DMs safe during the operations, the dry air flow injected in the bench has been increased to com- pensate the influx of humid air through the opening in the en- closure and stabilize the environment humidity and temperature in the turbulent airflow on testbed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Examples of power spectral density of the turbulence for the tip and tilt modes is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' They show a turbulent motion in the low frequency range below 1 Hz that we associate to turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Perturbations at higher frequencies around 10 Hz are also present and are as- sociated to vibrations of the testbed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' They are not specifically linked to the generated turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' The behavior is very similar with the other Zernike modes but almost two orders of magni- tude smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Article number, page 5 of 13 TTTIT LOWFC Segmented Science DM camera DM1 HOWFC Zernike Wavefront LS sensor o DM2 Focal plane maskA&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' paper Input perturbation on DM1 Jacobian Zernike poke 20 10 0 10 20 SM update 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='2 ZWFS update 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='1 Expected OPD (nm RMS) Projected response from LOWFC x 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='0 20 10 0 10 20 x 5268 20 10 0 10 20 x 339 20 10 0 10 20 Expected cmd (nm) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 4: Examples of different perturbations introduced on DM1 (top) and their corresponding LOWFC loop responses on DM1 (bottom), obtained by projecting the command on the controlled basis of the LOWFC loop and using the control matrix to estimate the feedback command.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' From left to right: the Zernike mode Z4 used for the interaction matrix calibration, a single update from SM after convergence of the DH digging, and a command update from the LOWFC loop to correct for residual turbulence in closed loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' In the bottom row, all the displayed commands are scaled to fit the same color bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' From left to right, the commands have been multiplied by 1, 5268 and 339 respectively to be displayed with the same color bar, and the respectively applied amplification factor is displayed on top of each command.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' The commands shown here expand over the whole controllable area of DM1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' The hexagons represent the projection of the segmented aperture onto DM1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' On DM1 Single act #1 Single act #2 HiCAT probe 1 HiCAT probe 2 10 0 10 Expected DM WFE (nm) Projected response from LOWFC 10 0 10 Projected cmd (nm) x 100 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 5: Examples of different PW probes introduced on DM1 (top) and their corresponding LOWFC loop responses on DM1 (bottom) obtained by projecting the command on the controlled basis of the LOWFC loop and using the control matrix to estimate the feedback command.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' From left to right: two examples of single-actuator probes poked by 10 nm in wavefront error;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' two of the 4 HiCAT probes designed to specifically modulate the electric field in the DH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' To match the top and bottom row color bars, the bottom commands have been scaled by a factor of 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' The commands shown here are over the whole controllable area of the DM1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' The hexagons represent the projection of the segmented aperture onto DM1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Article number, page 6 of 13 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Pourcelot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' : Low-order wavefront control using a Zernike sensor through Lyot coronagraphs for exoplanet imaging II Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 6: Vertical timeline of the parallel operations on HiCAT with three processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' The DMs are operated by a process that reads from all the different channels (LOWFC, PW and HOWFC here) and apply the shape corresponding to the sum of the differ- ent contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' The LOWFC process starts by taking an initial reference, and then reads images from the camera and computes corrections sent to the LOWFC DM channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' The HOWFC reads runs PW probing, using a dedicated DM channel, and computes DM corrections with SM for the HOWFC channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' It can also interrupt the LOWFC by clearing the channel for specific acqui- sitions without LOWFC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Each process runs independently from the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Data is transferred from cameras or to DMs through shared memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 10 2 10 1 100 101 10 7 10 5 10 3 10 1 101 PSD (nm2/Hz) Tip Closed enclosure Open enclosure 10 2 10 1 100 101 Frequency (Hz) 10 7 10 5 10 3 10 1 101 PSD (nm2/Hz) Tilt Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 7: Examples of power spectral density (PSD) of the tip (top) and tilt (bottom) perturbations at the level of the ZWFS on Hi- CAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' For both modes, a measurement was performed with all the panels closed (in blue) and with an open panel of the enclosure and a high dry air flow (orange).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Results The mean DH contrast is presented as a function of the iterations in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 8 and 9 for the single-actuator and HiCAT probe scenar- ios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Examples of DH images extracted from the two experiments are given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 10, with the image yielding the best DH con- trast for intervals (0) to (3) and the respective last images for intervals (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' The corresponding contrast statistics are detailed in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' For both probe types, the HOWFC loop manages to correct for most of the wavefront errors, yielding a fairly sta- ble contrast in the intervals (0) to (3), especially in the intervals (0) and (2) where no LOWFC loop is running.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Conversely, large drifts are visible in (4), where we observe a contrast loss of an order of magnitude compared to sections (0)-(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Consistently with the previous blind DH stabilization (Paper I), the LOWFC loop slows down the contrast drift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' From the difference between intervals (0) and (2), and intervals (1) and (3), we draw several conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' First, we show a HOWFC and LOWFC loop combination that manages to keep the DH stable at a contrast of ∼ 5 × 10−8 under large natural drifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Concerning the standard deviation σt, it is typically maintained at an order of magnitude lower than the averaged contrast, and with σt = 5 × 10−9 over 25 min of an ex- periment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' This asynchronous implementation does not introduce conflicts between the two controls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' At slightly better contrasts, the “Très Haute Dynamique 2” testbed is routinely operating a fast low-order correction loop (Galicher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' However, their approach uses the light reflected by the Lyot stop and it addresses only the control of tip and tilt on a separate steering mirror.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Contrasts deeper than 10-8 have also been obtained by RST/CGI (Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 2019, 2020) with a HOWFC loop includ- ing a LOWFC loop (Shi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 2016, 2017, 2018), but our ap- proach differs in several aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' (i) While RST relies on a mono- lithic primary mirror, HiCAT implements a segmented aperture Article number, page 7 of 13 Initial ref PW cmd Acquistion New cmd Probe 1 cmd Acquistion New cmd pW cmd cmd Acquistion Probe 2 New cmd Acquistion pW cmd New cmd .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' clear PW Solving electric field Stroke minimization cmd Coron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' image With LOWFC Clear LO Coron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' image No LOWFC Restore LO Iteration n+1A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' paper 0 100 200 300 400 500 600 700 Iteration # 2 × 10 8 4 × 10 8 7 × 10 8 10 7 2 × 10 7 Mean contrast in the DH 0 1 2 3 4 SM off SM only No control SM + ZWFS ZWFS only Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 8: Mean DH contrast while PW with single-actuator probes and SM are running, without LOWFC in blue in section (0) and (2), and with it in red in sections (1) and (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' SM is turned off in section (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' In (4), the mean contrast is measured while LOWFC is running.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Once per contrast measurement, the LOWFC is interrupted and reset to its initial command in (4) to get an open-loop measurement, plotted in black.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' When on, the HOWFC loop runs at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='4 Hz and the LOWFC loop at 80 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 0 100 200 300 400 500 600 700 Iteration # 2 × 10 8 4 × 10 8 7 × 10 8 10 7 2 × 10 7 Mean contrast in the DH 0 1 2 3 4 SM off SM only No control SM + ZWFS ZWFS only Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 9: Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 8 but using PW with HiCAT probes instead of single-actuator probes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' with an IrisAO deformable mirror.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' (ii) Although the tests on RST have been performed with a fast LOWFC loop at frequencies up to 1 kHz, this loop only addresses tip/tilt (Z2, Z3) at this speed with a steering mirror to control them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' The other modes are controlled at much slower rates, around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='2 Hz for defocus (Z4) with their DM2 and 5 mHz for other modes (Cady et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Shi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Seo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 2017, 2018) with their DM1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' At the same time, the HOWFC loop on RST is running on both DMs at around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='1 Hz, limiting the possible cross-talks with LOWFC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' The RST tests were also performed on a testbed located in a vac- uum, where it is only perturbed by known artificial drifts with a few Zernike modes while we operate our testbed in air and under strong turbulence with an opened enclosure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' (iii) The RST/CGI observation mode is designed to dig a DH on a bright star and Article number, page 8 of 13 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Pourcelot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' : Low-order wavefront control using a Zernike sensor through Lyot coronagraphs for exoplanet imaging II Table 2: Mean DH contrast statistics over the different intervals for the single-actuator probes experiment and HiCAT probes experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Situation Stat 0 1 2 3 4 with ZWFS 4 no ZWFS ZWFS off on off on on off SM on on on on off off Single-actuator probes Mean 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='5e-08 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='6e-08 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='0e-08 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='4e-08 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='4e-08 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='4e-07 σt 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='0e-09 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='6e-09 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='7e-09 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='3e-09 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='1e-08 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='3e-07 HiCAT probes Mean 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='5e-08 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='7e-08 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='3e-08 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='8e-08 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='0e-08 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='3e-07 σt 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='7e-09 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='6e-09 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='7e-09 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='9e-09 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='5e-08 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='3e-08 then slew to the science target, using the LOWFC loop only to correct for perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' In our case, we aim at demonstrating the continuous use of both loops while observing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Second, the contrast gain in both temporal average and stan- dard deviation from Table 2 are moderate but noticeable, espe- cially in the single-actuator probe experiment, reaching a mean contrast of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='4 × 10−8 and a value of σt of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='3 × 10−9 over the in- terval (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' While we expect the contrast with LOWFC to be bet- ter because of the turbulence that introduces a lot of low-order aberrations, the new implementation of the HOWFC control loop runs fast enough to handle a large part of the low-order cor- rections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' From our experience with electric field conjugation on HiCAT, digging a DH with faster iterations allows even deeper contrasts in the presence of turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Therefore, the contrast we observe is very likely to be driven by the internal turbulence that evolves faster than what the HOWFC loop can handle in the high-order modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Further experiments are required to fully understand the limitations in these conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Third, even if the temporal contrast averages are moderate, the ratio of contrast between the cases without and with LOWFC as a function of angular separation in the focal plane emphasizes the action of the loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' The corresponding curves are displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' For each probe, the plots show the ratio between the azimuthally averaged contrast over sections (0) and (2), and over sections (1) and (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' The maximum contrast improvement by a factor of up to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='5 is located at separations around 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='6 λ/Dpup, as expected - this corresponds to the edge of the focal-plane mask, and therefore the theoretical sensing limit of the ZWFS through the Lyot mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Since the LOWFC loop controls a limited number of 20 Zernike modes here, its impact is negligible at larger sepa- rations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' The same plot is also displayed for section (4) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 11 bottom plot, showing the same behavior, but with a much larger gain up, to a factor of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' This is consistent with the first point above: in this experiment, both loops control overlapping spatial frequencies of the wavefront error without cross-talk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' The gains at larger separations are negligible here as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Finally, the experiments have slightly better results with single-actuator probes, even though their interaction with the LOWFC loop was supposed to be greater with the HiCAT probes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' However, these single-shot experiments do not allow us to conclude on any significant difference between the two types of probes, the different behaviors in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 8 and 9 being possibly due to external factors such as turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' The real limitation due to loop interactions will probably appear when working at deeper contrasts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' We are still investigating the exact limitations on HiCAT of the electric field estimation by PW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' These inter- actions could degrade wavefront estimation that would result in reaching a contrast floor at some point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Therefore, using probes as invisible as possible to the ZWFS could prove of interest in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Experiment with slower high-order wavefront control loop The gain provided by the ZWFS depends not only on the in- put turbulence, but also on the amount of low-order aberrations the HOWFC loop corrects for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' By degrading the correction per- formed by the high-order controller we can emphasize the im- provement due to the parallel use of the ZWFS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' This also de- grades the contrast performance by an order of magnitude, and is not representative of the current operations of HiCAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' For this goal, we use HiCAT in a different setup, with a HOWFC loop running 8 times slower, at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='05 Hz and the LOWFC loop around 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='5 times slower at 30 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' In this exper- iment, the HiCAT probes are used for PW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' With the slower HOWFC and similar air turbulence, the contrast in the DH is stuck above 10-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' The speed difference between the two con- trollers now being much larger, we expect the cross-talk between them to be more challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' For this experiment, we have im- plemented a communication between the loops to offload the HOWFC loop commands to the LOWFC reference: each time PW or SM sends a command on the DMs, the command is sent to the LOWFC loop as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' The command is then projected on the controlled modes of the LOWFC loop and multiplied by the Ja- cobian matrix JZ to estimate the corresponding variations on the ZWFS signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' While there are many ways for improvement here, our code modifications for synchronisation add overheads in the computation and slow down the LOWFC loop from 80 Hz to 30 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' As a result we increase its gain from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='01 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='15, a value that empirically proved to be efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Considering the loop fre- quencies, the LOWFC loop produces around 600 corrections in a single HOWFC loop iteration, while with the previous setup it was only around 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' The contrast curve is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' In this configura- tion, the LOWFC loop proves very efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' While in interval (0) the contrast reaches a floor at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='9 × 10−7, turning on the LOWFC loop in (1) allows for an almost immediate contrast improvement by a factor of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' This behavior is similar with the intervals (2) and (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' During interval (4), when the HOWFC loop is turned off, the drift of the uncontrolled DH is immediate, while the LOWFC loop manages to slow it down, limited by the reduced number of controllable modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Overall, the contrast standard deviation over the intervals proves also to be greatly improved, from around 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='5 × 10−7 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='5 × 10−8 without and with LOWFC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' The az- imuthally averaged gain is represented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 13, showing a gain of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='5 and 6 with and without SM at short separations as in the previous experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' But it also shows a gain larger than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='5 at all separations, whether SM is on or off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Under these different experimental conditions, we also show a successful concurrent operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Similarly to the results in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='2, most of the improvement by the LOWFC is located at the inner radius of the DH, improving the contrast at short sep- arations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' By comparing the two setups, it is possible to extract Article number, page 9 of 13 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' paper SA probes Best (0) HiCAT probes Best (1) Best (2) Best (3) Last (4) 10 8 10 7 10 6 10 5 Contrast Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 10: Examples of focal-plane DH images for the experiment with single-actuator probes (top) and HiCAT probes (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' For intervals (0) to (3), this corresponds to the image yielding the best mean spatial contrast in the respective operational sections in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 8 and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' The image for interval (4) is the last one from the open-loop data, shown in black in both figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='0 Gain (No ZWFS / with ZWFS) IWA OWA With SM Single act probes HiCAT probes 4 6 8 10 12 14 Separation /D 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='0 Gain (No ZWFS / with ZWFS) No SM Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 11: Contrast gain due to ZWFS as a function of angular separation in the DH, for experiments with single-actuator and HiCAT probes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Solid lines represent the ratio of the contrast az- imutal average between the cases without and with LOWFC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' The filling colors show the standard deviation of the measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' The dashed horizontal lines are drawn at a gain of 1 in both plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' The top plot represents the values of the intervals (0) and (2) over the values of the intervals (1) and (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' The bottom plot corresponds to the values of the interval (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' The gain lower than 1 at the OWA is due to edge effects in the numerical computation of the contrast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' properties of the control loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' In both setups, when the control loops are off, the contrast drift is immediate, emphasizing the presence of the perturbation which is partially corrected when the LOWFC loop is on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' The remaining contrast drift is due to the higher order pertur- bation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' This perturbation is a main driver of the averaged contrast in the DH, and is out of reach for the LOWFS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' This is particularly visible when looking at the intervals (0) in both setups, where the contrast levels only depend on the HOWFC speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' In similar conditions, the faster setup reaches a contrast level deeper than the slower setup by almost an order of magnitude, showing the impact of loop speed with the air turbulence on HiCAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' With a faster speed than the HOWFC loop, the LOWFC loop can therefore correct for the low-order turbulence that evolves too fast for the high-order controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' As the leftover is more im- portant in the slower setup, the contrast gain due to the ZWFS is greater and more obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' On the contrary, with the faster setup, since the HOWFC is more efficient at correcting low-order aber- rations, the gains are more moderate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Overall, there is a gain in using the LOWFS in both setups, showing that the loop combi- nation does not generate cross-talks at the contrast levels in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Conclusions Wavefront error stability is a key parameter in the success of fu- ture space missions with exoplanet imaging capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' To im- prove the stellar light rejection by the coronagraph, WFSC ap- pears essential to alleviate the requirements on the observatory stability for high-contrast observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' In particular, WFSC has been proven efficient to correct for different kinds of aberrations such as low-order drifts, or improve the image plane DH by opti- mizing the DM shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' However, this was only done sequentially so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Following the demonstration of DH stabilization with a ZWFS-based LOWFC loop in the rejected light of a Lyot coron- agraph in Paper I, we have here studied its concurrent operation with a HOWFC loop using different WFSs but the same DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Using the HiCAT testbed for our experiments, we first study the impact of PW probes and SM commands on the LOWFC loop by projecting these commands on its controlled modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' We Article number, page 10 of 13 O0R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Pourcelot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' : Low-order wavefront control using a Zernike sensor through Lyot coronagraphs for exoplanet imaging II 0 200 400 600 800 1000 Iteration # 10 7 3 × 10 7 6 × 10 7 10 6 Mean DH contrast 0 1 2 3 4 SM off SM only No control SM + ZWFS ZWFS only Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 12: Evolution of the mean contrast in the DH during HOWFS (using SM) as a function of iterations during different intervals: without LOWFC in blue in section (0) and (2), and with it in red in sections (1) and (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' SM is turned off in section (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' In (4), the mean contrast is measured while LOWFC loop is running (green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Once per contrast measurement, the LOWFC loop is interrupted and reset to its initial command in (4) to get an open-loop measurement, in black.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' When on, the HOWFC loop runs at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='05 Hz and the LOWFC loop at 30 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 4 6 8 10 12 14 Separation /D 0 2 4 6 8 Gain (No ZWFS / with ZWFS) IWA OWA With SM No SM Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 13: Contrast gain as a function of angular separation in the DH when the HOWFC loop is slowed down to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='05 Hz on pur- pose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Solid lines represent the ratio of the azimuthally averaged contrast between the cases without and with LOWFC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' The fill- ing colors show the standard deviation of the measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' The dashed horizontal line is drawn at a gain of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' The blue curve represents the gain between the values of the intervals (0) and (2) over the values of the intervals (1) and (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' The yellow curve corresponds to intervals (4) values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' find a limited response, and considering the gain of the LOWFC loop of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='01 as well as the different loop frequencies on HiCAT, the loop response to a single command is unlikely to modify the original command by more than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' We find similar results for two kinds of probes for PW, whether single actuators pok- ing or HiCAT probes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Then we run the HOWFC and LOWFC loops to dig or stabilize a focal-plane DH under the conditions of increased perturbation on the testbed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' While the first loop is on, we maintain the contrast levels around 5 × 10−8, showing no cross-talk between the two control loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Even if the main DH contrast driver is due to high-order modes that evolve too fast for the HOWFC loop, it is possible to observe a contrast improve- ment with the LOWFC loop, with a factor of up to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='5, and a factor of 5 when the HOWFC loop is on and off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' To prove the ef- ficiency of the LOWFC loop, we compare the performances with a reduced speed of the HOWFC loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' With the same IWA of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='6 λ/Dpup this greatly degrades the achievable contrast, but em- phasizes the capability of both loops to work together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' A classi- cal Lyot coronagraph with a smaller focal-plane mask and there- fore a smaller IWA will be more sensitive to low-order aberra- tions, leading to an increased need for the LOWFC loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Further investigations are required to find the best functioning point be- tween the HOWFC and LOWFC loops for this setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Prelimi- nary tests have already been successfully performed on HiCAT with a knife-edge coronagraph, showing promising contrast sta- bility results at shorter separations which will be detailed in a forthcoming paper (Por in prep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' We validated our approach in a specific environment, in which the PW sensing runs at temporal frequencies very close to the LOWFC loop at 80 Hz with a loop gain of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='01 and a SM iteration every 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='5 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' On average, a single LOWFC loop correc- tion is applied at each PW probe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' However, when we reduce the HOWFC loop speed, we manage to achieve a contrast of 10−7 with a very simple loop synchronization to avoid cross-talk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' The asynchronous operations tend to converge toward DH solutions that are dependent on the LOWFC loop contribution, while we would prefer the two control loops to be as independent as pos- Article number, page 11 of 13 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' paper sible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' We currently operate HiCAT around contrasts of 5 × 10−8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' These loop interactions could create a bias in the wavefront es- timation, thus representing a possible limitation when pushing the contrast to lower values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Regarding these points, it might be worth re-evaluating the asynchronous operations and consider- ing an improved communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' This will enable adapting the LOWFC loop reference depending on the commands sent by the HOWFC loop, in a similar fashion as Guyon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' (2020b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' In the scope of combining more loops, the asynchronous approach will remain applicable as long as there is no cross-talk between the loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Further investigations on the loop communication will otherwise be required as the number of communications for of- floading grows quadratically with the number of loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' We conducted our experiments in idealized conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' First the tests are done in monochromatic light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' While the ZWFS is expected to work relatively well with broadband light (N’Diaye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 2013b), the DM DH commands will be different from those in monochromatic light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' In further studies, we will investigate the stability of the parallel loop operations in broadband light, pending the delivery of a new broadband source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Second, our work is made with enough photons to correct as much turbu- lence as possible, the main limitation being the camera max- imum frame rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Assuming a JWST-like primary mirror, the equivalent magnitude of the monochromatic source on HiCAT would be about -6 in the visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' This is clearly an ideal case and the impact of signal-to-noise ratio on ZWFS measurements re- mains to be explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' We will investigate these aspects further, following Sahoo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' (2022) who recently developed a novel ap- proach to determine the optimal wavefront sensor exposure time, for a given contrast requirement at a given stellar magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' A flux limitation will also likely increase the relevance of the LOWFS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' In a photon-limited regime, a reduced photon flux requires longer exposure times on the science camera to keep the signal-to-noise ratio constant, leading to a possible reduction of the HOWFC efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Since the ZWFS uses the light from the core of the source image, it collects more photons than the science camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Its associated loop will therefore be able to run faster and correct for the aberrations beyond the HOWFC tempo- ral bandpass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' As emphasized by the configuration in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='3, the low-order controller can then be complementary with the high- order controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Finally, these results have been obtained by degrading the HiCAT environment to generate larger drifts in an uncontrolled DH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' In this context, we do not reach the optimal performance of HiCAT that is way more stable in nominal conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' In particular, with the injected turbulence, here we are limited in contrast by the speed of the HOWFC loop that runs at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='4 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Using LOWFC, the contrast improvement is limited to separa- tions close to DH IWA, where the control regions of both loops is smoothly overlapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' It would be relevant to study how far out in the focal plane the LOWFC loop could effectively control by considering more than the current 20 Zernike modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' This could allow a gain in the temporal bandpass of correction as the LOWFC loop runs orders of magnitude faster than the HOWFC loop and alleviates the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' At the moment, our demonstration of the concurrent use of LOWFC and HOWFC loops at the 5× 10-8 contrast point repre- sent a first milestone towards concurrent operations with a 10-10 contrast goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Several aspects in our current experiment such as the selected control modes, the accuracy of the DM behavior modeling, or the sensor sensitivity, will be further explored in the regime of wavefront error fluctuations down to the picomet- ric level to further advance exo-Earth imaging with concurrent loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' This combination of several control loops is a necessary step toward a system-level demonstration of a future high-contrast in- strument for exoplanet imaging with a future large space obser- vatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' To stabilize the whole range of aberrations that can dis- turb the observations, these control loops have to be associated to others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' These include a loop dedicated to tip/tilt correction, with a dedicated steering mirror already implemented on HiCAT but currently unused, another to primary mirror segment alignment or one dedicated to vibration rejections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Fully operating and un- derstanding this setup will help develop future instrumentation for exoplanet and more particularly exo-Earth imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' acknowledges PhD scholarship funding from Région Provence-Alpes-Côte d’Azur and Thales Alenia Space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' The authors are espe- cially thankful to the extended HiCAT team (over 50 people) who have worked over the past several years to develop this testbed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' This work was supported in part by the National Aeronautics and Space Administration under Grant 80NSSC19K0120 issued through the Strategic Astrophysics Technology / Tech- nology Demonstration for Exoplanet Missions Program (SAT-TDEM;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' PI: R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' Soummer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' was supported by the NASA Hubble Fellowship grant HST- HF2-51467.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='001-A awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Incor- porated, under NASA contract NAS5-26555.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' acknowledges the support by a postdoctoral grant issued by the Centre National d’Études Spatiales 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 2020, in Society of Photo-Optical Instru- mentation Engineers (SPIE) Conference Series, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 11443, Society of Photo- Optical Instrumentation Engineers (SPIE) Conference Series, 114431W Zhou, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=', Krist, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=', Kern, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 2019, in Society of Photo-Optical Instru- mentation Engineers (SPIE) Conference Series, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} +page_content=' 11117, Society of Photo- Optical Instrumentation Engineers (SPIE) Conference Series, 111170H Article number, page 13 of 13' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE1T4oBgHgl3EQfhgSw/content/2301.03242v1.pdf'} diff --git a/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf b/o9AyT4oBgHgl3EQfY_f6/content/2301.00217v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..59f5bcf7943170af00da5177d76d1ca84acd07a4 --- /dev/null +++ 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novel chiral symmetry breaking +Paul Sutcliffe +Department of Mathematical Sciences, +Durham University, Durham DH1 3LE, United Kingdom. +Email: p.m.sutcliffe@durham.ac.uk +January 2023 +Abstract +An extension of the Skyrme model is presented in which derivative terms are added +that break chiral symmetry to isospin symmetry. The theory contains just one new +parameter and it reduces to the standard Skyrme model when this symmetry break- +ing parameter vanishes. The same Faddeev-Bogomolny energy bound applies for all +parameter values, but the parameter can be tuned so that the energy of the single +Skyrmion is much closer to the bound than in the standard Skyrme model. Apply- +ing the rational map approximation to multi-Skyrmions suggests that, for a suitable +value of the symmetry breaking parameter, binding energies in this theory may be +significantly more realistic than in the standard Skyrme model. +1 +arXiv:2301.03021v1 [hep-th] 8 Jan 2023 + +1 +Introduction +Skyrmions are topological solitons that model nuclei within a nonlinear theory of pions +[1]. There have been many developments on Skyrmions since Skyrme’s original work [2], and +during the last decade or so some attention has been directed towards modifications of the +Skyrme model that bring the energy of a single Skyrmion closer to the Faddeev-Bogomolny +energy bound [3]. The motivation for this is to address the issue that Skyrmions are too +tightly bound in comparison to nuclei. If the energy of a single Skyrmion is close to the bound +then clearly this places a tight restriction on multi-Skyrmion binding energies. However, in +the standard Skyrme model the single Skyrmion energy exceeds the bound by around 23% +and this leaves plenty of room for large binding energies, well in excess of those of nuclei, +that are no greater than around 1%. +Some examples of modifications to the standard Skyrme model to address this problem +include the BPS sextic model [4, 5], the lightly bound model [6, 7], the loosely bound +model [8], the dielectric model [9], the self-dual model [10], the quasi self-dual model [11], +the inclusion of ω mesons [12], and the addition of ρ mesons via Yang-Mills theory with +an extra dimension [13, 14, 15, 16]. Each theory has its own positive aspects that make +it worthy of investigation, together with negative aspects that mean that alternatives are +still being sought. The purpose of the present paper is to add a new model to the list of +modified theories that bring Skyrmion energies closer to the energy bound. This is achieved +by the introduction of derivative terms that break chiral symmetry to isospin symmetry. +The additional terms come with one new parameter that can be chosen so that the energy +of the single Skyrmion is reduced from 23% to 7% above the bound. An investigation of +multi-Skyrmions within this theory, using the rational map approximation [17], suggests that +more realistic binding energies may indeed be possible in this broken theory. +2 +A Skyrme model from Yang-Mills +The standard Skyrme model is a nonlinear theory of pions in which the pion fields +(π1, π2, π3) are used to construct the Skyrme field U ∈ SU(2) via +U = +� σ + iπ3 +iπ1 + π2 +iπ1 − π2 +σ − iπ3 +� +, +(2.1) +where the σ field imposes the constraint σ2 + π2 +1 + π2 +2 + π2 +3 = 1. In Skyrme units, the static +energy is given by +ESkyrme = +� +−Tr +�1 +2R2 +i + 1 +16[Ri, Rj]2 +� +d3x, +(2.2) +where the su(2)-valued right currents are Ri = ∂iU U −1. +This theory is invariant under chiral symmetry, given by multiplication of the Skyrme +field by an arbitrary constant SU(2) matrix on the left and another arbitrary constant SU(2) +matrix on the right, which together rotates the quartet of the σ and pion fields. Finite energy +2 + +imposes the boundary condition that the Skyrme field tends to a constant SU(2) matrix at +infinity, taken to be the identity matrix. This breaks chiral symmetry to isospin symmetry, +given by conjugation of the Skyrme field by an arbitrary constant SU(2) matrix, which +rotates the pions fields but leaves the σ field unchanged. +Chiral symmetry can also be +broken explicitly by the addition of a potential term to the energy (2.2). The most common +choice is to add the term 2m2(1 − σ), which gives the pions a mass m in Skyrme units. +However, this study will be restricted to the case of massless pions. +The boundary condition provides a compactification of space, therefore topologically U +is a map between three-spheres with an associated integer B, the topological degree, that +may be computed as +B = +� +1 +24π2εijkTr(RiRkRj) d3x. +(2.3) +This integer is identified with baryon number and is also referred to as the topological charge. +Skyrmions are minimizers of the energy for a given positive value of B and their energies +satisfy the Faddeev-Bogomolny energy bound [3] +ESkyrme ≥ 12π2B. +(2.4) +As mentioned above, for the single Skyrmion (B = 1) the energy exceeds this bound by +around 23% and an aim is to modify the theory to reduce this excess. +An interesting perspective on the Skyrme model is obtained via consideration of SU(2) +Yang-Mills theory with an additional spatial dimension. In four-dimensional Euclidean space +the Yang-Mills energy (with a non-standard normalization) is given by +E = −3 +4 +� +Tr(FIJFIJ) d4x, +(2.5) +where xI, with I = 1, .., 4, denote the spatial coordinates and FIJ = ∂IAJ − ∂JAI + [AI, AJ] +are the components of the su(2)-valued field strength. The instanton number N ∈ Z of the +gauge field provides a lower bound on the energy, which for positive N is +E ≥ 12π2 N. +(2.6) +Unlike the Skyrme model, this bound is attained for all positive N, by fields that are self-dual. +To make contact with the Skyrme model, fix the gauge A4 = 0 and restrict the remaining +three components Ai to have the form +Ai = −1 +2(1 + φ)Ri, +(2.7) +where φ = tanh x4 and Ri is the right current of the Skyrme field U, that depends upon +the remaining three spatial coordinates (x1, x2, x3). Substituting (2.7) into the Yang-Mills +energy (2.5) and performing the integration over x4 yields precisely the Skyrme energy (2.2). +Furthermore, as U is the holonomy of the gauge field along lines parallel to the x4-axis, the +baryon number of the Skyrme field is equal to the instanton number, B = N [18]. The +Faddeev-Bogomolny bound (2.4) therefore follows directly from the Yang-Mills bound (2.6). +3 + +From the Yang-Mills perspective, the Skyrmion excess energy above the bound (2.4) is +a measure of the failure of the restricted form (2.7) to describe a self-dual instanton. The +excess can be reduced by improving (2.7) through the addition of a term φ′Vi to the right- +hand-side. The su(2)-valued vector field Vi(x1, x2, x3) physically describes ρ mesons, the +lightest of the vector mesons. Performing the integration over x4 in the Yang-Mills energy +using the modified restricted form generates a modified Skyrme theory, coupling pions and +ρ mesons [13, 14, 15]. The Faddeev-Bogomolny bound of 12π2B remains intact, due to its +continued inheritance from the Yang-Mills bound, but now the single Skyrmion energy is +only around 6% above the bound. The symmetries and shapes of the Skyrmions retain the +same forms as in the standard Skyrme model, but binding energies are reduced to about one- +quarter of their previous values. Moreover, this flattening of the energy landscape means +that the addition of a pion mass term results in a more dramatic change in the structure of +Skyrmions, which for B > 4 have the cluster structure expected for light nuclei [16]. +The above comments highlight the positive aspects of the Skyrme model with pions and +ρ mesons, but there are also some negative aspects. In particular, with this modification the +number of independent field components increases from three to twelve, and in combination +with the large number of additional terms in the energy, this makes the theory difficult to +work with both analytically and numerically. This motivates the work in the following, to +try and produce a theory with similar results but without introducing any additional fields +to supplement the Skyrme field. +To derive the new variant of the Skyrme model, first define the twist operator + and its +inverse − by the following action on the current +R+ +i = (1 − U −1)Ri(1 − U) +2(1 − σ) +, +R− +i = (1 − U)Ri(1 − U −1) +2(1 − σ) +. +(2.8) +A quick calculation shows that a double application of the twist operator to the right current +yields the left current, R++ +i += U −1∂iU. Note that for the twisted current to be well-defined +as σ → 1, which of course corresponds to the Skyrme field approaching its vacuum value +U → 1, the current must vanish in this limit, Ri → 0. This condition, that needs to be +imposed on the Skyrme field, is the same condition that is imposed in the standard Skyrme +model at spatial infinity, by the requirement of finite energy. As Skyrmions in the standard +Skyrme model already satisfy this property then it seems that this requirement is not a +significant issue. +The new modified Skyrme model is derived by extending the restricted form (2.7) to +Ai = −1 +2(1 + φ)Ri + +χφ′ +√1 − σ(R− +i − R+ +i ) +(2.9) +where χ is a real constant parameter of the theory. The explicit appearance of the σ field +breaks chiral symmetry to isospin symmetry, in the same way as the inclusion of the pion +mass term, which also explicitly includes the σ field. The structure of this term is motivated +by symmetry considerations and the aim to have a surrogate for the vector field Vi that is +constructed from the Skyrme field and brings the restricted form closer to a self-dual field. +4 + +Substituting (2.9) into the Yang-Mills energy (2.5) and performing the integration over +x4 yields the following energy for a modified Skyrme model +(2.10) +E = +� +−Tr +�1 +2R2 +i + 1 +16[Ri, Rj]2 − χ +4 [R+ +i , R+ +j ]Hij + 4 +5χ3[Di, Dj]Hij + 24 +35χ4[Di, Dj]2 ++ χ2 +20 +� +32D2 +i + 5H2 +ij − 8[R+ +i , R+ +j ][Di, Dj] + ([R+ +i , Dj] − [R+ +j , Di])2 +�� +d3x. +For notational convenience, in the above the su(2)-valued quantities Di and Hij have been +introduced, where +Di = Ri − R++ +i +√1 − σ , +(2.11) +and Hij is the anti-symmetric tensor +Hij = 2[Ri, Rj] + 2[R++ +i +, R++ +j +] − [Di, Dj] +√1 − σ +− Di∂jσ − Dj∂iσ +1 − σ +. +(2.12) +If χ = 0 then the energy (2.10) reverts to that of the standard Skyrme model, but for χ ̸= 0 +it is a deformation via derivative terms that break chiral symmetry to isospin symmetry. +The Faddeev-Bogomolny energy bound, E ≥ 12π2B, holds for all χ, as it follows from the +Yang-Mills energy bound. +To get a feel for the structure of the additional terms it is perhaps useful to note that +in the first two terms, that agree with the standard Skyrme model, all currents Ri could +be replaced by their twisted versions R+ +i , because both these terms are invariant under +this replacement. Regarding Di as a variant of R+ +i , and Hij as a variant of [R+ +i , R+ +j ], the +structure of the remaining terms follows from the first two terms by making different possible +replacements of the twisted currents and their commutators by these variants. +3 +Skyrmions +To study the single Skyrmion in the theory (2.10), apply the hedgehog ansatz +σ = cos f, +πi = xi +r sin f, +(3.1) +with f(r) being a monotonic decreasing profile function that satisfies the boundary conditions +f(0) = π and f(∞) = 0. This yields a spherically symmetric energy density with the energy +5 + +given by +(3.2) +E = 4π +� ∞ +0 +� +r2f ′2 + 2(f ′2 + 1) sin2 f + sin4 f +r2 +− 8χ +� +f ′2(1 − 3 cos f) − 2 +r2 cos f sin2 f +� +sin f +� +1 + cos f ++ 8 +5χ2(1 + cos f) +� +f ′2(41 cos2 f − 30 cos f + 9) + 8sin2 f +r2 +(7 cos2 f − 2 + 2r2) +� +− 1024 +5r2 χ3(1 + cos f)3/2 sin3 f cos f + 6144 +35r2 χ4(1 + cos f)2 sin4 f +� +dr. +For each value of χ the energy is obtained by calculating the energy minimizing profile +function numerically using an annealing algorithm. The result is displayed in the left plot +in Fig.1, where the ratio of the energy to the bound is shown as a function of the symmetry +breaking parameter χ. It can be seen that as χ increases from zero, the excess energy initially +decreases from around 23% in the standard Skyrme model to around 7% for a critical value +of χ = 0.14 (to two decimal places) and then begins to increase again. Recall from an earlier +comment that this single Skyrmion excess energy must be greater than 6%, as it cannot be +lower than the value obtained in the ρ meson theory that allows an arbitrary unconstrained +vector field Vi in place of the specific form used in (2.9). +Figure 1: Left: The ratio of the single Skyrmion energy to the bound, as a function of the +symmetry breaking parameter χ. Right: The profile function f(r) for χ = 0 and χ = 0.14. +The energy minimizing profile function f(r) is displayed in the right plot in Fig.1 for +the standard Skyrme model (χ = 0, upper curve) and at the critical value (χ = 0.14, +lower curve). It can be seen that the change in the profile function is rather mild, with the +Skyrmion being a little smaller in the modified theory. +6 + +1.24 +1.22 +1.20 +1.18 +()/ +1.16 +1.14 +1.12 +1.10 +1.08 +1.06 +0.00 +0.02 +0.04 +0.06 +0.08 +0.10 +0.12 +0.14 +0.16 +0.18 +0.20 +X3.0 +X=0 +X = 0.14 +2.5 +2.0 +f +1.5 +1.0 - +0.5 - +0.0 + +0 +1 +2 +3 +4 +5 +6 +7 +8 +rThe fact that the energy at the critical value is close to the value in the ρ meson theory +reveals that the expression used in (2.9) is indeed a good surrogate for the inclusion of ρ +mesons and justifies this choice. It also implies that the new terms in (2.10) may be thought +of in a similar vein to terms that would be generated by integrating out the ρ mesons in +the theory with pions and ρ mesons, to leave an effective theory with just pions. A related +point of view is that the ρ mesons have been treated as excitations of pions, extending the +philosophy that all particles, mesons and baryons, are modelled as excitations of the pion +field. +In the standard Skyrme model the minimal energy multi-Skyrmions can be approximated +with a good accuracy using the rational map approximation [17]. It therefore seems reason- +able to suppose that this approximation is also valid in the modified model, at least for +small values of χ. This approximation uses spherical polar coordinates, r, θ, ϕ, with Rie- +mann sphere coordinate z = tan(θ/2)eiϕ. The input to create a charge B Skyrmion is W(z), +a degree B rational map in z. For B = 1, 2, 3, 4 these maps are given by +W = z, +W = z2, +W = z3 − +√ +3iz +√ +3iz2 − 1, +W = z4 + 2 +√ +3iz2 + 1 +z4 − 2 +√ +3iz2 + 1, +(3.3) +with the property that for each degree they minimize the angular integral +I = 1 +4π +� �(1 + |z|2)|W ′| +1 + |W|2 +�4 2i dzd¯z +(1 + |z|2)2, +(3.4) +with values given by 1.0, 5.8, 13.6, 20.7 respectively. +The rational map approximation is a generalization of the hedgehog ansatz (3.1) to +σ = cos f, +(π1, π2, π3) = +sin f +1 + |W|2(W + W, i(W − W), 1 − |W|2), +(3.5) +and indeed is identical to the hedgehog ansatz for W = z. Substituting (3.5) into the energy +(2.10) yields +E = 4π +� ∞ +0 +� +r2f ′2 + 2B(f ′2 + 1) sin2 f + I sin4 f +r2 +− 8χ +� +Bf ′2(1 − 3 cos f) − 2I +r2 cos f sin2 f +� +sin f +� +1 + cos f ++ 8 +5χ2(1 + cos f) +� +Bf ′2(41 cos2 f − 30 cos f + 9) + 8sin2 f +r2 +(I(7 cos2 f − 2) + 2Br2) +� +− 1024 +5r2 χ3I(1 + cos f)3/2 sin3 f cos f + 6144 +35r2 χ4I(1 + cos f)2 sin4 f +� +dr, +(3.6) +which is a generalization of (3.2), with the two being identical for B = I = 1. +For each value of χ an annealing algorithm is applied to calculate the energy minimizing +profile functions for B = 1, 2, 3, 4, and the resulting ratios of the energies to the bound +7 + +Figure 2: Left: The ratio of the rational map Skyrmion energies to the bound, as a function +of the symmetry breaking parameter χ, for baryon numbers one to four. Right: A similar +plot but using the estimated Skyrmion energies that are exact when χ = 0. +12π2B are plotted in the left image in Fig.2. Note that the points at which the B = 1 +curve crosses the other curves should not be interpreted as indicating that the corresponding +charge B Skyrmions are unstable, because for B > 1 the rational map approximation only +provides an upper bound on the Skyrmion energy. Indeed, in the standard Skyrme model +(χ = 0) numerical field theory calculations [19] reveal that the rational map approximation +typically overestimates the energy by something in the region of one to two percent, but for +low charges this can be over three percent. Such overestimates take on more significance in +modified models, as they are not small compared to binding energies. +An elementary attempt to address this overestimate issue is to use the previous numer- +ical field theory computations at χ = 0 to remove the overestimate in the rational map +approximation at this value, so that the intercepts of the energy curves are correct and +should therefore provide better estimates of the Skyrmion energies for small χ. The result is +displayed in the right image in Fig.2. This is encouraging, as the B = 1 curve now crosses +the other curves in a small interval of χ, suggesting that it may be possible to have small +binding energies for all B for a suitable value of χ. A caveat to this result is the assumption +that the symmetries given by the rational maps (3.3), namely axial, tetrahedral and cubic, +for B = 2, 3, 4 respectively, persist as symmetries of the minimal energy Skyrmions when +χ ̸= 0. This could be violated, for example by the B = 2 Skyrmion splitting into a pair of +distinct single Skyrmions, and this could lower the energy below the estimate presented in +this plot. To investigate such a possibility requires performing full numerical field theory +simulations, which is a significant task, but the results presented here provide evidence that +this would be a worthwhile endeavour for the future. +8 + +1.24 +B=1 +1.22 +B=2 +B=3 +1.20 +B=4 +1.18 +/(12π²B) +1.16 +1.14 +E +1.12 +1.10 +1.08 +1.06 +0.00 +0.02 +0.04 +0.06 +0.08 +0.10 +0.12 +0.14 +0.16 +0.18 +0.20 +X1.24 +B=1 +1.22 +B=2 +B=3 +1.20 +B=4 +1.18 +B +/(12元²1 +1.16 +1.14 +E +1.12 +1.10 +1.08 +1.06 +0.00 +0.02 +0.04 +0.06 +0.08 +0.10 +0.12 +0.14 +0.16 +0.18 +0.20 +X4 +Conclusion +By exploiting a connection between the Skyrme model and Yang-Mills theory with an extra +dimension, a variant of the Skyrme model has been presented in which derivative terms break +chiral symmetry to isospin symmetry. The new terms are controlled by a single parameter +χ and vanish when χ = 0. The main motivation for the new variant is the fact that χ can +be chosen so that the energy of a single Skyrmion is much closer to the topological energy +bound than when χ = 0. This is encouraging for obtaining more realistic binding energies, +with an analysis using the rational map approximation producing promising initial results. +Further investigations will require numerical field theory computations and the addition of +a pion mass term. It is expected that the structure of Skyrmions in this new variant of the +Skyrme model will be more sensitive to the pion mass for values of χ that yield low binding +energies, as this is the case in the model with ρ mesons that this new theory aims to mimic. +References +[1] N.S. Manton, Skyrmions: A theory of nuclei, New Jersey, World Scientific, 2022. +[2] T.H.R. Skyrme, A unified field theory of mesons and baryons, Nucl. Phys. 31, 556 +(1962). +[3] L.D. 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Livramento, A quasi self-dual Skyrme model, Phys. Rev. D106, +045003 (2022). +9 + +[12] S.B. Gudnason and J.M. Speight, Realistic classical binding energies in the ω-Skyrme +model, JHEP 2007, 184 (2020). +[13] P.M. Sutcliffe, Skyrmions, instantons and holography, JHEP 1008, 019 (2010). +[14] P.M. Sutcliffe, Skyrmions in a truncated BPS theory, JHEP 1104, 045 (2011). +[15] C. Naya and P.M. Sutcliffe, Skyrmions in models with pions and rho mesons, JHEP +1805, 174 (2018). +[16] C. Naya and P.M. Sutcliffe, Skyrmions and clustering in light nuclei, Phys. Rev. Lett. +121, 232002 (2018). +[17] C.J. Houghton, N.S. Manton and P.M. Sutcliffe, Rational maps, monopoles and +Skyrmions, Nucl. Phys. B510, 507 (1997). +[18] M.F. Atiyah and N.S. Manton, Skyrmions from instantons, Phys. Lett. B222, 438 +(1989). +[19] R. A. Battye and P. M. Sutcliffe, Skyrmions, fullerenes and rational maps, Rev. Math. +Phys. 14, 29 (2002). +10 + diff --git a/o9E1T4oBgHgl3EQfPAM1/content/tmp_files/load_file.txt b/o9E1T4oBgHgl3EQfPAM1/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7e25b632b20101aba103d344d8ba20c92c1582ea --- /dev/null +++ b/o9E1T4oBgHgl3EQfPAM1/content/tmp_files/load_file.txt @@ -0,0 +1,334 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf,len=333 +page_content='A Skyrme model with novel chiral symmetry breaking Paul Sutcliffe Department of Mathematical Sciences, Durham University, Durham DH1 3LE, United Kingdom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content=' Email: p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='sutcliffe@durham.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='uk January 2023 Abstract An extension of the Skyrme model is presented in which derivative terms are added that break chiral symmetry to isospin symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content=' The theory contains just one new parameter and it reduces to the standard Skyrme model when this symmetry break- ing parameter vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content=' The same Faddeev-Bogomolny energy bound applies for all parameter values, but the parameter can be tuned so that the energy of the single Skyrmion is much closer to the bound than in the standard Skyrme model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content=' Apply- ing the rational map approximation to multi-Skyrmions suggests that, for a suitable value of the symmetry breaking parameter, binding energies in this theory may be significantly more realistic than in the standard Skyrme model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='03021v1 [hep-th] 8 Jan 2023 1 Introduction Skyrmions are topological solitons that model nuclei within a nonlinear theory of pions [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content=' There have been many developments on Skyrmions since Skyrme’s original work [2], and during the last decade or so some attention has been directed towards modifications of the Skyrme model that bring the energy of a single Skyrmion closer to the Faddeev-Bogomolny energy bound [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content=' The motivation for this is to address the issue that Skyrmions are too tightly bound in comparison to nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content=' If the energy of a single Skyrmion is close to the bound then clearly this places a tight restriction on multi-Skyrmion binding energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content=' However, in the standard Skyrme model the single Skyrmion energy exceeds the bound by around 23% and this leaves plenty of room for large binding energies, well in excess of those of nuclei, that are no greater than around 1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content=' Some examples of modifications to the standard Skyrme model to address this problem include the BPS sextic model [4, 5], the lightly bound model [6, 7], the loosely bound model [8], the dielectric model [9], the self-dual model [10], the quasi self-dual model [11], the inclusion of ω mesons [12], and the addition of ρ mesons via Yang-Mills theory with an extra dimension [13, 14, 15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content=' Each theory has its own positive aspects that make it worthy of investigation, together with negative aspects that mean that alternatives are still being sought.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content=' The purpose of the present paper is to add a new model to the list of modified theories that bring Skyrmion energies closer to the energy bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content=' This is achieved by the introduction of derivative terms that break chiral symmetry to isospin symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content=' The additional terms come with one new parameter that can be chosen so that the energy of the single Skyrmion is reduced from 23% to 7% above the bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content=' An investigation of multi-Skyrmions within this theory, using the rational map approximation [17], suggests that more realistic binding energies may indeed be possible in this broken theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content=' 2 A Skyrme model from Yang-Mills The standard Skyrme model is a nonlinear theory of pions in which the pion fields (π1, π2, π3) are used to construct the Skyrme field U ∈ SU(2) via U = � σ + iπ3 iπ1 + π2 iπ1 − π2 σ − iπ3 � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='1) where the σ field imposes the constraint σ2 + π2 1 + π2 2 + π2 3 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content=' In Skyrme units, the static energy is given by ESkyrme = � −Tr �1 2R2 i + 1 16[Ri, Rj]2 � d3x, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='2) where the su(2)-valued right currents are Ri = ∂iU U −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content=' This theory is invariant under chiral symmetry, given by multiplication of the Skyrme field by an arbitrary constant SU(2) matrix on the left and another arbitrary constant SU(2) matrix on the right, which together rotates the quartet of the σ and pion fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content=' Finite energy 2 imposes the boundary condition that the Skyrme field tends to a constant SU(2) matrix at infinity, taken to be the identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content=' This breaks chiral symmetry to isospin symmetry, given by conjugation of the Skyrme field by an arbitrary constant SU(2) matrix, which rotates the pions fields but leaves the σ field unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content=' Chiral symmetry can also be broken explicitly by the addition of a potential term to the energy (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content=' The most common choice is to add the term 2m2(1 − σ), which gives the pions a mass m in Skyrme units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content=' However, this study will be restricted to the case of massless pions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content=' The boundary condition provides a compactification of space, therefore topologically U is a map between three-spheres with an associated integer B, the topological degree, that may be computed as B = � 1 24π2εijkTr(RiRkRj) d3x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='3) This integer is identified with baryon number and is also referred to as the topological charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content=' Skyrmions are minimizers of the energy for a given positive value of B and their energies satisfy the Faddeev-Bogomolny energy bound [3] ESkyrme ≥ 12π2B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='4) As mentioned above, for the single Skyrmion (B = 1) the energy exceeds this bound by around 23% and an aim is to modify the theory to reduce this excess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content=' An interesting perspective on the Skyrme model is obtained via consideration of SU(2) Yang-Mills theory with an additional spatial dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content=' In four-dimensional Euclidean space the Yang-Mills energy (with a non-standard normalization) is given by E = −3 4 � Tr(FIJFIJ) d4x, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='5) where xI, with I = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='., 4, denote the spatial coordinates and FIJ = ∂IAJ − ∂JAI + [AI, AJ] are the components of the su(2)-valued field strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content=' The instanton number N ∈ Z of the gauge field provides a lower bound on the energy, which for positive N is E ≥ 12π2 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='6) Unlike the Skyrme model, this bound is attained for all positive N, by fields that are self-dual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content=' To make contact with the Skyrme model, fix the gauge A4 = 0 and restrict the remaining three components Ai to have the form Ai = −1 2(1 + φ)Ri, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='7) where φ = tanh x4 and Ri is the right current of the Skyrme field U, that depends upon the remaining three spatial coordinates (x1, x2, x3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content=' Substituting (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='7) into the Yang-Mills energy (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='5) and performing the integration over x4 yields precisely the Skyrme energy (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content=' Furthermore, as U is the holonomy of the gauge field along lines parallel to the x4-axis, the baryon number of the Skyrme field is equal to the instanton number, B = N [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content=' The Faddeev-Bogomolny bound (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='4) therefore follows directly from the Yang-Mills bound (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content=' 3 From the Yang-Mills perspective, the Skyrmion excess energy above the bound (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='4) is a measure of the failure of the restricted form (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='7) to describe a self-dual instanton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content=' The excess can be reduced by improving (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='7) through the addition of a term φ′Vi to the right- hand-side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content=' The su(2)-valued vector field Vi(x1, x2, x3) physically describes ρ mesons, the lightest of the vector mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content=' Performing the integration over x4 in the Yang-Mills energy using the modified restricted form generates a modified Skyrme theory, coupling pions and ρ mesons [13, 14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content=' The Faddeev-Bogomolny bound of 12π2B remains intact, due to its continued inheritance from the Yang-Mills bound, but now the single Skyrmion energy is only around 6% above the bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content=' The symmetries and shapes of the Skyrmions retain the same forms as in the standard Skyrme model, but binding energies are reduced to about one- quarter of their previous values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content=' Moreover, this flattening of the energy landscape means that the addition of a pion mass term results in a more dramatic change in the structure of Skyrmions, which for B > 4 have the cluster structure expected for light nuclei [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content=' The above comments highlight the positive aspects of the Skyrme model with pions and ρ mesons, but there are also some negative aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content=' In particular, with this modification the number of independent field components increases from three to twelve, and in combination with the large number of additional terms in the energy, this makes the theory difficult to work with both analytically and numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content=' This motivates the work in the following, to try and produce a theory with similar results but without introducing any additional fields to supplement the Skyrme field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content=' To derive the new variant of the Skyrme model, first define the twist operator + and its inverse − by the following action on the current R+ i = (1 − U −1)Ri(1 − U) 2(1 − σ) , R− i = (1 − U)Ri(1 − U −1) 2(1 − σ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='8) A quick calculation shows that a double application of the twist operator to the right current yields the left current, R++ i = U −1∂iU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content=' Note that for the twisted current to be well-defined as σ → 1, which of course corresponds to the Skyrme field approaching its vacuum value U → 1, the current must vanish in this limit, Ri → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content=' This condition, that needs to be imposed on the Skyrme field, is the same condition that is imposed in the standard Skyrme model at spatial infinity, by the requirement of finite energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content=' As Skyrmions in the standard Skyrme model already satisfy this property then it seems that this requirement is not a significant issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content=' The new modified Skyrme model is derived by extending the restricted form (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='7) to Ai = −1 2(1 + φ)Ri + χφ′ √1 − σ(R− i − R+ i ) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='9) where χ is a real constant parameter of the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content=' The explicit appearance of the σ field breaks chiral symmetry to isospin symmetry, in the same way as the inclusion of the pion mass term, which also explicitly includes the σ field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content=' The structure of this term is motivated by symmetry considerations and the aim to have a surrogate for the vector field Vi that is constructed from the Skyrme field and brings the restricted form closer to a self-dual field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content=' 4 Substituting (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='9) into the Yang-Mills energy (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='5) and performing the integration over x4 yields the following energy for a modified Skyrme model (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='10) E = � −Tr �1 2R2 i + 1 16[Ri, Rj]2 − χ 4 [R+ i , R+ j ]Hij + 4 5χ3[Di, Dj]Hij + 24 35χ4[Di, Dj]2 + χ2 20 � 32D2 i + 5H2 ij − 8[R+ i , R+ j ][Di, Dj] + ([R+ i , Dj] − [R+ j , Di])2 �� d3x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content=' For notational convenience, in the above the su(2)-valued quantities Di and Hij have been introduced, where Di = Ri − R++ i √1 − σ , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='11) and Hij is the anti-symmetric tensor Hij = 2[Ri, Rj] + 2[R++ i , R++ j ] − [Di, Dj] √1 − σ − Di∂jσ − Dj∂iσ 1 − σ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='12) If χ = 0 then the energy (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='10) reverts to that of the standard Skyrme model, but for χ ̸= 0 it is a deformation via derivative terms that break chiral symmetry to isospin symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content=' The Faddeev-Bogomolny energy bound, E ≥ 12π2B, holds for all χ, as it follows from the Yang-Mills energy bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content=' To get a feel for the structure of the additional terms it is perhaps useful to note that in the first two terms, that agree with the standard Skyrme model, all currents Ri could be replaced by their twisted versions R+ i , because both these terms are invariant under this replacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content=' Regarding Di as a variant of R+ i , and Hij as a variant of [R+ i , R+ j ], the structure of the remaining terms follows from the first two terms by making different possible replacements of the twisted currents and their commutators by these variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content=' 3 Skyrmions To study the single Skyrmion in the theory (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='10), apply the hedgehog ansatz σ = cos f, πi = xi r sin f, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='1) with f(r) being a monotonic decreasing profile function that satisfies the boundary conditions f(0) = π and f(∞) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content=' This yields a spherically symmetric energy density with the energy 5 given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='2) E = 4π � ∞ 0 � r2f ′2 + 2(f ′2 + 1) sin2 f + sin4 f r2 − 8χ � f ′2(1 − 3 cos f) − 2 r2 cos f sin2 f � sin f � 1 + cos f + 8 5χ2(1 + cos f) � f ′2(41 cos2 f − 30 cos f + 9) + 8sin2 f r2 (7 cos2 f − 2 + 2r2) � − 1024 5r2 χ3(1 + cos f)3/2 sin3 f cos f + 6144 35r2 χ4(1 + cos f)2 sin4 f � dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content=' For each value of χ the energy is obtained by calculating the energy minimizing profile function numerically using an annealing algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content=' The result is displayed in the left plot in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='1, where the ratio of the energy to the bound is shown as a function of the symmetry breaking parameter χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content=' It can be seen that as χ increases from zero, the excess energy initially decreases from around 23% in the standard Skyrme model to around 7% for a critical value of χ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='14 (to two decimal places) and then begins to increase again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content=' Recall from an earlier comment that this single Skyrmion excess energy must be greater than 6%, as it cannot be lower than the value obtained in the ρ meson theory that allows an arbitrary unconstrained vector field Vi in place of the specific form used in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content=' Figure 1: Left: The ratio of the single Skyrmion energy to the bound, as a function of the symmetry breaking parameter χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content=' Right: The profile function f(r) for χ = 0 and χ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content=' The energy minimizing profile function f(r) is displayed in the right plot in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='1 for the standard Skyrme model (χ = 0, upper curve) and at the critical value (χ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='14, lower curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content=' It can be seen that the change in the profile function is rather mild, with the Skyrmion being a little smaller in the modified theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content=' 6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='24 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='22 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='18 ()/ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='16 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='14 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='08 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='20 X3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='0 X=0 X = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='14 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='0 f 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='0 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='5 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='0 + 0 1 2 3 4 5 6 7 8 rThe fact that the energy at the critical value is close to the value in the ρ meson theory reveals that the expression used in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='9) is indeed a good surrogate for the inclusion of ρ mesons and justifies this choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content=' It also implies that the new terms in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='10) may be thought of in a similar vein to terms that would be generated by integrating out the ρ mesons in the theory with pions and ρ mesons, to leave an effective theory with just pions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content=' A related point of view is that the ρ mesons have been treated as excitations of pions, extending the philosophy that all particles, mesons and baryons, are modelled as excitations of the pion field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content=' In the standard Skyrme model the minimal energy multi-Skyrmions can be approximated with a good accuracy using the rational map approximation [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content=' It therefore seems reason- able to suppose that this approximation is also valid in the modified model, at least for small values of χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content=' This approximation uses spherical polar coordinates, r, θ, ϕ, with Rie- mann sphere coordinate z = tan(θ/2)eiϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content=' The input to create a charge B Skyrmion is W(z), a degree B rational map in z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content=' For B = 1, 2, 3, 4 these maps are given by W = z, W = z2, W = z3 − √ 3iz √ 3iz2 − 1, W = z4 + 2 √ 3iz2 + 1 z4 − 2 √ 3iz2 + 1, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='3) with the property that for each degree they minimize the angular integral I = 1 4π � �(1 + |z|2)|W ′| 1 + |W|2 �4 2i dzd¯z (1 + |z|2)2, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='4) with values given by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='0, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='8, 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='6, 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='7 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content=' The rational map approximation is a generalization of the hedgehog ansatz (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='1) to σ = cos f, (π1, π2, π3) = sin f 1 + |W|2(W + W, i(W − W), 1 − |W|2), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='5) and indeed is identical to the hedgehog ansatz for W = z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content=' Substituting (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='5) into the energy (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='10) yields E = 4π � ∞ 0 � r2f ′2 + 2B(f ′2 + 1) sin2 f + I sin4 f r2 − 8χ � Bf ′2(1 − 3 cos f) − 2I r2 cos f sin2 f � sin f � 1 + cos f + 8 5χ2(1 + cos f) � Bf ′2(41 cos2 f − 30 cos f + 9) + 8sin2 f r2 (I(7 cos2 f − 2) + 2Br2) � − 1024 5r2 χ3I(1 + cos f)3/2 sin3 f cos f + 6144 35r2 χ4I(1 + cos f)2 sin4 f � dr, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='6) which is a generalization of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='2), with the two being identical for B = I = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content=' For each value of χ an annealing algorithm is applied to calculate the energy minimizing profile functions for B = 1, 2, 3, 4, and the resulting ratios of the energies to the bound 7 Figure 2: Left: The ratio of the rational map Skyrmion energies to the bound, as a function of the symmetry breaking parameter χ, for baryon numbers one to four.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content=' Right: A similar plot but using the estimated Skyrmion energies that are exact when χ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content=' 12π2B are plotted in the left image in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content=' Note that the points at which the B = 1 curve crosses the other curves should not be interpreted as indicating that the corresponding charge B Skyrmions are unstable, because for B > 1 the rational map approximation only provides an upper bound on the Skyrmion energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content=' Indeed, in the standard Skyrme model (χ = 0) numerical field theory calculations [19] reveal that the rational map approximation typically overestimates the energy by something in the region of one to two percent, but for low charges this can be over three percent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content=' Such overestimates take on more significance in modified models, as they are not small compared to binding energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content=' An elementary attempt to address this overestimate issue is to use the previous numer- ical field theory computations at χ = 0 to remove the overestimate in the rational map approximation at this value, so that the intercepts of the energy curves are correct and should therefore provide better estimates of the Skyrmion energies for small χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content=' The result is displayed in the right image in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content=' This is encouraging, as the B = 1 curve now crosses the other curves in a small interval of χ, suggesting that it may be possible to have small binding energies for all B for a suitable value of χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content=' A caveat to this result is the assumption that the symmetries given by the rational maps (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='3), namely axial, tetrahedral and cubic, for B = 2, 3, 4 respectively, persist as symmetries of the minimal energy Skyrmions when χ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content=' This could be violated, for example by the B = 2 Skyrmion splitting into a pair of distinct single Skyrmions, and this could lower the energy below the estimate presented in this plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content=' To investigate such a possibility requires performing full numerical field theory simulations, which is a significant task, but the results presented here provide evidence that this would be a worthwhile endeavour for the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content=' 8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='24 B=1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='22 B=2 B=3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='20 B=4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='18 /(12π²B) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='16 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='14 E 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='08 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='20 X1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='24 B=1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='22 B=2 B=3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='20 B=4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='18 B /(12元²1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='16 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='14 E 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='08 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content='20 X4 Conclusion By exploiting a connection between the Skyrme model and Yang-Mills theory with an extra dimension, a variant of the Skyrme model has been presented in which derivative terms break chiral symmetry to isospin symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content=' The new terms are controlled by a single parameter χ and vanish when χ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content=' The main motivation for the new variant is the fact that χ can be chosen so that the energy of a single Skyrmion is much closer to the topological energy bound than when χ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content=' This is encouraging for obtaining more realistic binding energies, with an analysis using the rational map approximation producing promising initial results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E1T4oBgHgl3EQfPAM1/content/2301.03021v1.pdf'} +page_content=' Further 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+1,2922 @@ +AN EFFICIENT QUADRATURE SEQUENCE AND SPARSIFYING +METHODOLOGY FOR MEAN-FIELD VARIATIONAL INFERENCE +JED A. DUERSCH∗ +Abstract. +This work proposes a quasirandom sequence of quadratures for high-dimensional +mean-field variational inference and a related sparsifying methodology. Each iterate of the sequence +contains two evaluations points that combine to correctly integrate all univariate quadratic functions, +as well as univariate cubics if the mean-field factors are symmetric. More importantly, averaging re- +sults over short subsequences achieves periodic exactness on a much larger space of multivariate +polynomials of quadratic total degree. +This framework is devised by first considering stochastic +blocked mean-field quadratures, which may be useful in other contexts. By replacing pseudoran- +dom sequences with quasirandom sequences, over half of all multivariate quadratic basis functions +integrate exactly with only 4 function evaluations, and the exactness dimension increases for longer +subsequences. Analysis shows how these efficient integrals characterize the dominant log-posterior +contributions to mean-field variational approximations, including diagonal Hessian approximations, +to support a robust sparsifying methodology in deep learning algorithms. A numerical demonstration +of this approach on a simple Convolutional Neural Network for MNIST retains high test accuracy, +96.9%, while training over 98.9% of parameters to zero in only 10 epochs, bearing potential to reduce +both storage and energy requirements for deep learning models. +Key words. variational inference, mean-field, quadrature, cubature, Hadamard basis, sparsity, +spike and slab, Hessian approximation +MSC codes. 62F30 65C05 65D32 65K10 +1. Introduction. Variational inference is an optimization-based approach to +discover parameter domains that dominate the Bayesian posterior with origins in sta- +tistical physics [28, 34, 4, 18, 5, 45]. One of the challenges with quantifying prediction +uncertainty for high-dimensional models is how to reliably characterize model uncer- +tainty as it evolves during training. For high-dimensional model classes, mean-field +distributions provide a simple and scalable method to track a component of model +uncertainty and thereby capture a useful contribution to uncertainty in predictions +at a reduced computational cost [1, 33]. In principle, optimizing the variational ob- +jective requires repeatedly integrating the log-likelihood of the training data as the +variational distribution changes. Thus, having a quadrature framework to efficiently +capture the primary contributions to the shape of a mean-field distribution, using only +a handful of function evaluations, reduces the computational burden of optimization. +The aim of this work is to improve the computational efficiency of variational +inference and related sparsifying methodologies by improving numerical integration. +This work proposes two blocking-based quadrature techniques that are suitable for +mean-field variational inference. The first approach, stochastic blocked mean-field +quadratures, may be useful for learning architectures that, based on the model’s +computational structure, allow us to identify key blocks of parameters that may con- +tain important correlations. Blocked quadratures allow these correlations to be feasi- +bly captured with high precision while still retaining scalability for high-dimensional +model classes. The second approach, derived as a quasirandom modification of the +∗Sandia National Laboratories, Livermore, CA (jaduers@sandia.gov). +Sandia National Laboratories is a multimission laboratory managed and operated by National Tech- +nology and Engineering Solutions of Sandia, LLC., a wholly owned subsidiary of Honeywell Inter- +national, Inc., for the U.S. Department of Energy’s National Nuclear Security Administration under +contract DE-NA-0003525. This paper describes objective technical results and analysis. Any sub- +jective views or opinions that might be expressed in the paper do not necessarily represent the views +of the U.S. Department of Energy or the United States Government. +1 +arXiv:2301.08374v1 [cs.LG] 20 Jan 2023 + +2 +J. A. DUERSCH +first, gains a striking additional property of periodic exactness on much larger bases. +To put this property in perspective, we might expect that a quadrature taking 4 func- +tion evaluations in d parameter dimensions, comprising 4(d + 1) degrees of freedom, +should only integrate the same number of basis functions. Yet, by averaging a consec- +utive pair of two-point quadratures from a quasirandom sequence, the result exactly +integrates 1+2d+ 1 +4d2, i.e. more than half, of all basis functions for quadratic total de- +gree polynomials. As d may easily surpass a million in modern learning architectures, +this provides a significant improvement in approximation quality at low cost. +Efficient numerical integration enables robust training methodologies for varia- +tional inference in high-dimensions. By analyzing the structure of optimal variational +distributions, this work illuminates the primary attributes of a fixed-point optimiza- +tion. Related linear functionals that act on gradients allow efficient approximation +of quadratic loss structure in each parameter. Not only does this support variational +inference with Gaussian mean-fields, it is also compatible with spike and slab distri- +butions that are suitable to induce sparsity during training. +1.1. Contributions. The key contributions of this work are 1. new numerical +integration schemes that are suitable for mean-field and blocked mean-field distri- +butions, 2. analysis of optimal variational distributions, bridging efficient integration +with a fixed-point optimization, and 3. a sparsifying methodology designed to over- +come practical implementation challenges for deep learning models. +1.1.1. Efficient Numerical Integration. Both of the proposed integration +approaches proceed by partitioning parameters into small blocks. The first approach +simply requires equal-weight sigma-point quadratures [43, 27] within each block. Pro- +vided all blocks use the same number of function evaluations, the evaluation coor- +dinates can be permuted uniformly at random and concatenated. Doing so retains +the same exactness property within each block, but also yields an expectation match- +ing the tensor product cubature over all blocks. This allows efficient integration of +blocked mean-field distributions, which could contain more comprehensive factor dis- +tributions that track correlations within each block, rather than only using products +of univariate distributions. This work does not further examine how to design and +implement blocked mean-field distributions for variational inference, only how to ef- +ficiently integrate them. +The second approach exchanges pseudorandom concatenation for quasirandom +sequences. +Each element of the sequence is a 2-point quadrature that exactly in- +tegrates all linear combinations of univariate quadratics. If the factor distributions +are symmetric, these are actually Gaussian quadratures and they integrate all linear +combinations of univariate cubics. Averaging over a subsequence that contains an in- +teger multiple of 2b iterates, where b = 1, 2, . . . , ⌈log2(d)⌉, yields exactness on an extra +d2 2b−1 +2b+1 dimensions of the function space comprising quadratic total degree polynomi- +als. This effect is due to the quasirandom sequence creating a hierarchy of overlapping +blocks that contain tensor product cubatures from the underlying quadratures. +1.1.2. Variational Fixed-Point Optimization. Analysis shows how updating +the variational distribution only requires projecting the log-posterior distribution onto +a compatible basis, provided the variational distribution has an exponential structure. +For Gaussian mean-field distributions, this analysis leads to update expressions that +are somewhat similar to the gradient averaging and variance-based scaling used in +ADAM[21], but quasirandom quadratures offer more efficient gradient and Hessian +projections against the mean-field distribution. Although a mean-field distribution, + +AN EFFICIENT QUADRATURE SEQUENCE AND SPARSIFYING METHODOLOGY +3 +q(θ), with mean, Eq(θ)[θ] = µ, and diagonal covariance, Varq(θ)[θ] = diag(σ)2, +cannot contain off-diagonal covariance terms, the projection quality is still affected +by whether off-diagonal covariance integrals correctly vanish, e.g. E[(θ1 − µ1)(θ2 − +µ2)] = 0. Thus, periodically enhanced exactness translates to more precise variational +updates. +Since the update expressions are derived from the structure of the variational ba- +sis, the same analytic framework also facilitates other update expressions for different +variational bases. A closely-related derivation provides updates for Dirac-Gauss mix- +tures, i.e. spike and slab distributions, which are needed to efficiently induce sparsity +by associating zeros with finite probabilities. +1.1.3. Sparsifying Methodology. By iterating through the quadrature se- +quence in concert with randomly permuted training data, we obtain a method that +averages errors, and converges to high accuracy when the distribution stabilizes and +more data are taken into account. A simple averaging technique using restarted sums +for recently processed data also allows old gradient and Hessian contributions to be +dropped more efficiently than the exponentially-damped averages used in ADAM. +This improves up-to-date approximations of the loss structure that drives sparsity to +support a sparsifying sieve that gradually filters parameters that are most suitable to +vanish from those that must be retained and readjusted. +Implementation challenges and the resulting sparsifying methodology are de- +scribed in detail. Numerical experiments demonstrate the ability to achieve extreme +sparsity, dropping 98.96% of parameters while retaining over 96.95% validation accu- +racy, and using only 10 epochs with just 4 prediction evaluations per training case. +This stands in sharp contrast to the 100s or 1000s of epochs that are required to +achieve convergence with other approaches. +1.2. Organization. Section 2 provides background on variational inference, +mean field distributions, related approaches to numerical integration, and recent +work on sparsification with variational inference. Section 3 proposes and analyzes +stochastic blocked mean-field quadratures and quasirandom quadrature sequences for +high-dimensional numerical integration. Section 4 analyzes a variational fixed-point +optimization that bridges efficient integration with Gaussian mean-field updates and +Dirac-Gauss mixtures. Section 5 outlines key implementation challenges and practi- +cal algorithmic design solutions for a sparsifying methodology, followed by numerical +experiments. Section 6 provides a brief discussion of potential follow-up work and a +final summary. +2. Background. Bayesian inference provides an attractive paradigm to quan- +tify prediction uncertainty by consistently resolving uncertainty in models that could +explain available data. Given a training dataset D, a model class with parameters θ, +prior belief p(θ), and likelihood p(D | θ), we obtain the posterior by applying Bayes’ +theorem, +p(θ | D) = p(D | θ)p(θ) +p(D) +, +where +p(D) = +� +dθ p(D | θ)p(θ) +is the model evidence. Unfortunately, when the likelihood function is complicated, +especially for high-dimensional architectures, capturing the shape of the posterior +becomes intractable due to limited computational resources. +Variational inference mitigates this issue by approximating the posterior with a +simpler distribution, q(θ | ϕ). Variational parameters ϕ characterize the approximate + +4 +J. A. DUERSCH +shape of a posterior-dominant region of the parameter domain. We discover such +domains by optimizing a variational objective, such as the maximizing the Evidence +Lower Bound (ELBO). Since the ELBO optimizer also minimizes the Kullback-Leibler +(KL) divergence [22], from the posterior to the variational distribution, this is a +practical minimization objective for training: +D[ q(θ | ϕ) ∥ p(θ | D) ] = +� +dθ q(θ | ϕ) log +� q(θ | ϕ) +p(θ | D) +� +. +(2.1) +It is also a principled objective in its own right, minimizing excess information created +by replacing the posterior distribution with a feasible approximation [12]. +Provided the dataset is composed of independent cases from the data-generating +process, D = {dc | c ∈ [nc]}, the optimizer of Equation (2.1) can be written as a sum +of integrals over each case c, +ϕ∗ = argmin +ϕ +D[ q(θ | ϕ) ∥ p(θ) ] − +nc +� +c=1 +� +dθ q(θ | ϕ) log p(dc | θ). +Notably, this construction does not depend on the model-evidence integral. Opti- +mization does, however, require repeatedly evaluating the log-likelihood integral as +the variational distribution evolves. +This work focuses on mean-field variational distributions, because they offer the +most scalable approach for high-dimensional model classes. Given d parameters, in- +dexed1 as i ∈ {0, 1, . . . , d − 1}, mean-field distributions take the form +q(θ | ϕ) = +d−1 +� +i=0 +q(θi | ϕi), +(2.2) +where each ϕi may contain several variational parameters that describe the shape +of each factor. This simple structure is what allows the efficient high-dimensional +numerical quadratures developed in Subsection 3.2. +2.1. Monte Carlo Versus Quadratures. There are generally two approaches +to account for probability in numerical integration, stochastic methods and determin- +istic methods. In the first case, randomized sampling matches integral contributions +in expectation over pseudorandom events that generate function evaluations. Monte +Carlo methods, including Markov Chain Monte Carlo (MCMC) [31, 16] as well as +tempered variations that achieve better posterior convergence [8, 23], are entirely sto- +chastic approaches, mapping pseudorandom numbers to a set of samples from the +posterior distribution. As the sample size ns increases, the scale of the integration +error drops as ε(ns) ≈ ε(1)ns−1/2. The drawback of sampling is that, although it will +eventually produce integral approximations with arbitrarily small error, the number +of function evaluations needed can be quite large. +Deterministic approaches are the domain of typical numerical quadrature formu- +las. For integration in multiple dimensions, these are often called cubature methods. +We solve a set of nq evaluation locations paired with weights, +� +(θ(q), wq) | q ∈ [nq] +� +, +that exactly integrate some basis of functions, Φ = {fℓ(·) | ℓ ∈ [nexact]} so that +Q [f] = +nq +� +q=1 +wqf(θ(q)) ≈ +� +dθ q(θ)f(θ) +where +Q [fℓ] = +� +dθ q(θ)fℓ(θ) +1We enumerate parameters from zero to be consistent with analysis in Section 3. + +AN EFFICIENT QUADRATURE SEQUENCE AND SPARSIFYING METHODOLOGY +5 +is exact for all ℓ ∈ [nexact]. Smolyak quadratures [39, 15, 35] are efficient formulas to +generate high-degree quadratures in a few dimensions. Novak and Ritter [32] show +that nq ≈ 2k +k dk quadrature nodes can be constructed to exactly integrate polynomials +of total degree 2k+1 in d dimensions against a fixed weight function with the structure +of Equation (2.2). +Unfortunately, even taking k = 1 is too expensive for current +learning architectures, where d often ranges from 105 to 1010 or more. Since optimizing +the variational distribution requires evaluating as many integrals as there are training +data, this approach is not feasible. We desire an approach that 1. only uses a few +function evaluations per integral, 2. exactly integrates basis functions that dominate +the shape of the mean-field distribution, and 3. efficiently suppresses the unavoidable +errors associated with basis functions that we cannot afford to integrate exactly. +2.2. Quasi-Monte Carlo Integration. Quasi-Monte Carlo methods find mid- +dle ground by incorporating both psuedorandom and deterministic aspects within the +sample-generating process. Caflisch [7] provides an overview targeting the perspec- +tive of applied mathematicians and numerical analysts. For example, we can use two +evaluation points, θ(1) = µ − δ and θ(2) = µ + δ, with equal weights, w1 = w2 = 1 +2, to +exactly integrate all affine functions against a distribution with mean µ. This is what +Caflisch calls an antithetic pair, i.e. a pair of evaluation points balanced about the +mean of the distribution. Other moment-matching methods extend this simple tech- +nique. If we know the mean of the distribution to be integrated, then we can adjust a +set of samples to exactly integrate all affine functions by applying a simple translation +of sample coordinates. Given a set of samples, Θ = +� +θ(q) ∼ q(θ) | q ∈ [ns] +� +, and a +known mean, Eq(θ)[θ] = µ, we can translate the samples as +θ(q)′ = θ(q) − ˆµ + µ +where +ˆµ = 1 +ns +ns +� +q=1 +θ(q). +Likewise, if we also have known diagonal covariance, e.g. Eq(θ)[(θ(q)−µ)(θ(q)−µ)T ] = +diag(σ)2, then we can use an affine transformation to exactly integrate all univariate +quadratic polynomials, and linear combinations of them, +θ(q)′ = (θ(q)′ − ˆµ) ∗ σ ⊘ ˆσ + µ +where +ˆσ2 = 1 +ns +ns +� +q +(θ(i) − ˆµ) ∗ (θ(i) − ˆµ). +The operator ⊘ indicates elementwise division and the operator ∗ represents the Ha- +damard product. As the set of samples becomes large, the sample moments converge +to the true moments, making the correction increasingly modest so that the same +convergence properties of Monte Carlo hold. Unfortunately, this approach requires +having the full set of sample locations before the correction can be made, whereas the +quasirandom quadrature sequences described in Subsection 3.2 can be accumulated +to match more basis functions as more function evaluation become available. +Beylkin [3] also examines numerical algorithms in high dimensions and Dick [11] +provides a recent overview of quasi-Monte Carlo methods for high-dimensional in- +tegration over the unit cube, [0, 1]d. Trefethen [41, 42] examines high-dimensional +integration methods that aim to subdue the curse of dimensionality, observing that +it is the special structure of certain problem-dependent integrands, deviating from +the anisotropy of the hypercube, that allows some quadrature formulas to avoid the +exponential increase in evaluation nodes needed for tensor-product cubatures. + +6 +J. A. DUERSCH +2.3. Latin Hypercube Sampling. Latin hypercube sampling [26] is another +quasi-Monte Carlo approach that closely relates to this work. It is a form of stratified +sampling that matches, in expectation, stratified sampling on the Cartesian product +of subsets in each dimension. If we break a single dimension of the integral domain +into subsets of equal probability, and ensure that an equal number of samples are +drawn from each subset, then we obtain a more precise integral approximation in that +dimension, because the samples equally represent components that are analytically +equivalent contributions. The key insight of Latin hypercube sampling is that if we +obtain such samples from each dimension independently within a mean-field distribu- +tion, permuting samples within each dimension uniformly at random, the expectation +of the result still matches the product of integrals over all dimensions. Figure 1 pro- +vides an illustration of this technique in two dimensions. The same idea drives the +development and analysis of stochastic blocked mean-field quadratures in Section 3. +Fig. 1. Latin hypercube illustration for 2D Gaussian separated into 10 equal-probability subin- +tervals for each dimension. +The quasirandom partition of samples produces a set of evaluation +points that are more evenly distributed in each dimension. Samples are composed by randomly per- +muting the source partitions in each coordinate and concatenating the results. If the distribution is +independent in each coordinate, the expectation matches the exact integral. +2.4. Sparsifying Variational Inference. Simplifying the complexity of learn- +ing models [40, 37, 17] is a fundamentally sounds objective [30, 13] in abstract learning +models and an experimentally demonstrated means to improve the generalization of +high-dimensional learning models [19]. A direct approach to achieving this is to in- +duce sparsity in model parameters, which may also benefit the operating speed and +power consumption of trained models. Combining spike and slab priors [29, 25] with +variational inference [9, 2, 20] allows finite probabilities to be assigned to parameter +zeros that can be controlled during training. +Polson and Roˇckov´a [36] developed Spike-and-Slab Deep Learning (SS-DL) mod- +els to improve generalizability by inducing sparsity. They show that this kind of regu- +larization has the ability to learn α-H¨older smooth functions efficiently, even when the +degree of smoothness is unknown. Schmidt-Hieber [38] also analyzes these learning +problems with DNNs comprised of ReLU activation functions and finds that sparsity +is a key tuning parameter affecting performance. Ch´erief-Abdellatif [9] derives gen- +eralization error bounds and automatic architecture optimization for nonparametric +regression of such functions using Deep Neural Networks (DNNs) with sparse varia- +tional inference. One of the key challenges identified in that work is how to design +efficient computational methods for spike and slab variational inference. +Bai, Song, and Cheng [2] address this challenge by proposing a sampling method- + +Latin Hypercube Sample for 2D Gaussian +2 +1 +2 +X +0 +-1 +-2 +-2 +-1 +0 +1 +2 +Parameter1AN EFFICIENT QUADRATURE SEQUENCE AND SPARSIFYING METHODOLOGY +7 +ology for spike and slab variational distributions that uses fully-realized sparsity pat- +terns in forward propagation. To support optimization with stochastic gradient meth- +ods, such as ADAM, they replace sparse samples with Gumbel-softmax probabilities +in backpropagation. Jantre, Bhattacharya, and Maiti [20] build on that approach +by associating the probability of specific nodes in a neural network, i.e. hidden-layer +neurons, with finite inclusion probabilities. This allows all of the connected edges +in the computational graph, i.e. weight-matrix elements, to be dropped at once, and +thereby yield more efficient processing in fully trained sparse networks. +To distinguish choices of a Gaussian spike [6], a Laplacian spike [10], or a Dirac +spike [2, 20], the explicit term, Dirac-Gauss mixture, will be used going forward in +this work. One key difference in this work is that quasirandom quadratures, Subsec- +tion 3.2, enable more efficient integration of gradients as the variational distribution +evolves. +The same gradient evaluations also yield approximate Hessian integrals, +Subsection 4.2. The projection-based analysis shows how this Hessian information +incorporates into the sparse variational updates Subsection 4.3. This is combined +with an empirically-developed sparsifying methodology that only required 10 epochs +to converge in numerical experiments, rather than hundreds or even thousands that +have been needed for other approaches. +3. Analysis of Blocked Mean-Field Quadratures. This section begins with +a discussion of blocked mean-field quadratures in Subsection 3.1. This line of reasoning +provides a stepping stone to the more powerful quasirandom sequences discussed in +Subsection 3.2. See Appendix A for numerical integration experiments for various +mean-field distributions with corresponding basis functions. +3.1. Stochastic Blocked Mean-Field Quadratures. Just as a mean-field +distribution is a product of distributions in each parameter, we can define a blocked +mean-field distribution as a product of distributions over whole blocks. +Formally, +when we define such a blocking structure, we have decomposed the parameter vector +space into a direct sum of orthogonal subspaces, +Θ = +nb +⊕ +b=1 Θb +so that +θb ∈ Θb +indicates the parameters with a single block b, rather than an individual parameter. +With an appropriate basis, any parameter state can be represented by concatenating +these components. This facilitates a modest generalization of mean-field distributions +to more comprehensive representations of parameter uncertainty within each block as +q(θ) = +nb +� +b=1 +q(θb) +where +θ = +� +θT +1 θT +2 · · · θT +nb +�T +∈ Θ, +while remaining scalable by disregarding correlations between blocks. Just as Latin +hypercube sampling concatenates stratified samples in each coordinate by using uni- +formly random permutations that match, in expectation, higher-dimensional strat- +ification, we apply the same insight to equal-weight quadratures within each block +to obtain composite quadratures with expectations that match the tensor product +cubature and still retain the exactness design within each block. +For each block b, we construct an equal-weight quadrature, Qb[·], that exactly +integrates a set of rb basis functions, Φb = +� +f (b) +k (θb) : Θb �→ R | k ∈ [rb] +� +, against the +corresponding block distribution q(θb). Given any function f(θb) ∈ span(Φb), we + +8 +J. A. DUERSCH +have exactly nq evaluation nodes, θ(q) +b +∈ Θb for q ∈ [nq], for which +Qb[f] = 1 +nq +nq +� +q=1 +f(θ(q) +b ) = +� +dθb q(θb)f(θb). +For example, a good choice would be Uhlmann’s sigma points [43, 27], Algorithm A.1. +We can then form a stochastic blocked mean-field quadrature by concatenating +independent uniformly random permutations, represented by permutation matrices +Pb for all b ∈ [nb], applied to the evaluation nodes in each block as +� +θ(1) +· · · +θ(nq)� += +� +����� +[θ(1) +1 +θ(2) +1 +· · · +θ(nq) +1 +]P1 +[θ(1) +2 +θ(2) +2 +· · · +θ(nq) +2 +]P2 +... +[θ(1) +nb +θ(2) +nb +· · · +θ(nq) +nb ]Pnb +� +����� +so +Q[f] = 1 +nq +nq +� +q=1 +f(θ(q)). +Theorem 3.1 (Expectation Exactness). Given a blocked mean-field distribution +and a stochastic equal-weight quadrature as described above, for any function that is +a product of exact functions within each block, +f(θ) = +nb +� +b=1 +f (b)(θb) +where +f (b) ∈ span(Φb), +the expectation of the quadrature is exact, +EP1,P2,...,Pnb Q[f] = +� +dθ q(θ)f(θ). +See Appendix B.1 for a short proof. The key idea is that by composing equal- +weight quadratures from each block of coordinates with concatenation, the result +retains the same exactness for functions restricted to each block. Since each quadra- +ture is equal-weight and the evaluation nodes are permuted uniformly at random, the +probability of concatenating any specific sequence of evaluations nodes is a constant +that coincides with the weight of each node in the tensor product cubature. +3.1.1. Extra Exactness. The exactness we obtain from this method goes be- +yond the partition of consecutive blocks. To understand this, consider this example +of concatenated 2-block sigma points (vertices of an equilateral triangle) in 4 pairs: +� +�� +θ(1)T +θ(2)T +θ(3)T +� +�� = +� +���� +√ +2 +0 +−1 +√ +2 +− +� +3 +2 +√ +2 +0 +−1 +√ +2 +� +3 +2 +−1 +√ +2 +− +� +3 +2 +√ +2 +0 +−1 +√ +2 +� +3 +2 +−1 +√ +2 +− +� +3 +2 +−1 +√ +2 +� +3 +2 +−1 +√ +2 +� +3 +2 +−1 +√ +2 +− +� +3 +2 +√ +2 +0 +� +���� . +As intended, these evaluation nodes contain sigma points in the 1-2 block, as well as +the 3-4, 5-6, and 7-8 blocks. However, we also obtained sigma points in the 1-6 and +2-5 blocks, since the permutations happen to have produced compatible pairings. +Figure 2 shows how extra exactness for mixed quadratic basis functions varies +with the block size of the partition. Within each block, the quadrature uses sigma +points composed of the simplex vertices. Since a larger block size increases the number +of permutations, the probability of realizing compatible permutations drops for larger + +AN EFFICIENT QUADRATURE SEQUENCE AND SPARSIFYING METHODOLOGY +9 +blocks. Thus, the reduction in extra exactness can outweigh the increase in exactness +within each block. This comparison also includes quadratures from Subsection 3.2, +cross-polytope vertices in the Hadamard basis, which are discussed next. Each nu- +merical experiment counts exact off-diagonal integrals on a 6000 × 6000 mean-field +covariance matrix, which should all be zero, and the average over 100 trials is shown. +Fig. 2. Left: Increasing the block dimension for the simplex-based quadratures does not neces- +sarily increase the number of exact mixed quadratic basis functions. Right: Using a block size of 2 +or 3 for the simplex-based quadratures, requiring 3 or 4 function evaluations, respectively, maximizes +the number of exact basis functions per evaluation. However, the cross-polytope sequence is even +more efficient, with the optimum at 4 function evaluations per quadrature. +3.2. Cross-Polytope Vertices in the Hadamard Basis. Unfortunately, we +often encounter a substantial difference between the integration error corresponding +to the expectation, i.e. the limit of averaging samples over increasingly long sequences, +and the error corresponding to a single quadrature or even an average of a few samples. +What follows is the result of trying to develop a quasirandom sequence to suppress +error more efficiently. +Consider concatenating a set of two-point equal-weight quadratures in each pa- +rameter dimension. This is the minimum number of evaluation nodes required to +exactly integrate all univariate quadratic functions, thus recovering both the mean +and diagonal covariance of a mean-field distribution. Rather than averaging several +quadratures composed by random permutations, we could just form tensor product cu- +batures in adjacent pairs. If the mean is Eq(θ) [θ] = µ and Eq(θ) +� +(θ − µ)(θ − µ)T � += +diag(σ)2, then these four evaluation nodes are +� +θ(1) +θ(2) +θ(3) +θ(4)� += µ + diag(σ) +� +������ +1 +−1 +1 +−1 +1 +−1 +−1 +1 +1 +−1 +1 +−1 +1 +−1 +−1 +1 +... +� +������ +. +Not only does this result in exact pairwise cubatures with only four function evalua- +tions, the exact cubature blocks actually include all d2 +4 pairs of dimensions containing +both an even parameter and an odd parameter. One can easily show that this is the +maximum number of exact blocks that can be obtained by switching some of the signs +of the second pair of nodes. +Building on this strategy, we can then partition coordinates into consecutive 4- +blocks. Repeating the evaluation nodes, but flipping the signs in the second pair, +results in a tensor product cubature in each 4-block. Iterating such node sequences to + +Mixed Quadratic Exactness Comparison +Mixed Quadratic Efficiency Comparison +cross-polytope vertices, Hadamard basis +simplex vertices +0.12 +evaluation +0.8 +0.1 +fraction +0.6 +0.08 +per +fraction r +Exact +0.06 +0.4 +Exact +0.04 +0.2 +0.02 +2 +4 +6 +8 +10 +12 +14 +16 +2 +4 +6 +8 +10 +12 +14 +16 +Function eyaluations +Function evaluations10 +J. A. DUERSCH +larger blocks to obtain cubatures of still higher dimensions yields the cross-polytope +vertices in the Hadamard basis. Algorithm 3.1 generates the signs needed to con- +struct an antithetic pair of vertices from the iterate index, q = 0, 1, . . . , 2⌈log2(d)⌉ − 1. +Each quadrature from this sequence still exactly integrates all linear combinations of +univariate quadratics, or 1 + 2d basis functions, with the mean-field distribution. If +the mean-field distribution is symmetric in each coordinate, all linear combinations +of univariate cubics are exact as well. +Algorithm 3.1 Cross-Polytope Vertex Sequence in Hadamard Basis +d is the number of parameter dimensions. +q is a non-negative integer index for the desired term of the sequence. +Output: s is the d×1 vector of signs needed to construct an antithetic pair, θ(2q+1) = +µ + s ∗ σ and θ(2q+2) = µ − s ∗ σ, for an equal-weight (w = 1 +2) quadrature for a +distribution with mean, µ, and covariance, diag(σ)2. +1: function s = quadrature sequence(d, q) +2: +Get the number of bits needed to index parameters, nb = ⌈log2(d)⌉. +3: +Compute the parity of each parameter dimension for iterate q, +pi = +nb +xor +j=1 [bitj(i) ∧ bitj(q)] +for all +i ∈ {0, 1, . . . , d − 1} . +4: +Return signs from parity, s = 2p − 1. +5: end function +Lemma 3.2 and Theorem 3.3 show how Algorithm 3.1 produces quadrature sub- +sequences that obtain periodic exactness within two-dimensional subspaces. See Ap- +pendix B.2 and Appendix B.3 for proofs. Figure 3 provides a visualization of the +cross-polytope sequence in the Hadamard basis (left) and the number of function +evaluations needed to correctly integrate specific covariance basis functions (right). +Lemma 3.2 (Relative Parity). +The relative parity, pi1 xor pi2, corresponding to +any two distinct parameters, θi1 and θi2, in Algorithm 3.1 determines the product +of corresponding signs in the result and it only depends on the iterate index, q, and +the binary string obtained by bitwise exclusive disjunction of binary representations of +parameter indices, +pi1 xor pi2 = +nb +xor +j=1 [xj ∧ bitj(q)] +where +xj = bitj(i1) xor bitj(i2) +for +j ∈ [nb]. +Theorem 3.3 (Exactness Periodicity). +Let θi1 and θi2 be any two distinct pa- +rameters. Let b ∈ [nb] indicate the position of the least-significant bit that is different +between both binary representations of their indices, i1 and i2. Every consecutive con- +tiguous quadrature subsequence of 2b antithetic pairs obtained by Algorithm 3.1, using +iterate indices q = z2b, z2b + 1, . . . , (z + 1)2b − 1 for z ∈ Z≥0, averages to the 4-node +tensor-product cubature over the corresponding two-dimensional subspace. +These subsequences also generate higher-order cubatures in up to nb + 1 dimen- +sions for some specific sets of parameters. This occurs when the binary representa- +tions of parameter indices only differ by a single bit, each, from a base index. For + +AN EFFICIENT QUADRATURE SEQUENCE AND SPARSIFYING METHODOLOGY 11 +Fig. 3. Left: Illustration of cross-polytope vertex evaluations in 8D, visualized with the nested +coordinate scaling x = 2 +5 ( 2 +5 θ1 + θ4) + θ7, y = 2 +5 ( 2 +5 θ2 + θ5) + θ8, and z = 2 +5 θ3 + θ6. The fine grid +points are hypercube vertices, a superset of the cross-polytope vertices in the Hadamard basis. Every +antithetic pair, e.g. in lavender, exactly integrates all univariate quadratics. Longer sequences— +shown in blue, green, and red—exactly integrate increasing sets of multivariate quadratics. +Right: Visualization of the number of function evaluations from this sequence needed to integrate +each product. Every four evaluations creates 2D cubatures for half of all coordinate pairs (exact on +lavender diagonal and blue checkerboard). +example, if parameters θi1, θi2, θi3, and θi4 are such that only bit1(i1) ̸= bit1(i4), +bit2(i2) ̸= bit2(i4), bit3(i3) ̸= bit3(i4), and all other bits are the same as i4, then +every 8 quadratures (16 evaluations) would result in a tensor-product cubature for +functions that only depend on these parameters. This easily follows by applying the +same reasoning as used in Appendix B.3. Unfortunately, this observation bears lit- +tle importance since errors in many two-dimensional subspaces will still persist, and +dominate, until 2nb quadratures have been averaged. +4. Variational Fixed-Point Optimization. Efficient numerical integration +enables a simple optimization procedure for mean-field variational inference as a +fixed-point iteration. Subsection 4.1 analyzes the relationship between an optimal +variational approximation and the corresponding integrals needed to project the pos- +terior distribution onto a variational basis. Subsection 4.2 shows how to modify the +preceding quadrature sequences to efficiently implement these posterior projections +onto a quadratic basis for Gaussian mean-field distributions. Subsection 4.3 provides +an extension to mean-field distributions that are suitable for capturing sparsity. +4.1. Basis Analysis for Fixed-Point Optimization. Let us consider a family +of variational distributions that may be written as an exponential of a linear combi- +nation of basis functions, fℓ(θ) for ℓ = 0, 1, . . . , nℓ. That is, +q(θ | ϕ) = exp +� nℓ +� +ℓ=0 +ϕℓfℓ(θ) +� +and +ϕ∗ = argmin +ϕ +D[ q(θ | ϕ) ∥ p(θ | D) ] +is the vector of coefficients corresponding to the optimizer. Clearly, we must restrict +feasible coefficients, ϕ, to those that yield proper, normalized, distributions. Since + +Cross-Polytope Vertex Sequence in 8D +Evaluations Needed for Exact Quadratic Integrals +D +Factor +Integrand +8 +and +econd +5 +2 +S +9 +and +3 +01 +2 +04 +05 +06 +, and +First Integrand Factor12 +J. A. DUERSCH +each variational distribution also induces an inner product, +⟨f, g⟩ϕ = +� +dθ q(θ | ϕ)f(θ)g(θ), +we can construct the basis to be orthogonal, ⟨fi, fj⟩ϕ∗ = 0 for all i ̸= j. This will +allow us to apply the calculus of variations to illuminate the relationship between +optimal coefficients and the posterior distribution, p(θ | D). +We can capture arbitrary infinitesimal perturbations in the vicinity of the opti- +mizer by using a differential element, ε, and a vector of perturbations, η, to write +Gateaux derivatives, q(θ | ϕ = ϕ∗ + εη). Analysis begins by constraining feasible +perturbations to only those that retain normalization. Going forward, it is useful to +define f0(θ) ≡ 1. Then, +∂ +∂ε +�� +dθ q(θ | ϕ = ϕ∗ + εη) +� +ε=0 += +� +dθ q(θ | ϕ∗) +nℓ +� +ℓ=0 +ηℓfℓ(θ) += +nℓ +� +ℓ=0 +ηℓ⟨f0, fℓ⟩ϕ∗ = η0 = 0. +Thus, by using an orthogonal basis and capturing the normalization coefficient with +f0, normalization-preserving perturbation directions only require fixing η0 = 0. +Provided both the variational optimizer and the posterior distribution have full +support over the parameter domain, we can rewrite the posterior distribution by +factoring out the optimizer and defining what remains with the residual, r(θ), so that +p(θ | D) = exp +� +r(θ) + +n +� +ℓ=0 +ϕ∗ +ℓfℓ(θ) +� +. +Since arbitrary perturbations must satisfy the variational principle, we have +∂ +∂ε +�� +dθ q(θ | ϕ∗ + εη) log +�q(θ | ϕ∗ + εη) +p(θ | D) +�� +ε=0 += +� +dθ q(θ | ϕ∗) +� nℓ +� +ℓ=1 +ηℓfℓ(θ) +� +[1 − r(θ)] = − +nℓ +� +ℓ=1 +ηℓ⟨fℓ, r(θ)⟩ϕ∗ = 0. +As each remaining ηℓ is arbitrary, all inner products must vanish. It follows that the +residual must be orthogonal to the span of the variational basis, excluding the nor- +malizing component in f0, which is typically unknown. Thus, the optimal coefficients +are a fixed point that we obtain by projecting the log-posterior onto the span of the +variational basis, disregarding normalization, +ϕ∗ +ℓ = +⟨fℓ, log(p(D | θ)p(θ))⟩ϕ∗ +⟨fℓ, fℓ⟩ϕ∗ +for +ℓ = 1, 2, . . . , nℓ. +4.2. Gradient and Hessian Extraction. Perhaps the simplest mean-field dis- +tribution that is amenable to this approach uses a quadratic basis in each coordinate, +yielding a mean-field Gaussian. Since we typically think about minimizing loss during +optimization of learning algorithms, we will frame analysis in terms of the negative + +AN EFFICIENT QUADRATURE SEQUENCE AND SPARSIFYING METHODOLOGY 13 +log-posterior. Let µ(t) be the expansion point for a quadratic basis, J0 is the con- +stant offset, g is the average gradient, h is the Hessian diagonal, and the residual r(θ) +contains all other terms so that the loss is +J (θ | D) = − log (p(D | θ)p(θ)) += J0 + (θ − µ(t))T g + 1 +2(θ − µ(t))T diag(h)(θ − µ(t)) − r(θ). +To maintain stability and normalizability, the Hessian diagonal must be constrained +as ˆhi = max(hi, hmin) for some hmin > 0. This allows us to compute the update, +µ(t+1) = µ(t) − g ⊘ ˆh and σ(t+1) +i += ˆh +−1/2 +i +to obtain +q(θ | ϕ(t+1)) = N(θ | µ(t+1), diag(σ(t+1))2). +In principle, we could use our quadrature scheme to approximate these updates by +evaluating orthonormal projections, +q(θ | ϕ(t+1)) ∝ exp +� +− +nℓ +� +ℓ=1 +fℓ(θ) +⟨fℓ, − log p(D | θ)p(θ)⟩ϕ(t) +⟨fℓ, fℓ⟩ϕ(t) +� +, +but a much more efficient approach leverages loss gradients. +A quadrature may be generally understood as a linear combination of linear +functionals (e.g. point-wise evaluations) that approximates another linear functional +(e.g. integration against the mean-field distribution). If the approximation is exact +for some basis, then the quadrature must also be exact for the entire span by ap- +plying linearity. While the quadratures we have considered so far employ point-wise +evaluations of a function, point-wise evaluations of the gradient also comprise linear +functionals, d of them, and allow us to evaluate many basis coefficients simultaneously. +The gradient of the negative log posterior is +−∇θ log p(θ | D) = g + h ∗ (θ − µ(t)) − ∇θ r(θ). +Note that if the posterior is locally smooth, the gradient of the residual will be dom- +inated by off-diagonal Hessian components. If H is the full Hessian matrix, then +−∇θ r(θ) = (H − diag(h))(θ − µ(t))+ higher-order terms. Averaging antithetic pairs +easily captures g, canceling all Hessian contributions as well as some higher-order +terms in concert with Theorem 3.3. +We can also extract the Hessian diagonals by adjusting the signs of each term in +a second sum. These signs correspond to multiplying gradients by first-order basis +functions in each coordinate. For example, if we evaluate the Hessian contribution to +gradients from two antithetic pairs, multiplied elementwise by (θ − µ(t)) ⊘ σ(t), and +average the results, we obtain the diagonal +�1 +1 +� +∗ +��h11 +h12 +h21 +h22 +� �1 +−1 +1 +−1 +� � +1/4 +−1/4 +�� ++ +� 1 +−1 +� +∗ +��h11 +h12 +h21 +h22 +� � 1 +−1 +−1 +1 +� � +1/4 +−1/4 +�� += 1 +2 +�� +1 +1 +� +∗ +�h11 + h12 +h21 + h22 +� ++ +� 1 +−1 +� +∗ +�h11 − h12 +h21 − h22 +�� += +�h11 +h22 +� +, +demonstrating how off-diagonal terms vanish according to the exactness periodicity +of Theorem 3.3. Algorithm 4.1 shows how to implement this approach to construct a +quadratic approximation of the loss, i.e. the negative log-posterior, from gradients. + +14 +J. A. DUERSCH +Algorithm 4.1 Quadratic Loss Approximation +Input: µ is a d × 1 vector of means, where d is the parameter dimension. +σ is a d × 1 vector of standard deviations of the mean-field distribution. +q1 is an integer indicating the next position within the quadrature sequence. +nq indicates the number of antithetic pairs to use from the quadrature sequence. +J (·) is a loss function, returning both the value and gradient. +Output: J, g, and h so that J (θ) ≈ J + (θ − µ)T g + 1 +2(θ − µ)T diag(h)(θ − µ). +1: function (J, g, h) = quadratic approx(µ, σ, q1, nq, J (·)) +2: +J = 0; +g = 0d×1; +h = 0d×1 +▷ Initialize accumulators. +3: +for q = q1, q1 + 1, . . . , q1 + nq − 1 do +4: +s = quadrature sequence(d, k) +▷ Get quadrature signs. +5: +[J+, g+] = J (µ + σ ∗ s) +▷ Operator ∗ is Hadamard product. +6: +[J−, g−] = J (µ − σ ∗ s) +7: +J ← J + J+ + J−; +g ← g + g+ + g− +▷ Ordinary integration. +8: +h ← h + (g+ − g−) ∗ s +▷ Integrate product against first-order basis. +9: +end for +10: +g ← +1 +2nq g +11: +h ← h ⊘ (2nqσ) +▷ Operator ⊘ is elementwise right division. +12: +J ← +J +2nq − hT σ2 +2 +13: end function +4.3. Dirac-Gauss Mixtures. Now we will consider a mean-field framework +that is capable of capturing a finite probability that any individual coordinate, θi, is +zero. By introducing a Bernoulli-distributed random variable, zi, we can write each +mean-field factor as Dirac-Gauss mixture by marginalizing over zi as +q(θi) = +� +zi∈{0,1} +q(θi, zi) = +� +zi∈{0,1} +q(zi)q(θi | zi) +(4.1) += q(zi)δε(θi) + q(¬zi)Nε(θi | νi, τ 2 +i ). +This expression uses shorthand, q(zi) ≡ q(zi = 1) and q(¬zi) ≡ q(zi = 0), and +conditional distributions that have been constructed to be non-overlapping by carving +out a small interval, (− ε +2, ε +2) for some ε > 0, +δε(θi) = +� +ε−1 +|θi| < ε +2 +0 +else +and +Nε(θi | νi, τ 2 +i ) = +� +0 +|θi| < ε +2 +N(θi | νi, τ 2 +i ) +else +. +In the limit ε → 0, exact normalization of Nε(θi | νi, τ 2 +i ) is unnecessary, because +doing so only multiplies the scaling factor by 1 + O(ε), a vanishing change. We can +also construct a spike-and-slab prior with the same structure, +p(θi) = p(zi)δε(θi) + p(¬zi)Nε(θi | 0, hp +−1), +where hp is the prior precision associated with a nonzero. + +AN EFFICIENT QUADRATURE SEQUENCE AND SPARSIFYING METHODOLOGY 15 +Note the important distinction between the mean and standard deviations re- +quired by the quadrature formulas in Algorithm 4.1 versus the mean and variance of +the normal distribution in Equation (4.1), which is conditioned on a nonzero. The +correct moments of this mean-field distribution are: +µi = q(¬zi)νi +and +σ2 +i = q(zi)q(¬zi)ν2 +i + q(¬zi)τ 2 +i . +(4.2) +To apply the analysis from Subsection 4.1, we need to write the variational family +as an exponential, so it is useful to formulate the probabilities of zeros as logits, +ζp = log +� p(zi) +p(¬zi) +� +, +p(zi) = +exp(ζp) +1 + exp(ζp), +ζi = log +� q(zi) +q(¬zi) +� +, +and +q(zi) = +exp(ζi) +1 + exp(ζi). +Since the conditional distributions are disjoint, we can easily compute +log p(θi) = − log (1 + exp(ζp)) + +� +ζp − log(ε) +|θi| < ε +2 +1 +2 log( hp +2π) − 1 +2hpθ2 +i +else +(4.3) += const − hpθ2 +i +2 ++ +� +ζp + 1 +2 log( 2π +hp ) − log(ε) +|θi| < ε +2 +0 +else +Likewise, each variational factor can be written +log q(θi) = const − (θi − νi)2 +2τ 2 +i ++ +� +ζi + 1 +2 log +� +2πτ 2 +i +� ++ ν2 +i +2τ 2 +i − log(ε) +|θi| < ε +2 +0 +else +. +(4.4) +Since Algorithm 4.1 allows us to project the loss onto a quadratic univariate basis, we +only need to add the prior discontinuity term in Equation (4.3) to continuous quadratic +projections of the remaining log posterior components. +For example, adding the +quadratic terms from the negative log-prior and negative log-likelihood, with gradient +g and Hessian diagonal h at the expansion point µ(t), gives +hp +2 θ2 +i + gi(θi − µ(t)) + hi +2 (θi − µ(t))2 = const + (θi − νi)2 +2τ 2 +i +where +νi = hiµ(t) − gi +hp + hi +and +τ 2 +i = (hp + hi)−1. +Including the remaining prior terms and absorbing constants into the residual, we +have +log p(θ | D) = r(θ) + +� +i +� +−(θi − νi)2 +2τ 2 +i ++ +� +ζp + 1 +2 log +� +2π +hp +� +− log(ε) +|θi| < ε +2 +0 +else +� +. +(4.5) +Matching Equation (4.5) to Equation (4.4) gives the logit of the zero probability, +ζi = ζp + 1 +2 +� +log +�hp + hi +hp +� +− (hp + hi)ν2 +i +� +. +(4.6) +In practice, the continuous component of the negative log prior can be included in +the loss projection. See Subsection 5.1.2. + +16 +J. A. DUERSCH +5. Sparsifying Methodology. Although the principles governing the fixed- +point optimizations outlined in Section 4 are fairly simple to derive, many practi- +cal complications must be addressed to efficiently discover high-posterior domains for +deep learning models with a specified sparsity target. The primary purpose of this +content is to demonstrate that the quadrature sequence may be efficiently combined +with sparsifying variational inference to achieve good predictions from a sparse model. +Subsection 5.1 examines practical implementation challenges for the complicated loss +landscapes that typify deep learning models. Subsection 5.2 provides detailed algo- +rithms to address these challenges, and Subsection 5.3 includes numerical experiments. +5.1. Challenges. The methodology that follows draws upon empirical testing +that illuminated key difficulties with sparse optimization and simple approaches to +address them. To benefit future work on sparsifying methodologies, we first examine +these challenges and how the methodology seeks to address them. +5.1.1. Negative Hessian Eigenvalues. The approximate Hessian diagonals +of the loss function, the negative log-posterior, are often negative and there is no +reason to suspect that true Hessian diagonals would not be also. Negative eigenvalues +indicate increasing probability density as parameters drift from the critical point +along the corresponding eigenvectors. Not only is such a distribution unnormalizable, +it becomes untrustworthy as we move outside of the dominant region of integration. +We can easily solve this problem by distinguishing the log-variational distribution +from the log-posterior approximation, and then setting a minimum positive curvature +to maintain a coherent distribution. Yet, a problem still remains with stable gradient +updates from the underlying quadratic approximation of the log-posterior. +Negative eigenvalues also create vicious feedback between small perturbations to +the quadratic expansion point and the corresponding gradients. Testing showed that +it is better to apply the same curvature limitation to the quadratic approximation of +the local log-posterior structure to suppress this effect, which may be interpreted as +enforcing a local trust region that only allows gradient zeros that are both stable and +nearby. Safely moving beyond the trust region simply requires reevaluating the log- +posterior projection. Thus, it will be useful to quickly replace obsolete contributions. +5.1.2. Posterior Annealing with Restarted Sums. Accumulating too many +log-posterior contributions initially can result in large Hessian diagonals and obsolete +gradient information. Since high curvature suppresses zeros, it becomes more difficult +for training to identify parameters to send to zero. A simulated annealing strategy +with restarted sums solves this by preventing the log-posterior approximation from +suppressing potential zeros too early or keeping old contributions too long. +To derive the moment-matching formula, suppose we have a set of independent +identically-distributed random variables, xi for i ∈ N, that we would like to sum. +Let the expectation and variance be E[xi] = α and Var[xi] = v, respectively. If we +accumulate n0 samples in an initial sum, and n1 more samples in a restarted sum, +a0 = +n0 +� +i=1 +xi +and +a1 = +n0+n1 +� +i=n0+1 +xi +where n0 + n1 < N, then we have E[a0] = n0α, E[a1] = n1α, Var[a0] = n0v, and +Var[a1] = n1v. It easily follows that the linear combination comprising a hybrid sum, +ˆa = max +� +0, n0 − n1 +n0 + n1 +� +a0 + max +� +1, +2n0 +n0 + n1 +� +a0, +(5.1) + +AN EFFICIENT QUADRATURE SEQUENCE AND SPARSIFYING METHODOLOGY 17 +preserves both the mean and variance of the first sum until the number of samples in +the restarted sum overtakes it, i.e. E[ˆa] = max(n0, n1)α and Var[ˆa] = max(n0, n1)v. +We can use this formula to implement simulated annealing and keep the log- +posterior approximation current with the mean-field distribution. Old terms vanish +as quickly as possible without reducing the number of effective terms accumulated +thus far, and we can control increases in the number of data points with n1. +Annealing both posterior factors in Equation (2.1), as p(D | θ)αp(θ)α, rather +than just the likelihood, or p(D | θ)αp(θ), allows the mean of the Gaussian compo- +nents, ν, to remain stable as the annealing exponent increases to α = 1. To implement +this easily, when each negative log-likelihood term is computed from a training case, +we also just add the corresponding fraction of the negative log prior, disregarding the +Dirac delta contribution: +J (θ | D, α) = −α (log p(D | θ) + log p(θ)) +≈ +αnc +� +c=1 +J (θ | dc) +where +J (θ | dc) = − log p(dc | θ) − 1 +nc +log p(θ). +From each quadrature projection of a loss term, we obtain the loss value, gradient, +and Hessian diagonals at the expansion point, +J (θ | dc) = − log p(dc | θ) − 1 +nc +log p(θ) +≈ J + gT (θ − µ) + 1 +2(θ − µ)T diag(h)(θ − µ). +These coefficients are then added into their respective sums J1, g1, and h1, and the +counter n1 increments. By combining these current sums with the previous counter- +parts using Equation (5.1), we obtain running quadratic approximations that drop +old contributions efficiently. +When the current sum reaches the annealing target, +n1 = ntgt = αnc, all values are moved to the index-0 counterparts and the current +sums restart. Letting the present epoch be indexed as e = 1, 2, . . . , ne, the following +exponential schedule works well for ne = 10 epochs: +ntgt = +� +nc2e−ne� +. +5.1.3. Stable Quadrature Domains. The initial discovery of a basin of at- +traction in the loss function is difficult if the mean-field variance is large. It is better +to start with small localized quadratures that yield trustworthy gradients and grad- +ually increase the mean-field variance to match that of the loss structure, or until +the lower bound on curvature is attained. Regarding the previous point, not only +does enforcing a positive lower bound on Hessian diagonals support stable gradient +updates, it also suppresses the mean-field variance and, thus, benefits the quality of +local gradient approximations. +Similarly, uncontrolled changes in zero probabilities, q(zi), can heavily interfere +with discovery of, and adherence to, local basins of attraction. This is due to the +variance contribution, q(zi)q(¬zi)ν2 +i , in Equation (4.2). A small increase in a zero +probability may only slightly change the parameter mean, but it can significantly +increase the mean-field variance from a random initialization. Again, if the quadra- +ture scale is too large, gradient quality deteriorates, because the long-distance average +no longer matches the vicinity of the expansion point. This is the reason the algo- +rithm that follows completes a full epoch over the training data while keeping all zero +probabilities very small to support initial discovery. + +18 +J. A. DUERSCH +Another tool to increase the stability of variational updates is to set an effective +maximum learning rate on the Gaussian mean, ν. Rather than immediately jumping +to the critical point of the quadratic loss approximation with every new term, we can +first identify the gradient at ν and then apply a more restrictive trust region in the +Newton step that would yield the zero: +gν = ˆg + ˆh ∗ (ν − µ) +and +ν ← ν − gν ⊘ max +� +ˆh, max(n0, n1)hmin +� +. +Scaling the lower bound on the Hessian by the number of terms in the present stage +of annealing is equivalent to setting a maximum perturbation scale as a function of +the average gradient over the current stage of annealing. That is, +max(n0, n1)gavg + max(n0, n1)hminδavg +implies +δavg = − gavg +hmin +. +We can still track gradient updates in log-posterior approximations and the effect +averages new gradient contributions as data accumulate. See Algorithm 5.3. +5.1.4. Stable Parameter Readjustment. Building on the previous point, +small changes in the zero probabilities of some parameters may also require significant +shifts in the nonzero parameters. This effect becomes more pronounced as the num- +ber of remaining nonzero parameters decreases, thus decreasing the flexibility of the +model to account for new zeros. A gentle approach, changing the zero probabilities +gradually, allows these corrections to stabilize and stay within a high posterior region. +These corrections also support better sparsity; as some nonzero parameters become +more constrained, others become less so and, thus, trend to zero more easily. +We can control the rate of change in zero probabilities by using a continuous +piecewise-affine map on ζ, computed from Equation (4.6). Since we will control the +constant offsets directly with the map that follows, there is no need to include the +prior contribution, ζp. Given ne epochs and a training progress indicator, t ∈ [0, ne], +we will set a schedule for s(0)(t) representing the fraction of parameters that we would +like to hold near zero, i.e. below a lower threshold on the probability of being nonzero, +such as p(0) +NZ = 0.001. Likewise, let s(1)(t) represent a fraction of parameters that we +would like to hold above an upper threshold, e.g. p(1) +NZ = 0.999. +By sorting ζ after each update to the variational parameters, we can easily identify +the set of s(0) largest logits, Z(0), and the set of the s(1) smallest logits, Z(1), to obtain +the domain boundaries +ζ(0) = min +ζ∈Z(0) ζ +and +ζ(1) = max +ζ∈Z(1) ζ. +These boundaries obviously depend on the training progress, t, which is left implied +going forward to simplify notation. The range boundaries corresponding to the non- +zero probabilities we desire are ˆζ +(0) = −logit(p(0) +NZ) and ˆζ +(1) = −logit(p(1) +NZ), giving the +continuous piecewise-affine map that appears in Algorithm 5.3. This mapping serves +as a sieve on zeros, gradually realizing plausible zeros while letting the remaining pa- +rameters readjust until the sparsity target is achieved. Empirical testing shows that, +after the initial epoch, the sieve schedule can proceed quickly, but it most slow down +as fewer nonzero parameters remain. Otherwise, training and validation accuracies +deteriorate. Given end-of-training targets, s(0) +tgt and s(1) +tgt, a simple schedule is +s(0)(t) = max +� +0, min +� +1, 1 − 21−t +1 − 22−ne +�� +s(0) +tgt + +AN EFFICIENT QUADRATURE SEQUENCE AND SPARSIFYING METHODOLOGY 19 +and s(1)(t) = s(1) +tgt + s(0) +tgt − s(0)(t), removing nearly half of the remaining parameters +that are targeted to vanish per epoch, while keeping the intermediate fraction of +parameters fixed. The sparsity target is achieved when t = ne − 1, saving the last +epoch for final adjustments on a fully realized sparsity pattern. +5.2. Training Algorithms. While there are many potential improvements to +this sparsifying methodology, these basic algorithms provide a sufficient means to drive +strong sparsity in high-dimensional models. As there are many variational parameters +and hyperparameters to track during training, we will simply pass a training data +structure, T , between the following subroutines. Table 1 lists its contents. +Table 1 +Training Data Structure Hyperparameters and Variational Attributes +Hyperpara. +Description +Attr. +Description +d +Model parameter dimension +ζ +Logits of zero probabilities +s(0) +tgt = 0.97 +Target fraction of zeros +ν +Means of nonzero Gaussians +s(1) +tgt = 0.01 +Target fraction of nonzeros +τ +Standard deviations of nonzeros +ne = 10 +Number of epochs +µ +Mean of mean-field +e +Current epochs +σ +Standard deviations of mean-field +nq = 2 +Antithetic pairs per quadrature +n0, n1 +Number of terms in quadratic sums +q +Current quadrature index +J0, J1 +Loss at quadratic expansion point +αmax = 0.1 +Maximum learning rate +g0, g1 +Gradient at expansion point +α0 = 10−5 +Initial learning rate +h0, h1 +Hessian diagonals +τmax = 0.3 +Maximum Gaussian standard dev. +ˆζ +(0), ˆζ +(1) +Target logits for sieve +hmin +Minimum Hessian for var. update +Jtrain +Training loss over current epoch +p(0) +NZ, p(1) +NZ +Sieve targets (0.001 and 0.999) +The main training loop, Algorithm 5.1, begins by randomly initializing the index +of the quadrature sequence as well as the quadratic loss sums comprised of n0, g0, +and h0. By setting the curvature in h0 from α0 and the number of terms, n0, that +are needed until the next quadratic loss approximation achieves a full replacement, +Algorithm 5.1 Main Training Loop +Input: T contains the parameters listed above as well as the training dataset. +Output: (µ, σ) trained mean-field mean and standard deviations. +1: function (µ, σ) = train(T ) +2: +q = +� +uniform(0, 1) d +nq +� +nq. +▷ Randomize initial quadrature index. +3: +n0 = ⌊nc21−ne⌋; +g0 = 0d×1; +h0 = +1 +α0 1d×1 +▷ Initialize quadratic sums. +4: +Randomly initialize parameters as usual. Place into expansion point, µ. +5: +σ = h−1/2 +0 +▷ Set consistent standard deviations (elementwise). +6: +hmin = (n0αmax)−1. +▷ Minimum Hessian will scale with annealing schedule. +7: +ˆζ +(0) = −logit(p(0) +NZ); +ˆζ +(1) = −logit(p(1) +NZ) +▷ Set target logits for sieve. +8: +for e = 1, 2, . . . , ne do +9: +T = variational epoch(T ) +10: +end for +11: +Return (µ, σ). +12: end function + +20 +J. A. DUERSCH +Algorithm 5.2 Variational Epoch +1: function T = variational epoch(T ) +2: +n1 = 0; +g1 = 0d×1; +h1 = 0d×1 +▷ Initialize current sums. +3: +Jtrain = 0 +▷ Track training loss over this epoch. +4: +ntgt = ⌊nc2e−ne⌋ +▷ Set target number of terms for each restart. +5: +if +e = ne +then +rNZ ← round(pNZ) +▷ Realize sparsity pattern. +6: +Randomly permute training data, dc for c ∈ [nc]. +7: +for c = 1, 2, . . . , nc do +8: +(J, g, h) = quadratic approx (µ, σ, q, nq, J (θ | dc)). +9: +Jtrain ← Jtrain + J +10: +q ← q + nq +▷ Increment quadrature index. +11: +t ← e − 1 + +c +nc +▷ Update training progress. +12: +T ← variational update(T , g, h). +13: +if n1 = ntgt then +14: +n0 ← n1; +g0 ← g1; +h0 ← h1 +▷ Overwrite previous sums. +15: +n1 ← 0; +g1 ← 0d×1; +h1 ← 0d×1 +▷ Restart current sums. +16: +end if +17: +end for +18: +Return T . +19: end function +n1 = n0, we obtain an effective limitation on the initial learning rate. +Likewise, +hmin limits the maximum learning rate by setting a lower bound on the curvature +contribution of each loss contribution. +The algorithm then loops over the desired +number of epochs. +Algorithm 5.2 performs each epoch by first initializing the quadratic loss approxi- +mation coefficients and the number of target terms, ntgt, needed for the current stage +of simulated annealing. At the start of the final epoch, this algorithm also realizes the +most probable sparsity pattern. For each training case, the new coefficients for the +quadratic projection are obtained by numerical integration, Algorithm 4.1, and the +results are accumulated by Algorithm 5.3. Every time the targeted number of terms +is reached, the previous quadratic sums are overwritten and the current sums restart. +The variational update, Algorithm 5.3, accumulates the quadratic approxima- +tions, which always share the same expansion point. Next, the gradient at Gaussian +means, ν, is computed from the hybrid quadratic approximation that uses moment +matching. This allows a limited quasi-Newton step to control perturbations to ν. The +sieve, Lines 10 through 15, drives the mean-field distribution to a particular sparsity +pattern by filtering the parameters that are most suitable zeros from those that must +remain free to adjust. Finally, the mean-field moments are updated in µ and σ and +the gradients are updated to the new expansion point. +5.3. Experimental Results. These experimental results are performed using +a convolutional neural network for MNIST [24] shown in Appendix A.3 with ternary + +AN EFFICIENT QUADRATURE SEQUENCE AND SPARSIFYING METHODOLOGY 21 +Algorithm 5.3 Variational Update +1: function T = variational update(T , g, h) +2: +n1 ← n1 + 1; +g1 ← g1 + g +▷ Update current quadratic coefficient sums. +3: +h1 ← max(h1 + h, τmax +−2) +4: +α0 = max(0, n0−n1 +n0+n1 ); +α1 = max(1, +2n0 +n0+n1 ) +▷ Get moment-matched sums. +5: +ˆg = α0g0 + α1g1; +ˆh = α1h0 + α1h1 +6: +gν = ˆg + ˆh ∗ (ν − µ) +▷ Get gradient at ν. +7: +ν ← ν − gν ⊘ max +� +ˆh, max(n0, n1)hmin +� +▷ Limit effective learning rate. +8: +τ = ˆh +−1/2 +▷ Get standard deviations of nonzeros. +9: +if e < ne then +10: +ζ = 1 +2 +� +log +� +ˆhτmax2� +− ˆh ∗ ν2� +▷ Get initial logits of zero probabilities. +11: +s(0) = max +� +0, min +� +1, +1−21−t +1−22−ne +�� +s(0) +tgt +▷ Update sparsity schedule. +12: +s(1) = s(1) +tgt + s(0) +tgt − s(0)(t) +13: +Sort ζ and obtain boundaries ζ(0) and ζ(1) from s(0) and s(1). +14: +Apply piecewise map, +ˆζi = +� +� +� +� +� +� +� +� +� +� +� +ζi − ζ(0) + ˆζ +(0) +ζi ≤ ζ(0) +ˆζ +(0) + (ζi−ζ(0)) +� +ˆζ +(1)−ˆζ +(0)� +ζ(1)−ζ(0) +ζ(0) < ζi ≤ ζ(1) +ζi − ζ(1) + ˆζ +(1) +ζi > ζ(1) +15: +pNZ = +� +1 + exp(ˆζ) +�−1 +16: +else +pNZ = rNZ +17: +end if +18: +ˆµ = pNZ ∗ ν; +δ = ˆµ − µ +▷ Get new mean and perturbation. +19: +σ ← +� +pNZ ∗ (1 − pNZ) ∗ ν2 + pNZ ∗ τ 2�−1/2 +▷ Update standard deviations. +20: +µ ← ˆµ; +g0 ← g0 + h0 ∗ δ; +g1 ← g1 + h1 ∗ δ +▷ Move expansion point. +21: end function +Fig. 4. These histograms show sparsification progress during training. Note the logarithmic +scaling of the vertical axis (only the left and right peaks appear with linear scaling). The sieve holds +1% of parameters between the low (0.001) and high (0.999) probabilities of a nonzero. The last epoch +uses the fixed sparsity pattern given by rounding, i.e. the maximum variational realization of z. + +Progress of Sparsifying Sieve During Training +105 +104 +103 +103 +102 +ie. +10 +10° +Probability of Nonzero, Epoch 1 +Probability of Nonzero, Epoch 2 +Probability of Nonzero, Epoch 3 +Probability of Nonzero, Epoch 4 Probability of Nonzero, Epoch 5 +105 +104 +Cou +103 +102 +Par +10 +100 +0 +0.5 +0 +0.5 +0.5 +0 +0.5 +0 +0.5 +ProbabilityofNonzero.Epoch6 +ProbabilityofNonzero.Epoch7 +Probability of Nonzero, Epoch 8 +ProbabilityofNonzero.Epoch9 +ProbabilityofNonzero.Epoch1022 +J. A. DUERSCH +logical activation functions [14]. The model was trained with 60, 000 cases and 10 +epochs. Figure 4 illustrates the progress of the sparsity sieve on the first training trail. +The trained model achieves over 98.9% zeros upon completion, while retaining 96.9% +accuracy on 12, 000 test cases. See Appendix A.3 for additional details on the structure +of the network and numerical integration comparisons for various quadratures applied +to the mean-field distribution at the start and at the end of training. +6. Conclusion. Viewed from the context of tensors, mean-field distributions +are equivalent to rank-1 functions [41]. We can compare a rank-r Canonical Polyadic +(CP) approximation, a rank-r tensor, to a mean-field mixture as +Xj ≈ +r +� +k=1 +λk +d−1 +� +m=0 +A(m) +jmk +p(θ) ≈ +r +� +k=1 +q(k) +d−1 +� +i=0 +q(θi | k). +Although this work does not explore higher-rank variational distributions, the re- +sulting mean-field mixtures would also be scalable and feasible to integrate, only +multiplying the integration cost by the rank, i.e. the number of mixture components. +Although the analysis in Section 4 only examines Gaussian mean-field distribu- +tions and Dirac-Gauss mixtures, a related approach may be suitable for Laplacian +mean-field distributions or perhaps even Dirac-Laplace mixtures. One difficulty, how- +ever, is that the quadratic basis functions, (θ − ν)2, would have to be replaced by +absolute value functions, |θ − ν|. Unfortunately, this would require an adaptive basis, +rather than just algebraically manipulating a fixed basis. +It may be possible to improve sparsity further by taking correlated parameter +structures into account by referencing the computational graph in a manor similar to +the method by Jantre, Bhattacharya, and Maiti. That said, a key benefit of this work +is its generality. Many deep learning architectures do not adhere to an elementary +chain of matrix multiplications and elementwise activation functions. For example, +the computational graph dependencies in transformer architectures [44] may span +multiple layers. If, however, the probability of a parameter zero can be efficiently tied +to such dependencies, it would be possible to control entire swaths of parameters at +once and improve computational efficiency. +6.1. Summary. This work began by investigating stochastic blocked mean-field +quadratures in order to ensure numerical integration would retain high accuracy +within specific parameter blocks, while also ensuring many samples converge to a +tensor-product cubature. By considering quasirandom sequences to speed up con- +vergence, we arrived at quadrature sequences composed of antithetic evaluation pairs +from the cross-polytope sequence in the Hadamard basis. In d dimensions, this method +exactly integrates approximately 1 +4d2 multivariate quadratic basis functions using only +4 function evaluations, which gives the highest exactness efficiency for all the methods +tested. Theorem 3.3 shows how every doubling of the number of evaluations increases +exactness to include half of the remaining quadratic basis functions. +We then examined the optimal structure of variational distributions that can be +written as an exponential of a linear combination of variational basis functions, which +ties efficient numerical integration to a fixed-point iteration for variational updates. +Adjusting the quadrature sequences to act on gradients allows us to approximate +Hessian diagonals for Gaussian mean-field distributions, as well as reconstruct the +probabilities of specific parameters realizing to zero in Dirac-Gauss mixtures. +Finally, a practical sparsifying methodology was devised to overcome several op- +timization challenges for the complicated loss structures that typify deep learning. + +AN EFFICIENT QUADRATURE SEQUENCE AND SPARSIFYING METHODOLOGY 23 +Numerical experiments demonstrate the ability to achieve strong sparsity while re- +taining high validation accuracy. +This work opens new approaches to reduce both storage and operating energy +requirements for trained machine learning models by allowing dense matrix multi- +plications and tensor contractions to be replaced with more efficient sparse versions. +With regard to previous work on logical activation functions, sparsity allows us to +suppress logical complexity of predictions during training in pursuit of better gener- +alization. +Acknowledgements. My sincere thanks to Erin Acquesta, Tommie Catanach, +Jaideep Ray, and Cosmin Safta for providing early feedback. Tommie suggested an +exponential annealing schedule and log-likelihood integration experiments. Jaideep +noted that by limiting the diameter of evaluation nodes, these quadrature sequences +may provide an additional benefit to prediction models that cannot support large +parameter perturbations, such as physics models that cannot accept unphysical states. +By probing these challenges from many perspectives, these conversations provide an +important means to deepen understanding. +Appendix A. Numerical Integration Experiments. +This set of experiments, Figures 5 to 7, shows the typical range of integration +errors for several numerical approaches. +The first approach is pure Monte Carlo +integration via sampling the mean-field distribution. The second and third approaches +are quasi-Monte Carlo, translating the set of samples to match the mean (second) and +also scaling to match both the mean and the variance (third). +The fourth approach demonstrates stochastic blocked mean-field quadratures with +a block size of 2. This requires 3 sigma points, given by the simplex vertices from +Algorithm A.1, within in each 2D block. Within each block, the evaluation nodes are +then permuted uniformly at random and concatenated. +The fifth approach shows antithetic pairs of cross-polytope vertices in the Hada- +mard basis, Algorithm 3.1. Signed errors are stored and sorted for each integration +method from 5000 trials to obtain the 90% confidence intervals. +A.1. Symmetric Distributions. The first two mean-field distributions are +Gaussian and Laplacian, Figure 5 and Figure 6, respectively. Both are both symmetric +Algorithm A.1 Simplex Polytope Sigma Points +Input: d is the number of dimensions in which the desired sigma points are embedded. +Output: X is d × d + 1 matrix of evaluation nodes and corresponding weights, w. +1: function (X, w) = simplex quadrature(d) +2: +r = +√ +d +3: +X = 0d×d+1 +4: +for i = 1, 2, . . . , d do +5: +Xii = r and Xij = +−r +d+1−i for j = i + 1, i + 2, . . . , d + 1 +6: +r ← r +√ +(d+1−i)2−1 +d+1−i +7: +end for +8: +Return X and shared weight, w = +1 +d+1. +9: end function + +24 +J. A. DUERSCH +Quadrature Error for Mean-Field Gaussian +Fig. 5. Selected quadrature errors for, q(θi) ≡ �d +i=0 N(θi | 0, 1). Each plot shows the quadra- +ture error for products of orthonormal polynomials, ϕd(·), where d indicates the degree. +Row 1: Univariate basis functions show how quasi-Monte Carlo methods achieve exactness on first +and second degree polynomials. These methods do not necessarily reduce error on higher-order basis +functions; compare to Figure 7. +The stochastic 2-block mean-field quadrature does not correctly +integrate 3rd-order basis functions. The cross-polytope sequence is exact for these functions. +Row 2: Quasi-Monte Carlo methods offer little improvement to these multivariate quadratic in- +tegrals. Stochastic blocking retains exactness for quadratics within each block, column 1, but not +between blocks. Finally, the cross-polytope sequence shows the exactness periodicity we expect. +Row 3: Stochastic blocking appears to reduces the error for these 3rd-order functions. The cross- +polytope sequence produces Gauss points, roots of ϕ2(θ0), causing all products to vanish. +Row 4: The cross-polytope sequence generates odd pairs of evaluations for each odd basis function. +The product of an odd number of such evaluations remains odd, thus correctly summing to zero. +This is why both quasi-Monte Carlo methods integrate to zero with two samples. +and demonstrate higher-order exactness for the cross-polytope quadrature sequences. +The univariate basis functions in Row 1 show how quasi-Monte Carlo methods +achieve exactness on first and second degree polynomials by transforming samples to +match leading moments. These methods may also reduce error on higher-order basis +functions, but not always. See Figure 7. The stochastic blocked mean-field quadrature +does not correctly integrate 3rd-order basis functions. In contrast, the cross-polytope +sequence in the Hadamard basis, always produces a pair of Gauss-points in each +dimension, thus integrating all univariate 3rd-order polynomials exactly. +The multivariate quadratics in Row 2 show that even variance-matched quasi- + +1 +p1(0o) +P2(00) +P3(0) +...... Monte Carlo. +.-.. Mean-MatchedQuasi-MonteCarlo +---X---.Variance-MatchedQuasi-MonteCarlo +0.5 +----.. Stochastic 2-Block Mean-Field +-.. Cross-Polytope Vertex Sequence +Error +0 +***+***** +0.5 +1 +4 +8 +12 +16 +20 +24 +28 +32 +4 +8 +12 +20 +24 +28 +32 +4 +8 +12 +16 +20 +24 +28 +32 +4 +8 +12 +20 +24 +28 +32 +P1(00)p1(01) +P1(00)P1(02) +P1(00)P1(04) +P1(00)p1(07) +0.5 +众众 +0 +A +-0.5 +X +4 +8 +12 +16 +20 +24 +28 +32 +4 +8 +12 +16 +20 +24 +28 +32 +4 +8 +12 +16 +20 +24 +28 +32 +4 +8 +12 +16 +20 +24 +28 +32 +2(0)1(01) +2(00)P1(02) +P2(00)1(04) +(10) 1()3) +0.5 +Error +F +ned +Sign +国 +-0.5 +4 +8 +12 +16 +20 +24 +28 +32 +4 +8 +12 +16 +20 +24 +28 +32 +4 +8 +12 +16 +20 +24 +28 +32 +4 +8 +12 +16 +20 +24 +28 +32 +P1(00)P1(01)1(02) +P1(00)91(01)91(04) +1(00)P1(03)01(04) +P1(00)1(01)1(07) +0.5 +ror +S +-0.5 +4 +8 +12 +16 +20 +24 +28 +32 +4 +8 +12 +16 +20 +24 +28 +32 +4 +8 +12 +16 +20 +24 +28 +32 +4 +8 +12 +16 +20 +24 +28 +32 +Function Evaluations +Function Evaluations +Function Evaluations +Function EvaluationsAN EFFICIENT QUADRATURE SEQUENCE AND SPARSIFYING METHODOLOGY 25 +Quadrature Error for Mean-Field Laplace +Fig. 6. Selected quadrature errors for, q(θi) ≡ �d +i=0 Laplace(θi | 0, 1). The critical property +of the mean-field distribution that determines the characteristics of these error plots is symmetry. +Since Laplace distributions are symmetric, we observe the same structures as the Gaussian case. +The only notable difference is the scale of errors for some of the stochastic methods. For example, +errors associated with ϕ3(θ0) (top-right) are significantly smaller in this case, as are the errors for +the multivariate cubics in row 3. The cross-polytope sequence quadratures are exact for all the same +cases as before. +Monte Carlo may not significantly improve mixed second-order integrals. Stochastic +blocking retains exactness for quadratics within each block, column 1, but not be- +tween blocks. Finally, the cross-polytope sequence shows the exactness periodicity +we expect, based on the leading mismatched bit between each pair of parameter in- +dices. For example, ϕ1(θ0)ϕ1(θ7) has the same exactness periodicity as ϕ1(θ0)ϕ1(θ1) +because the bit strings, 000 and 111, differ in the leading bit, just as 000 and 001. +In Row 3, the stochastic mean-field quadrature reduces error for these 3rd-order +functions. Since the cross-polytope sequence produces Gauss points in each dimension, +the zeros of ϕ2(θ0), all of these products correctly vanish. +The cross-polytope sequence performs well in Row 4 because it generates odd +evaluation pairs for each odd function. Since the product of an odd number of such +evaluations is still odd, the average sums to zero. This also explains why both moment- +matching methods integrate to zero when they only contain two samples. + +1 +p1(0o) +P2(00) +P3(0) +...... Monte Carlo +..... Mean-Matched Quasi-Monte Carlo +.---X---. Variance-Matched Quasi-Monte Carlo +0.5 +----.. Stochastic 2-Block Mean-Field +--. Cross-Polytope Vertex Sequence +Error +0 +XXXX +0.5 +-1 +4 +8 +12 +20 +24 +28 +32 +4 +8 +12 +20 +24 +28 +32 +4 +8 +12 +16 +20 +24 +28 +32 +4 +8 +12 +20 +24 +28 +32 +P1(00)p1(01) +P1(00)P1(02) +P1(00)P1(04) +P1(00)p1(07) +0.5 +众众 +** +0 +A +-0.5 +4 +8 +12 +16 +20 +24 +28 +32 +4 +8 +12 +16 +20 +24 +28 +32 +4 +8 +12 +16 +20 +24 +28 +32 +4 +8 +12 +16 +20 +24 +28 +32 +2(0)1(01) +2(00)P1(02) +2(00)P1(04) +2(00)P1(07) +0.5 +-0.5 +-1 +4 +8 +12 +16 +20 +24 +28 +32 +4 +8 +12 +16 +20 +24 +28 +32 +4 +8 +12 +16 +20 +24 +28 +32 +4 +8 +12 +16 +20 +24 +28 +32 +P1(00)P1(01)1(02) +1(00)91(01)91(04) +1(00)P1(03)1(04) +P1(00)1(01)1(07) +0.5 +ror +S +-0.5 +4 +8 +12 +16 +20 +24 +28 +32 +4 +8 +12 +16 +20 +24 +28 +32 +4 +8 +12 +16 +20 +24 +28 +32 +4 +8 +12 +16 +20 +24 +28 +32 +Function Evaluations +Function Evaluations +Function Evaluations +Function Evaluations26 +J. A. DUERSCH +Quadrature Error for Mean-Field Dirac-Gauss Mixture +Fig. 7. Selected quadrature errors for, q(θi) ≡ �d +i=0 +� 1 +2 δ(θi) + 1 +2 N(θi | 2, 1) +� +. +Row 1: Note the conspicuous errors for the few-sample variance-matched quasi-Monte Carlo cases +with ϕ2(θ0). +Since the cross-polytope sequence does not produce Gaussian quadratures in each +coordinate, we no longer obtain 3rd-order exactness. +Row 2: The cross-polytope sequence still operates as designed to integrate multivariate quadratics +with the same periodicity as the previous cases. +Row 3: The cross-polytope sequence is no longer always exact for these cases because it no longer +evaluates at roots of ϕ2(θi), but we still obtain the same exactness periodicity as Row 2. +Row 4: The cross-polytope sequence is still exact for these cases for the same reason as before. +A.2. Dirac-Gauss Mixture. The third mean-field distribution tested is a spike +and slab, Figure 7, which is not symmetric. As a consequence of the asymmetry, it +is not possible for the cross-polytope sequence to generate Gaussian quadrature pairs +that also have equal-weights, the essential property that allowed the cross-polytope +sequence to generate 3rd-order cubatures earlier. We can still construct equal-weight +quadratures for this purpose, but they are only 2nd-order in each dimension, thus only +becoming second-order cubatures with the exactness periodicity of Theorem 3.3. +The errors in Row 1 for the few-sample variance-matched quasi-Monte Carlo cases +with ϕ2(θ0) occur because each factor distribution contains finite probability mass at +θi = 0. With only a few samples, a specific coordinate is often zero for all samples, +meaning it is not possible to match the sample variance to the distribution variance. +Only being able to match the mean causes these results. +Again, since the cross- +polytope sequence does not produce Gaussian quadratures in each coordinate, we no + +1 +p1(0o) +p2(0) +(p3(00) +...... Monte Carlo +-... Mean-Matched Quasi-Monte Carlo +.--X---. Variance-MatchedQuasi-MonteCarlo +0.5 +---.Stochastic 2-Block Mean-Field +-.. Cross-Polytope Vertex Sequence +Error +-0.5 +1 +4 +8 +12 +16 +20 +24 +28 +32 +4 +8 +12 +16 +20 +24 +28 +32 +4 +8 +12 +16 +20 +24 +28 +32 +4 +8 +12 +20 +24 +28 +32 +P1(00)p1(01) +P1(00)P1(02) +(0) (0) 1 +(10) () ) +0.5 +众众 +0 +-0.5 +X +X +4 +8 +12 +16 +20 +24 +28 +32 +4 +8 +12 +16 +20 +24 +28 +32 +4 +8 +12 +16 +20 +24 +28 +32 +4 +8 +12 +16 +20 +24 +28 +32 +2(0)1(01) +p2(00)p1(02) +P2(00)1(04) +2(00)P1(07) +0.5 +Error +-0.5 +4 +8 +12 +16 +20 +24 +28 +32 +4 +8 +12 +16 +20 +24 +28 +32 +4 +8 +12 +16 +20 +24 +28 +32 +4 +8 +12 +16 +20 +24 +28 +32 +P1(00)1(01)1(02) +P1(00)1(01)P1(04) +1(00)P1(03)1(04) +P1(00)1(01)1(07) +0.5 +2. +ror +-0.5 +4 +8 +12 +16 +20 +24 +28 +32 +4 +8 +12 +16 +20 +24 +28 +32 +4 +8 +12 +16 +20 +24 +28 +32 +4 +8 +12 +16 +20 +24 +28 +32 +Function Evaluations +Function Evaluations +Function Evaluations +Function EvaluationsAN EFFICIENT QUADRATURE SEQUENCE AND SPARSIFYING METHODOLOGY 27 +Fig. 8. +Left: The initial, untrained, mean-field distribution exhibits high loss, i.e. negative +log-likelihood. Numerical integration with cross-polytope vertices in the Hadamard basis produces +slightly tighter confidence intervals. Right: At the end of training, the model has converged to a +sparse Gaussian mean-field. Both the stochastic 2-block simplex sigma points and the cross polytope +sequence produce significantly tighter confidence intervals on the test data. +longer obtain cubic exactness. +Row 2 shows that the cross-polytope sequence still operates as designed to inte- +grate multivariate quadratics with the same periodicity as the previous cases. Row 3, +however, shows that the cross-polytope sequence is no longer always exact for these +cases, since it no longer evaluates roots of ϕ2(θi). Instead, we revert to the same ex- +actness periodicity as seen in Row 2. In Row 4, the same reasoning for odd products +of odd function evaluations still holds, so these are still exact. +A.3. MNIST CNN Negative Log-Likelihood. This simple network uses +ternary logical activation functions, which were designed to forge a relationship be- +tween parameter sparsity and logical complexity [14]. The specific layer structure is +shown in Table 2. At the beginning of training, we have a Dirac-Gauss mixture to +support inference of a sparsity pattern. At the end of the final epoch, the model +has converged to a full realization of a specific sparsity pattern, leaving the mean- +field distribution as a Gaussian in each nonzero parameter. Both the initial and final +mean-field integrals are shown on the left and right, respectively, of Figure 8. These +results use a fixed sequence of test cases and the 90% confidence intervals are taken +by sorting outcomes from 3000 trials. For each trial, the random number generator is +seeded with the trial index before constructing each quadrature sequence. +Table 2 +Convolutional Neural Network with Ternary Logic Activations +Layer +Channels In +Channels Out +Kernel +Stride +Parameters +1 +Convolution +1 +12 +2 × 2 +2 +48 +2 +Ternary +12 +4 +32 +3 +Convolution +4 +48 +2 × 2 +2 +768 +4 +Ternary +48 +16 +128 +5 +Convolution +16 +192 +2 × 2 +2 +12288 +6 +Ternary +192 +64 +512 +7 +Convolution +64 +192 +2 × 2 +2 +49152 +8 +Ternary +192 +64 +512 +9 +Convolution +64 +192 +2 × 2 +2 +49152 +10 +Ternary +192 +64 +512 +11 +Linear +64 +10 +640 +12 +SoftMax +10 +10 +0 +We see that, initially, the cross-polytope sequence in the Hadamard basis provides +a marginal improvement on the integrals as we average over several data samples. At + +MNIST CNN, Initial Dirac-Gauss Mean-Field +Final Sparse Gaussian Mean-Field +0.16 +. Monte Carlo +2.4 +..... +Mean-Matched Quasi-Monte Carlo +Loss +0.14 +X.... +:Variance-Matched Quasi-Monte Carlo +Stochastic 2-Block Mean-Field +Cross-Polytope Vertex Sequence +0.12 +0.1 +for +2.3 +Intervals +果果贝果果 +0.08 +[] +2.25 + Confidence +0.06 +0 +0 +2.2 +0 +0.04 +%06 +88881 +0.02 +2.15 +0 +4 +8 +12 +16 +20 +24 +28 +32 +4 +8 +12 +16 +20 +24 +28 +32 +Test Data Count +Test Data Count28 +J. A. DUERSCH +the end of training, however, both the 2-blocked simplex sigma points and the cross- +polytope sequence produce much tighter integral bounds than the other approaches. +Note that the third test case on the left has an atypically large loss, i.e. a poor +prediction, that demonstrates how the average loss integrals respond to typical, albeit +intermittent, perturbations to the average integrals. +Appendix B. Proofs. +B.1. Proof of Theorem 3.1. Since all permutations are equally likely, we have +EP1,P2,...,Pnb f(θ(1)) = +1 +nqnb +nq +� +q1=1 +nq +� +q2=1 +· · · +nq +� +qnb=1 +nb +� +b=1 +f (b)(θ(qb) +b +) += +� +1 +nq +nq +� +q1=1 +f (1)(θ(q1) +1 +) +� � +1 +nq +nq +� +q2=1 +f (2)(θ(q2) +2 +) +� +· · · +� +� 1 +nq +nq +� +qnb=1 +f (nb)(θ +(qnb) +nb +) +� +� += +�� +dθ1 q(θ1)f (1)(θ1) +� �� +dθ2 q(θ2)f (2)(θ2) +� +· · · +�� +dθnb q(θnb)f (nb)(θnb) +� += +� +dθ q(θ)f(θ). +Since this holds for all evaluation nodes, the average yields the same result□ +B.2. Proof of Lemma 3.2. +pi1 xor pi2 = +� nb +xor +j=1 [bitj(i1) ∧ bitj(q)] +� +xor +� nb +xor +j=1 [bitj(i2) ∧ bitj(q)] +� += +nb +xor +j=1 [(bitj(i1) ∧ bitj(q)) xor (bitj(i2) ∧ bitj(q))] += +nb +xor +j=1 [(bitj(i1) xor bitj(i2)) ∧ bitj(q)] = +nb +xor +j=1 [xj ∧ bitj(q)]□ +B.3. Proof of Theorem 3.3. This result easily follows from Lemma 3.2. As +q increases, the relative parity, pi1 xor pi2, can only switch when at least one bit, +bitj(q), at a position j ≥ b flips. 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Mandt, Advances in variational inference, +IEEE transactions on pattern analysis and machine intelligence, (2018). + diff --git a/oNE_T4oBgHgl3EQf8RxN/content/tmp_files/load_file.txt b/oNE_T4oBgHgl3EQf8RxN/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7918b3dcdd5d3518ce2342c061546147e24f628b --- /dev/null +++ b/oNE_T4oBgHgl3EQf8RxN/content/tmp_files/load_file.txt @@ -0,0 +1,1278 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf,len=1277 +page_content='AN EFFICIENT QUADRATURE SEQUENCE AND SPARSIFYING METHODOLOGY FOR MEAN-FIELD VARIATIONAL INFERENCE JED A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' DUERSCH∗ Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' This work proposes a quasirandom sequence of quadratures for high-dimensional mean-field variational inference and a related sparsifying methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Each iterate of the sequence contains two evaluations points that combine to correctly integrate all univariate quadratic functions, as well as univariate cubics if the mean-field factors are symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' More importantly, averaging re- sults over short subsequences achieves periodic exactness on a much larger space of multivariate polynomials of quadratic total degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' This framework is devised by first considering stochastic blocked mean-field quadratures, which may be useful in other contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' By replacing pseudoran- dom sequences with quasirandom sequences, over half of all multivariate quadratic basis functions integrate exactly with only 4 function evaluations, and the exactness dimension increases for longer subsequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Analysis shows how these efficient integrals characterize the dominant log-posterior contributions to mean-field variational approximations, including diagonal Hessian approximations, to support a robust sparsifying methodology in deep learning algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' A numerical demonstration of this approach on a simple Convolutional Neural Network for MNIST retains high test accuracy, 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='9%, while training over 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='9% of parameters to zero in only 10 epochs, bearing potential to reduce both storage and energy requirements for deep learning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Key words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' variational inference, mean-field, quadrature, cubature, Hadamard basis, sparsity, spike and slab, Hessian approximation MSC codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' 62F30 65C05 65D32 65K10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Variational inference is an optimization-based approach to discover parameter domains that dominate the Bayesian posterior with origins in sta- tistical physics [28, 34, 4, 18, 5, 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' One of the challenges with quantifying prediction uncertainty for high-dimensional models is how to reliably characterize model uncer- tainty as it evolves during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' For high-dimensional model classes, mean-field distributions provide a simple and scalable method to track a component of model uncertainty and thereby capture a useful contribution to uncertainty in predictions at a reduced computational cost [1, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' In principle, optimizing the variational ob- jective requires repeatedly integrating the log-likelihood of the training data as the variational distribution changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Thus, having a quadrature framework to efficiently capture the primary contributions to the shape of a mean-field distribution, using only a handful of function evaluations, reduces the computational burden of optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' The aim of this work is to improve the computational efficiency of variational inference and related sparsifying methodologies by improving numerical integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' This work proposes two blocking-based quadrature techniques that are suitable for mean-field variational inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' The first approach, stochastic blocked mean-field quadratures, may be useful for learning architectures that, based on the model’s computational structure, allow us to identify key blocks of parameters that may con- tain important correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Blocked quadratures allow these correlations to be feasi- bly captured with high precision while still retaining scalability for high-dimensional model classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' The second approach, derived as a quasirandom modification of the ∗Sandia National Laboratories, Livermore, CA (jaduers@sandia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='gov).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Sandia National Laboratories is a multimission laboratory managed and operated by National Tech- nology and Engineering Solutions of Sandia, LLC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=', a wholly owned subsidiary of Honeywell Inter- national, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=', for the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Department of Energy’s National Nuclear Security Administration under contract DE-NA-0003525.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' This paper describes objective technical results and analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Any sub- jective views or opinions that might be expressed in the paper do not necessarily represent the views of the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Department of Energy or the United States Government.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='08374v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='LG] 20 Jan 2023 2 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' DUERSCH first, gains a striking additional property of periodic exactness on much larger bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' To put this property in perspective, we might expect that a quadrature taking 4 func- tion evaluations in d parameter dimensions, comprising 4(d + 1) degrees of freedom, should only integrate the same number of basis functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Yet, by averaging a consec- utive pair of two-point quadratures from a quasirandom sequence, the result exactly integrates 1+2d+ 1 4d2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' more than half, of all basis functions for quadratic total de- gree polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' As d may easily surpass a million in modern learning architectures, this provides a significant improvement in approximation quality at low cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Efficient numerical integration enables robust training methodologies for varia- tional inference in high-dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' By analyzing the structure of optimal variational distributions, this work illuminates the primary attributes of a fixed-point optimiza- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Related linear functionals that act on gradients allow efficient approximation of quadratic loss structure in each parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Not only does this support variational inference with Gaussian mean-fields, it is also compatible with spike and slab distri- butions that are suitable to induce sparsity during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' The key contributions of this work are 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' new numerical integration schemes that are suitable for mean-field and blocked mean-field distri- butions, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' analysis of optimal variational distributions, bridging efficient integration with a fixed-point optimization, and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' a sparsifying methodology designed to over- come practical implementation challenges for deep learning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Efficient Numerical Integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Both of the proposed integration approaches proceed by partitioning parameters into small blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' The first approach simply requires equal-weight sigma-point quadratures [43, 27] within each block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Pro- vided all blocks use the same number of function evaluations, the evaluation coor- dinates can be permuted uniformly at random and concatenated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Doing so retains the same exactness property within each block, but also yields an expectation match- ing the tensor product cubature over all blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' This allows efficient integration of blocked mean-field distributions, which could contain more comprehensive factor dis- tributions that track correlations within each block, rather than only using products of univariate distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' This work does not further examine how to design and implement blocked mean-field distributions for variational inference, only how to ef- ficiently integrate them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' The second approach exchanges pseudorandom concatenation for quasirandom sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Each element of the sequence is a 2-point quadrature that exactly in- tegrates all linear combinations of univariate quadratics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' If the factor distributions are symmetric, these are actually Gaussian quadratures and they integrate all linear combinations of univariate cubics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Averaging over a subsequence that contains an in- teger multiple of 2b iterates, where b = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' , ⌈log2(d)⌉, yields exactness on an extra d2 2b−1 2b+1 dimensions of the function space comprising quadratic total degree polynomi- als.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' This effect is due to the quasirandom sequence creating a hierarchy of overlapping blocks that contain tensor product cubatures from the underlying quadratures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Variational Fixed-Point Optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Analysis shows how updating the variational distribution only requires projecting the log-posterior distribution onto a compatible basis, provided the variational distribution has an exponential structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' For Gaussian mean-field distributions, this analysis leads to update expressions that are somewhat similar to the gradient averaging and variance-based scaling used in ADAM[21], but quasirandom quadratures offer more efficient gradient and Hessian projections against the mean-field distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Although a mean-field distribution, AN EFFICIENT QUADRATURE SEQUENCE AND SPARSIFYING METHODOLOGY 3 q(θ), with mean, Eq(θ)[θ] = µ, and diagonal covariance, Varq(θ)[θ] = diag(σ)2, cannot contain off-diagonal covariance terms, the projection quality is still affected by whether off-diagonal covariance integrals correctly vanish, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' E[(θ1 − µ1)(θ2 − µ2)] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Thus, periodically enhanced exactness translates to more precise variational updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Since the update expressions are derived from the structure of the variational ba- sis, the same analytic framework also facilitates other update expressions for different variational bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' A closely-related derivation provides updates for Dirac-Gauss mix- tures, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' spike and slab distributions, which are needed to efficiently induce sparsity by associating zeros with finite probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Sparsifying Methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' By iterating through the quadrature se- quence in concert with randomly permuted training data, we obtain a method that averages errors, and converges to high accuracy when the distribution stabilizes and more data are taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' A simple averaging technique using restarted sums for recently processed data also allows old gradient and Hessian contributions to be dropped more efficiently than the exponentially-damped averages used in ADAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' This improves up-to-date approximations of the loss structure that drives sparsity to support a sparsifying sieve that gradually filters parameters that are most suitable to vanish from those that must be retained and readjusted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Implementation challenges and the resulting sparsifying methodology are de- scribed in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Numerical experiments demonstrate the ability to achieve extreme sparsity, dropping 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='96% of parameters while retaining over 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='95% validation accu- racy, and using only 10 epochs with just 4 prediction evaluations per training case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' This stands in sharp contrast to the 100s or 1000s of epochs that are required to achieve convergence with other approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Section 2 provides background on variational inference, mean field distributions, related approaches to numerical integration, and recent work on sparsification with variational inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Section 3 proposes and analyzes stochastic blocked mean-field quadratures and quasirandom quadrature sequences for high-dimensional numerical integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Section 4 analyzes a variational fixed-point optimization that bridges efficient integration with Gaussian mean-field updates and Dirac-Gauss mixtures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Section 5 outlines key implementation challenges and practi- cal algorithmic design solutions for a sparsifying methodology, followed by numerical experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Section 6 provides a brief discussion of potential follow-up work and a final summary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Bayesian inference provides an attractive paradigm to quan- tify prediction uncertainty by consistently resolving uncertainty in models that could explain available data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Given a training dataset D, a model class with parameters θ, prior belief p(θ), and likelihood p(D | θ), we obtain the posterior by applying Bayes’ theorem, p(θ | D) = p(D | θ)p(θ) p(D) , where p(D) = � dθ p(D | θ)p(θ) is the model evidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Unfortunately, when the likelihood function is complicated, especially for high-dimensional architectures, capturing the shape of the posterior becomes intractable due to limited computational resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Variational inference mitigates this issue by approximating the posterior with a simpler distribution, q(θ | ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Variational parameters ϕ characterize the approximate 4 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' DUERSCH shape of a posterior-dominant region of the parameter domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' We discover such domains by optimizing a variational objective, such as the maximizing the Evidence Lower Bound (ELBO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Since the ELBO optimizer also minimizes the Kullback-Leibler (KL) divergence [22], from the posterior to the variational distribution, this is a practical minimization objective for training: D[ q(θ | ϕ) ∥ p(θ | D) ] = � dθ q(θ | ϕ) log � q(θ | ϕ) p(θ | D) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='1) It is also a principled objective in its own right, minimizing excess information created by replacing the posterior distribution with a feasible approximation [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Provided the dataset is composed of independent cases from the data-generating process, D = {dc | c ∈ [nc]}, the optimizer of Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='1) can be written as a sum of integrals over each case c, ϕ∗ = argmin ϕ D[ q(θ | ϕ) ∥ p(θ) ] − nc � c=1 � dθ q(θ | ϕ) log p(dc | θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Notably, this construction does not depend on the model-evidence integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Opti- mization does, however, require repeatedly evaluating the log-likelihood integral as the variational distribution evolves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' This work focuses on mean-field variational distributions, because they offer the most scalable approach for high-dimensional model classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Given d parameters, in- dexed1 as i ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' , d − 1}, mean-field distributions take the form q(θ | ϕ) = d−1 � i=0 q(θi | ϕi), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='2) where each ϕi may contain several variational parameters that describe the shape of each factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' This simple structure is what allows the efficient high-dimensional numerical quadratures developed in Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Monte Carlo Versus Quadratures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' There are generally two approaches to account for probability in numerical integration, stochastic methods and determin- istic methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' In the first case, randomized sampling matches integral contributions in expectation over pseudorandom events that generate function evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Monte Carlo methods, including Markov Chain Monte Carlo (MCMC) [31, 16] as well as tempered variations that achieve better posterior convergence [8, 23], are entirely sto- chastic approaches, mapping pseudorandom numbers to a set of samples from the posterior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' As the sample size ns increases, the scale of the integration error drops as ε(ns) ≈ ε(1)ns−1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' The drawback of sampling is that, although it will eventually produce integral approximations with arbitrarily small error, the number of function evaluations needed can be quite large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Deterministic approaches are the domain of typical numerical quadrature formu- las.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' For integration in multiple dimensions, these are often called cubature methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' We solve a set of nq evaluation locations paired with weights, � (θ(q), wq) | q ∈ [nq] � , that exactly integrate some basis of functions, Φ = {fℓ(·) | ℓ ∈ [nexact]} so that Q [f] = nq � q=1 wqf(θ(q)) ≈ � dθ q(θ)f(θ) where Q [fℓ] = � dθ q(θ)fℓ(θ) 1We enumerate parameters from zero to be consistent with analysis in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' AN EFFICIENT QUADRATURE SEQUENCE AND SPARSIFYING METHODOLOGY 5 is exact for all ℓ ∈ [nexact].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Smolyak quadratures [39, 15, 35] are efficient formulas to generate high-degree quadratures in a few dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Novak and Ritter [32] show that nq ≈ 2k k dk quadrature nodes can be constructed to exactly integrate polynomials of total degree 2k+1 in d dimensions against a fixed weight function with the structure of Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Unfortunately, even taking k = 1 is too expensive for current learning architectures, where d often ranges from 105 to 1010 or more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Since optimizing the variational distribution requires evaluating as many integrals as there are training data, this approach is not feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' We desire an approach that 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' only uses a few function evaluations per integral, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' exactly integrates basis functions that dominate the shape of the mean-field distribution, and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' efficiently suppresses the unavoidable errors associated with basis functions that we cannot afford to integrate exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Quasi-Monte Carlo Integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Quasi-Monte Carlo methods find mid- dle ground by incorporating both psuedorandom and deterministic aspects within the sample-generating process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Caflisch [7] provides an overview targeting the perspec- tive of applied mathematicians and numerical analysts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' For example, we can use two evaluation points, θ(1) = µ − δ and θ(2) = µ + δ, with equal weights, w1 = w2 = 1 2, to exactly integrate all affine functions against a distribution with mean µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' This is what Caflisch calls an antithetic pair, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' a pair of evaluation points balanced about the mean of the distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Other moment-matching methods extend this simple tech- nique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' If we know the mean of the distribution to be integrated, then we can adjust a set of samples to exactly integrate all affine functions by applying a simple translation of sample coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Given a set of samples, Θ = � θ(q) ∼ q(θ) | q ∈ [ns] � , and a known mean, Eq(θ)[θ] = µ, we can translate the samples as θ(q)′ = θ(q) − ˆµ + µ where ˆµ = 1 ns ns � q=1 θ(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Likewise, if we also have known diagonal covariance, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Eq(θ)[(θ(q)−µ)(θ(q)−µ)T ] = diag(σ)2, then we can use an affine transformation to exactly integrate all univariate quadratic polynomials, and linear combinations of them, θ(q)′ = (θ(q)′ − ˆµ) ∗ σ ⊘ ˆσ + µ where ˆσ2 = 1 ns ns � q (θ(i) − ˆµ) ∗ (θ(i) − ˆµ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' The operator ⊘ indicates elementwise division and the operator ∗ represents the Ha- damard product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' As the set of samples becomes large, the sample moments converge to the true moments, making the correction increasingly modest so that the same convergence properties of Monte Carlo hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Unfortunately, this approach requires having the full set of sample locations before the correction can be made, whereas the quasirandom quadrature sequences described in Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='2 can be accumulated to match more basis functions as more function evaluation become available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Beylkin [3] also examines numerical algorithms in high dimensions and Dick [11] provides a recent overview of quasi-Monte Carlo methods for high-dimensional in- tegration over the unit cube, [0, 1]d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Trefethen [41, 42] examines high-dimensional integration methods that aim to subdue the curse of dimensionality, observing that it is the special structure of certain problem-dependent integrands, deviating from the anisotropy of the hypercube, that allows some quadrature formulas to avoid the exponential increase in evaluation nodes needed for tensor-product cubatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' 6 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' DUERSCH 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Latin Hypercube Sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Latin hypercube sampling [26] is another quasi-Monte Carlo approach that closely relates to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' It is a form of stratified sampling that matches, in expectation, stratified sampling on the Cartesian product of subsets in each dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' If we break a single dimension of the integral domain into subsets of equal probability, and ensure that an equal number of samples are drawn from each subset, then we obtain a more precise integral approximation in that dimension, because the samples equally represent components that are analytically equivalent contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' The key insight of Latin hypercube sampling is that if we obtain such samples from each dimension independently within a mean-field distribu- tion, permuting samples within each dimension uniformly at random, the expectation of the result still matches the product of integrals over all dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Figure 1 pro- vides an illustration of this technique in two dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' The same idea drives the development and analysis of stochastic blocked mean-field quadratures in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Latin hypercube illustration for 2D Gaussian separated into 10 equal-probability subin- tervals for each dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' The quasirandom partition of samples produces a set of evaluation points that are more evenly distributed in each dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Samples are composed by randomly per- muting the source partitions in each coordinate and concatenating the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' If the distribution is independent in each coordinate, the expectation matches the exact integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Sparsifying Variational Inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Simplifying the complexity of learn- ing models [40, 37, 17] is a fundamentally sounds objective [30, 13] in abstract learning models and an experimentally demonstrated means to improve the generalization of high-dimensional learning models [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' A direct approach to achieving this is to in- duce sparsity in model parameters, which may also benefit the operating speed and power consumption of trained models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Combining spike and slab priors [29, 25] with variational inference [9, 2, 20] allows finite probabilities to be assigned to parameter zeros that can be controlled during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Polson and Roˇckov´a [36] developed Spike-and-Slab Deep Learning (SS-DL) mod- els to improve generalizability by inducing sparsity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' They show that this kind of regu- larization has the ability to learn α-H¨older smooth functions efficiently, even when the degree of smoothness is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Schmidt-Hieber [38] also analyzes these learning problems with DNNs comprised of ReLU activation functions and finds that sparsity is a key tuning parameter affecting performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Ch´erief-Abdellatif [9] derives gen- eralization error bounds and automatic architecture optimization for nonparametric regression of such functions using Deep Neural Networks (DNNs) with sparse varia- tional inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' One of the key challenges identified in that work is how to design efficient computational methods for spike and slab variational inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Bai, Song, and Cheng [2] address this challenge by proposing a sampling method- Latin Hypercube Sample for 2D Gaussian 2 1 2 X 0 1 2 2 1 0 1 2 Parameter1AN EFFICIENT QUADRATURE SEQUENCE AND SPARSIFYING METHODOLOGY 7 ology for spike and slab variational distributions that uses fully-realized sparsity pat- terns in forward propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' To support optimization with stochastic gradient meth- ods, such as ADAM, they replace sparse samples with Gumbel-softmax probabilities in backpropagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Jantre, Bhattacharya, and Maiti [20] build on that approach by associating the probability of specific nodes in a neural network, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' hidden-layer neurons, with finite inclusion probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' This allows all of the connected edges in the computational graph, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' weight-matrix elements, to be dropped at once, and thereby yield more efficient processing in fully trained sparse networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' To distinguish choices of a Gaussian spike [6], a Laplacian spike [10], or a Dirac spike [2, 20], the explicit term, Dirac-Gauss mixture, will be used going forward in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' One key difference in this work is that quasirandom quadratures, Subsec- tion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='2, enable more efficient integration of gradients as the variational distribution evolves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' The same gradient evaluations also yield approximate Hessian integrals, Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' The projection-based analysis shows how this Hessian information incorporates into the sparse variational updates Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' This is combined with an empirically-developed sparsifying methodology that only required 10 epochs to converge in numerical experiments, rather than hundreds or even thousands that have been needed for other approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Analysis of Blocked Mean-Field Quadratures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' This section begins with a discussion of blocked mean-field quadratures in Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' This line of reasoning provides a stepping stone to the more powerful quasirandom sequences discussed in Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' See Appendix A for numerical integration experiments for various mean-field distributions with corresponding basis functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Stochastic Blocked Mean-Field Quadratures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Just as a mean-field distribution is a product of distributions in each parameter, we can define a blocked mean-field distribution as a product of distributions over whole blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Formally, when we define such a blocking structure, we have decomposed the parameter vector space into a direct sum of orthogonal subspaces, Θ = nb ⊕ b=1 Θb so that θb ∈ Θb indicates the parameters with a single block b, rather than an individual parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' With an appropriate basis, any parameter state can be represented by concatenating these components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' This facilitates a modest generalization of mean-field distributions to more comprehensive representations of parameter uncertainty within each block as q(θ) = nb � b=1 q(θb) where θ = � θT 1 θT 2 · · · θT nb �T ∈ Θ, while remaining scalable by disregarding correlations between blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Just as Latin hypercube sampling concatenates stratified samples in each coordinate by using uni- formly random permutations that match, in expectation, higher-dimensional strat- ification, we apply the same insight to equal-weight quadratures within each block to obtain composite quadratures with expectations that match the tensor product cubature and still retain the exactness design within each block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' For each block b, we construct an equal-weight quadrature, Qb[·], that exactly integrates a set of rb basis functions, Φb = � f (b) k (θb) : Θb �→ R | k ∈ [rb] � , against the corresponding block distribution q(θb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Given any function f(θb) ∈ span(Φb), we 8 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' DUERSCH have exactly nq evaluation nodes, θ(q) b ∈ Θb for q ∈ [nq], for which Qb[f] = 1 nq nq � q=1 f(θ(q) b ) = � dθb q(θb)f(θb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' For example, a good choice would be Uhlmann’s sigma points [43, 27], Algorithm A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' We can then form a stochastic blocked mean-field quadrature by concatenating independent uniformly random permutations, represented by permutation matrices Pb for all b ∈ [nb], applied to the evaluation nodes in each block as � θ(1) · · θ(nq)� = � ����� [θ(1) 1 θ(2) 1 · · θ(nq) 1 ]P1 [θ(1) 2 θ(2) 2 · · θ(nq) 2 ]P2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' [θ(1) nb θ(2) nb · · θ(nq) nb ]Pnb � ����� so Q[f] = 1 nq nq � q=1 f(θ(q)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='1 (Expectation Exactness).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Given a blocked mean-field distribution and a stochastic equal-weight quadrature as described above, for any function that is a product of exact functions within each block, f(θ) = nb � b=1 f (b)(θb) where f (b) ∈ span(Φb), the expectation of the quadrature is exact, EP1,P2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=',Pnb Q[f] = � dθ q(θ)f(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' See Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='1 for a short proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' The key idea is that by composing equal- weight quadratures from each block of coordinates with concatenation, the result retains the same exactness for functions restricted to each block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Since each quadra- ture is equal-weight and the evaluation nodes are permuted uniformly at random, the probability of concatenating any specific sequence of evaluations nodes is a constant that coincides with the weight of each node in the tensor product cubature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Extra Exactness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' The exactness we obtain from this method goes be- yond the partition of consecutive blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' To understand this, consider this example of concatenated 2-block sigma points (vertices of an equilateral triangle) in 4 pairs: � �� θ(1)T θ(2)T θ(3)T � �� = � ���� √ 2 0 −1 √ 2 − � 3 2 √ 2 0 −1 √ 2 � 3 2 −1 √ 2 − � 3 2 √ 2 0 −1 √ 2 � 3 2 −1 √ 2 − � 3 2 −1 √ 2 � 3 2 −1 √ 2 � 3 2 −1 √ 2 − � 3 2 √ 2 0 � ���� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' As intended, these evaluation nodes contain sigma points in the 1-2 block, as well as the 3-4, 5-6, and 7-8 blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' However, we also obtained sigma points in the 1-6 and 2-5 blocks, since the permutations happen to have produced compatible pairings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Figure 2 shows how extra exactness for mixed quadratic basis functions varies with the block size of the partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Within each block, the quadrature uses sigma points composed of the simplex vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Since a larger block size increases the number of permutations, the probability of realizing compatible permutations drops for larger AN EFFICIENT QUADRATURE SEQUENCE AND SPARSIFYING METHODOLOGY 9 blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Thus, the reduction in extra exactness can outweigh the increase in exactness within each block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' This comparison also includes quadratures from Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='2, cross-polytope vertices in the Hadamard basis, which are discussed next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Each nu- merical experiment counts exact off-diagonal integrals on a 6000 × 6000 mean-field covariance matrix, which should all be zero, and the average over 100 trials is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Left: Increasing the block dimension for the simplex-based quadratures does not neces- sarily increase the number of exact mixed quadratic basis functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Right: Using a block size of 2 or 3 for the simplex-based quadratures, requiring 3 or 4 function evaluations, respectively, maximizes the number of exact basis functions per evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' However, the cross-polytope sequence is even more efficient, with the optimum at 4 function evaluations per quadrature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Cross-Polytope Vertices in the Hadamard Basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Unfortunately, we often encounter a substantial difference between the integration error corresponding to the expectation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' the limit of averaging samples over increasingly long sequences, and the error corresponding to a single quadrature or even an average of a few samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' What follows is the result of trying to develop a quasirandom sequence to suppress error more efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Consider concatenating a set of two-point equal-weight quadratures in each pa- rameter dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' This is the minimum number of evaluation nodes required to exactly integrate all univariate quadratic functions, thus recovering both the mean and diagonal covariance of a mean-field distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Rather than averaging several quadratures composed by random permutations, we could just form tensor product cu- batures in adjacent pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' If the mean is Eq(θ) [θ] = µ and Eq(θ) � (θ − µ)(θ − µ)T � = diag(σ)2, then these four evaluation nodes are � θ(1) θ(2) θ(3) θ(4)� = µ + diag(σ) � ������ 1 −1 1 −1 1 −1 −1 1 1 −1 1 −1 1 −1 −1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' � ������ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Not only does this result in exact pairwise cubatures with only four function evalua- tions, the exact cubature blocks actually include all d2 4 pairs of dimensions containing both an even parameter and an odd parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' One can easily show that this is the maximum number of exact blocks that can be obtained by switching some of the signs of the second pair of nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Building on this strategy, we can then partition coordinates into consecutive 4- blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Repeating the evaluation nodes, but flipping the signs in the second pair, results in a tensor product cubature in each 4-block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Iterating such node sequences to Mixed Quadratic Exactness Comparison Mixed Quadratic Efficiency Comparison cross-polytope vertices, Hadamard basis simplex vertices 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='12 evaluation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='1 fraction 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='08 per fraction r Exact 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='4 Exact 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='02 2 4 6 8 10 12 14 16 2 4 6 8 10 12 14 16 Function eyaluations Function evaluations10 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' DUERSCH larger blocks to obtain cubatures of still higher dimensions yields the cross-polytope vertices in the Hadamard basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='1 generates the signs needed to con- struct an antithetic pair of vertices from the iterate index, q = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' , 2⌈log2(d)⌉ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Each quadrature from this sequence still exactly integrates all linear combinations of univariate quadratics, or 1 + 2d basis functions, with the mean-field distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' If the mean-field distribution is symmetric in each coordinate, all linear combinations of univariate cubics are exact as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='1 Cross-Polytope Vertex Sequence in Hadamard Basis d is the number of parameter dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' q is a non-negative integer index for the desired term of the sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Output: s is the d×1 vector of signs needed to construct an antithetic pair, θ(2q+1) = µ + s ∗ σ and θ(2q+2) = µ − s ∗ σ, for an equal-weight (w = 1 2) quadrature for a distribution with mean, µ, and covariance, diag(σ)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' 1: function s = quadrature sequence(d, q) 2: Get the number of bits needed to index parameters, nb = ⌈log2(d)⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' 3: Compute the parity of each parameter dimension for iterate q, pi = nb xor j=1 [bitj(i) ∧ bitj(q)] for all i ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' , d − 1} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' 4: Return signs from parity, s = 2p − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' 5: end function Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='2 and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='3 show how Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='1 produces quadrature sub- sequences that obtain periodic exactness within two-dimensional subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' See Ap- pendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='2 and Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='3 for proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Figure 3 provides a visualization of the cross-polytope sequence in the Hadamard basis (left) and the number of function evaluations needed to correctly integrate specific covariance basis functions (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='2 (Relative Parity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' The relative parity, pi1 xor pi2, corresponding to any two distinct parameters, θi1 and θi2, in Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='1 determines the product of corresponding signs in the result and it only depends on the iterate index, q, and the binary string obtained by bitwise exclusive disjunction of binary representations of parameter indices, pi1 xor pi2 = nb xor j=1 [xj ∧ bitj(q)] where xj = bitj(i1) xor bitj(i2) for j ∈ [nb].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='3 (Exactness Periodicity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Let θi1 and θi2 be any two distinct pa- rameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Let b ∈ [nb] indicate the position of the least-significant bit that is different between both binary representations of their indices, i1 and i2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Every consecutive con- tiguous quadrature subsequence of 2b antithetic pairs obtained by Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='1, using iterate indices q = z2b, z2b + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' , (z + 1)2b − 1 for z ∈ Z≥0, averages to the 4-node tensor-product cubature over the corresponding two-dimensional subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' These subsequences also generate higher-order cubatures in up to nb + 1 dimen- sions for some specific sets of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' This occurs when the binary representa- tions of parameter indices only differ by a single bit, each, from a base index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' For AN EFFICIENT QUADRATURE SEQUENCE AND SPARSIFYING METHODOLOGY 11 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Left: Illustration of cross-polytope vertex evaluations in 8D, visualized with the nested coordinate scaling x = 2 5 ( 2 5 θ1 + θ4) + θ7, y = 2 5 ( 2 5 θ2 + θ5) + θ8, and z = 2 5 θ3 + θ6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' The fine grid points are hypercube vertices, a superset of the cross-polytope vertices in the Hadamard basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Every antithetic pair, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' in lavender, exactly integrates all univariate quadratics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Longer sequences— shown in blue, green, and red—exactly integrate increasing sets of multivariate quadratics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Right: Visualization of the number of function evaluations from this sequence needed to integrate each product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Every four evaluations creates 2D cubatures for half of all coordinate pairs (exact on lavender diagonal and blue checkerboard).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' example, if parameters θi1, θi2, θi3, and θi4 are such that only bit1(i1) ̸= bit1(i4), bit2(i2) ̸= bit2(i4), bit3(i3) ̸= bit3(i4), and all other bits are the same as i4, then every 8 quadratures (16 evaluations) would result in a tensor-product cubature for functions that only depend on these parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' This easily follows by applying the same reasoning as used in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Unfortunately, this observation bears lit- tle importance since errors in many two-dimensional subspaces will still persist, and dominate, until 2nb quadratures have been averaged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Variational Fixed-Point Optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Efficient numerical integration enables a simple optimization procedure for mean-field variational inference as a fixed-point iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='1 analyzes the relationship between an optimal variational approximation and the corresponding integrals needed to project the pos- terior distribution onto a variational basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='2 shows how to modify the preceding quadrature sequences to efficiently implement these posterior projections onto a quadratic basis for Gaussian mean-field distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='3 provides an extension to mean-field distributions that are suitable for capturing sparsity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Basis Analysis for Fixed-Point Optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Let us consider a family of variational distributions that may be written as an exponential of a linear combi- nation of basis functions, fℓ(θ) for ℓ = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' , nℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' That is, q(θ | ϕ) = exp � nℓ � ℓ=0 ϕℓfℓ(θ) � and ϕ∗ = argmin ϕ D[ q(θ | ϕ) ∥ p(θ | D) ] is the vector of coefficients corresponding to the optimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Clearly, we must restrict feasible coefficients, ϕ, to those that yield proper, normalized, distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Since Cross-Polytope Vertex Sequence in 8D Evaluations Needed for Exact Quadratic Integrals D Factor Integrand 8 and econd 5 2 S 9 and 3 01 2 04 05 06 , and First Integrand Factor12 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' DUERSCH each variational distribution also induces an inner product, ⟨f, g⟩ϕ = � dθ q(θ | ϕ)f(θ)g(θ), we can construct the basis to be orthogonal, ⟨fi, fj⟩ϕ∗ = 0 for all i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' This will allow us to apply the calculus of variations to illuminate the relationship between optimal coefficients and the posterior distribution, p(θ | D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' We can capture arbitrary infinitesimal perturbations in the vicinity of the opti- mizer by using a differential element, ε, and a vector of perturbations, η, to write Gateaux derivatives, q(θ | ϕ = ϕ∗ + εη).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Analysis begins by constraining feasible perturbations to only those that retain normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Going forward, it is useful to define f0(θ) ≡ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Then, ∂ ∂ε �� dθ q(θ | ϕ = ϕ∗ + εη) � ε=0 = � dθ q(θ | ϕ∗) nℓ � ℓ=0 ηℓfℓ(θ) = nℓ � ℓ=0 ηℓ⟨f0, fℓ⟩ϕ∗ = η0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Thus, by using an orthogonal basis and capturing the normalization coefficient with f0, normalization-preserving perturbation directions only require fixing η0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Provided both the variational optimizer and the posterior distribution have full support over the parameter domain, we can rewrite the posterior distribution by factoring out the optimizer and defining what remains with the residual, r(θ), so that p(θ | D) = exp � r(θ) + n � ℓ=0 ϕ∗ ℓfℓ(θ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Since arbitrary perturbations must satisfy the variational principle, we have ∂ ∂ε �� dθ q(θ | ϕ∗ + εη) log �q(θ | ϕ∗ + εη) p(θ | D) �� ε=0 = � dθ q(θ | ϕ∗) � nℓ � ℓ=1 ηℓfℓ(θ) � [1 − r(θ)] = − nℓ � ℓ=1 ηℓ⟨fℓ, r(θ)⟩ϕ∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' As each remaining ηℓ is arbitrary, all inner products must vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' It follows that the residual must be orthogonal to the span of the variational basis, excluding the nor- malizing component in f0, which is typically unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Thus, the optimal coefficients are a fixed point that we obtain by projecting the log-posterior onto the span of the variational basis, disregarding normalization, ϕ∗ ℓ = ⟨fℓ, log(p(D | θ)p(θ))⟩ϕ∗ ⟨fℓ, fℓ⟩ϕ∗ for ℓ = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' , nℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Gradient and Hessian Extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Perhaps the simplest mean-field dis- tribution that is amenable to this approach uses a quadratic basis in each coordinate, yielding a mean-field Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Since we typically think about minimizing loss during optimization of learning algorithms, we will frame analysis in terms of the negative AN EFFICIENT QUADRATURE SEQUENCE AND SPARSIFYING METHODOLOGY 13 log-posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Let µ(t) be the expansion point for a quadratic basis, J0 is the con- stant offset, g is the average gradient, h is the Hessian diagonal, and the residual r(θ) contains all other terms so that the loss is J (θ | D) = − log (p(D | θ)p(θ)) = J0 + (θ − µ(t))T g + 1 2(θ − µ(t))T diag(h)(θ − µ(t)) − r(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' To maintain stability and normalizability, the Hessian diagonal must be constrained as ˆhi = max(hi, hmin) for some hmin > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' This allows us to compute the update, µ(t+1) = µ(t) − g ⊘ ˆh and σ(t+1) i = ˆh −1/2 i to obtain q(θ | ϕ(t+1)) = N(θ | µ(t+1), diag(σ(t+1))2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' In principle, we could use our quadrature scheme to approximate these updates by evaluating orthonormal projections, q(θ | ϕ(t+1)) ∝ exp � − nℓ � ℓ=1 fℓ(θ) ⟨fℓ, − log p(D | θ)p(θ)⟩ϕ(t) ⟨fℓ, fℓ⟩ϕ(t) � , but a much more efficient approach leverages loss gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' A quadrature may be generally understood as a linear combination of linear functionals (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' point-wise evaluations) that approximates another linear functional (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' integration against the mean-field distribution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' If the approximation is exact for some basis, then the quadrature must also be exact for the entire span by ap- plying linearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' While the quadratures we have considered so far employ point-wise evaluations of a function, point-wise evaluations of the gradient also comprise linear functionals, d of them, and allow us to evaluate many basis coefficients simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' The gradient of the negative log posterior is −∇θ log p(θ | D) = g + h ∗ (θ − µ(t)) − ∇θ r(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Note that if the posterior is locally smooth, the gradient of the residual will be dom- inated by off-diagonal Hessian components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' If H is the full Hessian matrix, then −∇θ r(θ) = (H − diag(h))(θ − µ(t))+ higher-order terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Averaging antithetic pairs easily captures g, canceling all Hessian contributions as well as some higher-order terms in concert with Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' We can also extract the Hessian diagonals by adjusting the signs of each term in a second sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' These signs correspond to multiplying gradients by first-order basis functions in each coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' For example,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' if we evaluate the Hessian contribution to gradients from two antithetic pairs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' multiplied elementwise by (θ − µ(t)) ⊘ σ(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' and average the results,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' we obtain the diagonal �1 1 � ∗ ��h11 h12 h21 h22 � �1 −1 1 −1 � � 1/4 −1/4 �� + � 1 −1 � ∗ ��h11 h12 h21 h22 � � 1 −1 −1 1 � � 1/4 −1/4 �� = 1 2 �� 1 1 � ∗ �h11 + h12 h21 + h22 � + � 1 −1 � ∗ �h11 − h12 h21 − h22 �� = �h11 h22 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' demonstrating how off-diagonal terms vanish according to the exactness periodicity of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Algorithm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='1 shows how to implement this approach to construct a quadratic approximation of the loss, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' the negative log-posterior, from gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' 14 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' DUERSCH Algorithm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='1 Quadratic Loss Approximation Input: µ is a d × 1 vector of means, where d is the parameter dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' σ is a d × 1 vector of standard deviations of the mean-field distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' q1 is an integer indicating the next position within the quadrature sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' nq indicates the number of antithetic pairs to use from the quadrature sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' J (·) is a loss function, returning both the value and gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Output: J, g, and h so that J (θ) ≈ J + (θ − µ)T g + 1 2(θ − µ)T diag(h)(θ − µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' 1: function (J, g, h) = quadratic approx(µ, σ, q1, nq, J (·)) 2: J = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' g = 0d×1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' h = 0d×1 ▷ Initialize accumulators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' 3: for q = q1, q1 + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' , q1 + nq − 1 do 4: s = quadrature sequence(d, k) ▷ Get quadrature signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' 5: [J+, g+] = J (µ + σ ∗ s) ▷ Operator ∗ is Hadamard product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' 6: [J−, g−] = J (µ − σ ∗ s) 7: J ← J + J+ + J−;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' g ← g + g+ + g− ▷ Ordinary integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' 8: h ← h + (g+ − g−) ∗ s ▷ Integrate product against first-order basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' 9: end for 10: g ← 1 2nq g 11: h ← h ⊘ (2nqσ) ▷ Operator ⊘ is elementwise right division.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' 12: J ← J 2nq − hT σ2 2 13: end function 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Dirac-Gauss Mixtures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Now we will consider a mean-field framework that is capable of capturing a finite probability that any individual coordinate, θi, is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' By introducing a Bernoulli-distributed random variable, zi, we can write each mean-field factor as Dirac-Gauss mixture by marginalizing over zi as q(θi) = � zi∈{0,1} q(θi, zi) = � zi∈{0,1} q(zi)q(θi | zi) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='1) = q(zi)δε(θi) + q(¬zi)Nε(θi | νi, τ 2 i ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' This expression uses shorthand, q(zi) ≡ q(zi = 1) and q(¬zi) ≡ q(zi = 0), and conditional distributions that have been constructed to be non-overlapping by carving out a small interval, (− ε 2, ε 2) for some ε > 0, δε(θi) = � ε−1 |θi| < ε 2 0 else and Nε(θi | νi, τ 2 i ) = � 0 |θi| < ε 2 N(θi | νi, τ 2 i ) else .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' In the limit ε → 0, exact normalization of Nε(θi | νi, τ 2 i ) is unnecessary, because doing so only multiplies the scaling factor by 1 + O(ε), a vanishing change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' We can also construct a spike-and-slab prior with the same structure, p(θi) = p(zi)δε(θi) + p(¬zi)Nε(θi | 0, hp −1), where hp is the prior precision associated with a nonzero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' AN EFFICIENT QUADRATURE SEQUENCE AND SPARSIFYING METHODOLOGY 15 Note the important distinction between the mean and standard deviations re- quired by the quadrature formulas in Algorithm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='1 versus the mean and variance of the normal distribution in Equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='1), which is conditioned on a nonzero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' The correct moments of this mean-field distribution are: µi = q(¬zi)νi and σ2 i = q(zi)q(¬zi)ν2 i + q(¬zi)τ 2 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='2) To apply the analysis from Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='1, we need to write the variational family as an exponential, so it is useful to formulate the probabilities of zeros as logits, ζp = log � p(zi) p(¬zi) � , p(zi) = exp(ζp) 1 + exp(ζp), ζi = log � q(zi) q(¬zi) � , and q(zi) = exp(ζi) 1 + exp(ζi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Since the conditional distributions are disjoint, we can easily compute log p(θi) = − log (1 + exp(ζp)) + � ζp − log(ε) |θi| < ε 2 1 2 log( hp 2π) − 1 2hpθ2 i else (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='3) = const − hpθ2 i 2 + � ζp + 1 2 log( 2π hp ) − log(ε) |θi| < ε 2 0 else Likewise, each variational factor can be written log q(θi) = const − (θi − νi)2 2τ 2 i + � ζi + 1 2 log � 2πτ 2 i � + ν2 i 2τ 2 i − log(ε) |θi| < ε 2 0 else .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='4) Since Algorithm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='1 allows us to project the loss onto a quadratic univariate basis, we only need to add the prior discontinuity term in Equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='3) to continuous quadratic projections of the remaining log posterior components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' For example, adding the quadratic terms from the negative log-prior and negative log-likelihood, with gradient g and Hessian diagonal h at the expansion point µ(t), gives hp 2 θ2 i + gi(θi − µ(t)) + hi 2 (θi − µ(t))2 = const + (θi − νi)2 2τ 2 i where νi = hiµ(t) − gi hp + hi and τ 2 i = (hp + hi)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Including the remaining prior terms and absorbing constants into the residual, we have log p(θ | D) = r(θ) + � i � −(θi − νi)2 2τ 2 i + � ζp + 1 2 log � 2π hp � − log(ε) |θi| < ε 2 0 else � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='5) Matching Equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='5) to Equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='4) gives the logit of the zero probability, ζi = ζp + 1 2 � log �hp + hi hp � − (hp + hi)ν2 i � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='6) In practice, the continuous component of the negative log prior can be included in the loss projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' See Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' 16 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' DUERSCH 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Sparsifying Methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Although the principles governing the fixed- point optimizations outlined in Section 4 are fairly simple to derive, many practi- cal complications must be addressed to efficiently discover high-posterior domains for deep learning models with a specified sparsity target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' The primary purpose of this content is to demonstrate that the quadrature sequence may be efficiently combined with sparsifying variational inference to achieve good predictions from a sparse model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='1 examines practical implementation challenges for the complicated loss landscapes that typify deep learning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='2 provides detailed algo- rithms to address these challenges, and Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='3 includes numerical experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' The methodology that follows draws upon empirical testing that illuminated key difficulties with sparse optimization and simple approaches to address them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' To benefit future work on sparsifying methodologies, we first examine these challenges and how the methodology seeks to address them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Negative Hessian Eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' The approximate Hessian diagonals of the loss function, the negative log-posterior, are often negative and there is no reason to suspect that true Hessian diagonals would not be also.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Negative eigenvalues indicate increasing probability density as parameters drift from the critical point along the corresponding eigenvectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Not only is such a distribution unnormalizable, it becomes untrustworthy as we move outside of the dominant region of integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' We can easily solve this problem by distinguishing the log-variational distribution from the log-posterior approximation, and then setting a minimum positive curvature to maintain a coherent distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Yet, a problem still remains with stable gradient updates from the underlying quadratic approximation of the log-posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Negative eigenvalues also create vicious feedback between small perturbations to the quadratic expansion point and the corresponding gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Testing showed that it is better to apply the same curvature limitation to the quadratic approximation of the local log-posterior structure to suppress this effect, which may be interpreted as enforcing a local trust region that only allows gradient zeros that are both stable and nearby.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Safely moving beyond the trust region simply requires reevaluating the log- posterior projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Thus, it will be useful to quickly replace obsolete contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Posterior Annealing with Restarted Sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Accumulating too many log-posterior contributions initially can result in large Hessian diagonals and obsolete gradient information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Since high curvature suppresses zeros, it becomes more difficult for training to identify parameters to send to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' A simulated annealing strategy with restarted sums solves this by preventing the log-posterior approximation from suppressing potential zeros too early or keeping old contributions too long.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' To derive the moment-matching formula, suppose we have a set of independent identically-distributed random variables, xi for i ∈ N, that we would like to sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Let the expectation and variance be E[xi] = α and Var[xi] = v, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' If we accumulate n0 samples in an initial sum, and n1 more samples in a restarted sum, a0 = n0 � i=1 xi and a1 = n0+n1 � i=n0+1 xi where n0 + n1 < N, then we have E[a0] = n0α, E[a1] = n1α, Var[a0] = n0v, and Var[a1] = n1v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' It easily follows that the linear combination comprising a hybrid sum, ˆa = max � 0, n0 − n1 n0 + n1 � a0 + max � 1, 2n0 n0 + n1 � a0, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='1) AN EFFICIENT QUADRATURE SEQUENCE AND SPARSIFYING METHODOLOGY 17 preserves both the mean and variance of the first sum until the number of samples in the restarted sum overtakes it, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' E[ˆa] = max(n0, n1)α and Var[ˆa] = max(n0, n1)v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' We can use this formula to implement simulated annealing and keep the log- posterior approximation current with the mean-field distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Old terms vanish as quickly as possible without reducing the number of effective terms accumulated thus far, and we can control increases in the number of data points with n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Annealing both posterior factors in Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='1), as p(D | θ)αp(θ)α, rather than just the likelihood, or p(D | θ)αp(θ), allows the mean of the Gaussian compo- nents, ν, to remain stable as the annealing exponent increases to α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' To implement this easily, when each negative log-likelihood term is computed from a training case, we also just add the corresponding fraction of the negative log prior, disregarding the Dirac delta contribution: J (θ | D, α) = −α (log p(D | θ) + log p(θ)) ≈ αnc � c=1 J (θ | dc) where J (θ | dc) = − log p(dc | θ) − 1 nc log p(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' From each quadrature projection of a loss term, we obtain the loss value, gradient, and Hessian diagonals at the expansion point, J (θ | dc) = − log p(dc | θ) − 1 nc log p(θ) ≈ J + gT (θ − µ) + 1 2(θ − µ)T diag(h)(θ − µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' These coefficients are then added into their respective sums J1, g1, and h1, and the counter n1 increments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' By combining these current sums with the previous counter- parts using Equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='1), we obtain running quadratic approximations that drop old contributions efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' When the current sum reaches the annealing target, n1 = ntgt = αnc, all values are moved to the index-0 counterparts and the current sums restart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Letting the present epoch be indexed as e = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' , ne, the following exponential schedule works well for ne = 10 epochs: ntgt = � nc2e−ne� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Stable Quadrature Domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' The initial discovery of a basin of at- traction in the loss function is difficult if the mean-field variance is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' It is better to start with small localized quadratures that yield trustworthy gradients and grad- ually increase the mean-field variance to match that of the loss structure, or until the lower bound on curvature is attained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Regarding the previous point, not only does enforcing a positive lower bound on Hessian diagonals support stable gradient updates, it also suppresses the mean-field variance and, thus, benefits the quality of local gradient approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Similarly, uncontrolled changes in zero probabilities, q(zi), can heavily interfere with discovery of, and adherence to, local basins of attraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' This is due to the variance contribution, q(zi)q(¬zi)ν2 i , in Equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' A small increase in a zero probability may only slightly change the parameter mean, but it can significantly increase the mean-field variance from a random initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Again, if the quadra- ture scale is too large, gradient quality deteriorates, because the long-distance average no longer matches the vicinity of the expansion point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' This is the reason the algo- rithm that follows completes a full epoch over the training data while keeping all zero probabilities very small to support initial discovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' 18 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' DUERSCH Another tool to increase the stability of variational updates is to set an effective maximum learning rate on the Gaussian mean, ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Rather than immediately jumping to the critical point of the quadratic loss approximation with every new term, we can first identify the gradient at ν and then apply a more restrictive trust region in the Newton step that would yield the zero: gν = ˆg + ˆh ∗ (ν − µ) and ν ← ν − gν ⊘ max � ˆh, max(n0, n1)hmin � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Scaling the lower bound on the Hessian by the number of terms in the present stage of annealing is equivalent to setting a maximum perturbation scale as a function of the average gradient over the current stage of annealing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' That is, max(n0, n1)gavg + max(n0, n1)hminδavg implies δavg = − gavg hmin .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' We can still track gradient updates in log-posterior approximations and the effect averages new gradient contributions as data accumulate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' See Algorithm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Stable Parameter Readjustment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Building on the previous point, small changes in the zero probabilities of some parameters may also require significant shifts in the nonzero parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' This effect becomes more pronounced as the num- ber of remaining nonzero parameters decreases, thus decreasing the flexibility of the model to account for new zeros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' A gentle approach, changing the zero probabilities gradually, allows these corrections to stabilize and stay within a high posterior region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' These corrections also support better sparsity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' as some nonzero parameters become more constrained, others become less so and, thus, trend to zero more easily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' We can control the rate of change in zero probabilities by using a continuous piecewise-affine map on ζ, computed from Equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Since we will control the constant offsets directly with the map that follows, there is no need to include the prior contribution, ζp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Given ne epochs and a training progress indicator, t ∈ [0, ne], we will set a schedule for s(0)(t) representing the fraction of parameters that we would like to hold near zero, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' below a lower threshold on the probability of being nonzero, such as p(0) NZ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Likewise, let s(1)(t) represent a fraction of parameters that we would like to hold above an upper threshold, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' p(1) NZ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' By sorting ζ after each update to the variational parameters, we can easily identify the set of s(0) largest logits, Z(0), and the set of the s(1) smallest logits, Z(1), to obtain the domain boundaries ζ(0) = min ζ∈Z(0) ζ and ζ(1) = max ζ∈Z(1) ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' These boundaries obviously depend on the training progress, t, which is left implied going forward to simplify notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' The range boundaries corresponding to the non- zero probabilities we desire are ˆζ (0) = −logit(p(0) NZ) and ˆζ (1) = −logit(p(1) NZ), giving the continuous piecewise-affine map that appears in Algorithm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' This mapping serves as a sieve on zeros, gradually realizing plausible zeros while letting the remaining pa- rameters readjust until the sparsity target is achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Empirical testing shows that, after the initial epoch, the sieve schedule can proceed quickly, but it most slow down as fewer nonzero parameters remain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Otherwise, training and validation accuracies deteriorate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Given end-of-training targets, s(0) tgt and s(1) tgt, a simple schedule is s(0)(t) = max � 0, min � 1, 1 − 21−t 1 − 22−ne �� s(0) tgt AN EFFICIENT QUADRATURE SEQUENCE AND SPARSIFYING METHODOLOGY 19 and s(1)(t) = s(1) tgt + s(0) tgt − s(0)(t), removing nearly half of the remaining parameters that are targeted to vanish per epoch, while keeping the intermediate fraction of parameters fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' The sparsity target is achieved when t = ne − 1, saving the last epoch for final adjustments on a fully realized sparsity pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Training Algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' While there are many potential improvements to this sparsifying methodology, these basic algorithms provide a sufficient means to drive strong sparsity in high-dimensional models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' As there are many variational parameters and hyperparameters to track during training, we will simply pass a training data structure, T , between the following subroutines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Table 1 lists its contents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Table 1 Training Data Structure Hyperparameters and Variational Attributes Hyperpara.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Description Attr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Description d Model parameter dimension ζ Logits of zero probabilities s(0) tgt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='97 Target fraction of zeros ν Means of nonzero Gaussians s(1) tgt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='01 Target fraction of nonzeros τ Standard deviations of nonzeros ne = 10 Number of epochs µ Mean of mean-field e Current epochs σ Standard deviations of mean-field nq = 2 Antithetic pairs per quadrature n0, n1 Number of terms in quadratic sums q Current quadrature index J0, J1 Loss at quadratic expansion point αmax = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='1 Maximum learning rate g0, g1 Gradient at expansion point α0 = 10−5 Initial learning rate h0, h1 Hessian diagonals τmax = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='3 Maximum Gaussian standard dev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' ˆζ (0), ˆζ (1) Target logits for sieve hmin Minimum Hessian for var.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' update Jtrain Training loss over current epoch p(0) NZ, p(1) NZ Sieve targets (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='001 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='999) The main training loop, Algorithm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='1, begins by randomly initializing the index of the quadrature sequence as well as the quadratic loss sums comprised of n0, g0, and h0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' By setting the curvature in h0 from α0 and the number of terms, n0, that are needed until the next quadratic loss approximation achieves a full replacement, Algorithm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='1 Main Training Loop Input: T contains the parameters listed above as well as the training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Output: (µ, σ) trained mean-field mean and standard deviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' 1: function (µ, σ) = train(T ) 2: q = � uniform(0, 1) d nq � nq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' ▷ Randomize initial quadrature index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' 3: n0 = ⌊nc21−ne⌋;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' g0 = 0d×1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' h0 = 1 α0 1d×1 ▷ Initialize quadratic sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' 4: Randomly initialize parameters as usual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Place into expansion point, µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' 5: σ = h−1/2 0 ▷ Set consistent standard deviations (elementwise).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' 6: hmin = (n0αmax)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' ▷ Minimum Hessian will scale with annealing schedule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' 7: ˆζ (0) = −logit(p(0) NZ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' ˆζ (1) = −logit(p(1) NZ) ▷ Set target logits for sieve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' 8: for e = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' , ne do 9: T = variational epoch(T ) 10: end for 11: Return (µ, σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' 12: end function 20 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' DUERSCH Algorithm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='2 Variational Epoch 1: function T = variational epoch(T ) 2: n1 = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' g1 = 0d×1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' h1 = 0d×1 ▷ Initialize current sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' 3: Jtrain = 0 ▷ Track training loss over this epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' 4: ntgt = ⌊nc2e−ne⌋ ▷ Set target number of terms for each restart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' 5: if e = ne then rNZ ← round(pNZ) ▷ Realize sparsity pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' 6: Randomly permute training data, dc for c ∈ [nc].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' 7: for c = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' , nc do 8: (J, g, h) = quadratic approx (µ, σ, q, nq, J (θ | dc)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' 9: Jtrain ← Jtrain + J 10: q ← q + nq ▷ Increment quadrature index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' 11: t ← e − 1 + c nc ▷ Update training progress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' 12: T ← variational update(T , g, h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' 13: if n1 = ntgt then 14: n0 ← n1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' g0 ← g1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' h0 ← h1 ▷ Overwrite previous sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' 15: n1 ← 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' g1 ← 0d×1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' h1 ← 0d×1 ▷ Restart current sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' 16: end if 17: end for 18: Return T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' 19: end function n1 = n0, we obtain an effective limitation on the initial learning rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Likewise, hmin limits the maximum learning rate by setting a lower bound on the curvature contribution of each loss contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' The algorithm then loops over the desired number of epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Algorithm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='2 performs each epoch by first initializing the quadratic loss approxi- mation coefficients and the number of target terms, ntgt, needed for the current stage of simulated annealing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' At the start of the final epoch, this algorithm also realizes the most probable sparsity pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' For each training case, the new coefficients for the quadratic projection are obtained by numerical integration, Algorithm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='1, and the results are accumulated by Algorithm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Every time the targeted number of terms is reached, the previous quadratic sums are overwritten and the current sums restart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' The variational update, Algorithm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='3, accumulates the quadratic approxima- tions, which always share the same expansion point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Next, the gradient at Gaussian means, ν, is computed from the hybrid quadratic approximation that uses moment matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' This allows a limited quasi-Newton step to control perturbations to ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' The sieve, Lines 10 through 15, drives the mean-field distribution to a particular sparsity pattern by filtering the parameters that are most suitable zeros from those that must remain free to adjust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Finally, the mean-field moments are updated in µ and σ and the gradients are updated to the new expansion point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Experimental Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' These experimental results are performed using a convolutional neural network for MNIST [24] shown in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='3 with ternary AN EFFICIENT QUADRATURE SEQUENCE AND SPARSIFYING METHODOLOGY 21 Algorithm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='3 Variational Update 1: function T = variational update(T , g, h) 2: n1 ← n1 + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' g1 ← g1 + g ▷ Update current quadratic coefficient sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' 3: h1 ← max(h1 + h, τmax −2) 4: α0 = max(0, n0−n1 n0+n1 );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' α1 = max(1, 2n0 n0+n1 ) ▷ Get moment-matched sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' 5: ˆg = α0g0 + α1g1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' ˆh = α1h0 + α1h1 6: gν = ˆg + ˆh ∗ (ν − µ) ▷ Get gradient at ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' 7: ν ← ν − gν ⊘ max � ˆh, max(n0, n1)hmin � ▷ Limit effective learning rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' 8: τ = ˆh −1/2 ▷ Get standard deviations of nonzeros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' 9: if e < ne then 10: ζ = 1 2 � log � ˆhτmax2� − ˆh ∗ ν2� ▷ Get initial logits of zero probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' 11: s(0) = max � 0, min � 1, 1−21−t 1−22−ne �� s(0) tgt ▷ Update sparsity schedule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' 12: s(1) = s(1) tgt + s(0) tgt − s(0)(t) 13: Sort ζ and obtain boundaries ζ(0) and ζ(1) from s(0) and s(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' 14: Apply piecewise map, ˆζi = � � � � � � � � � � � ζi − ζ(0) + ˆζ (0) ζi ≤ ζ(0) ˆζ (0) + (ζi−ζ(0)) � ˆζ (1)−ˆζ (0)� ζ(1)−ζ(0) ζ(0) < ζi ≤ ζ(1) ζi − ζ(1) + ˆζ (1) ζi > ζ(1) 15: pNZ = � 1 + exp(ˆζ) �−1 16: else pNZ = rNZ 17: end if 18: ˆµ = pNZ ∗ ν;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' δ = ˆµ − µ ▷ Get new mean and perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' 19: σ ← � pNZ ∗ (1 − pNZ) ∗ ν2 + pNZ ∗ τ 2�−1/2 ▷ Update standard deviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' 20: µ ← ˆµ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' g0 ← g0 + h0 ∗ δ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' g1 ← g1 + h1 ∗ δ ▷ Move expansion point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' 21: end function Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' These histograms show sparsification progress during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Note the logarithmic scaling of the vertical axis (only the left and right peaks appear with linear scaling).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' The sieve holds 1% of parameters between the low (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='001) and high (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='999) probabilities of a nonzero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' The last epoch uses the fixed sparsity pattern given by rounding, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' the maximum variational realization of z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Progress of Sparsifying Sieve During Training 105 104 103 103 102 ie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' 10 10° Probability of Nonzero, Epoch 1 Probability of Nonzero, Epoch 2 Probability of Nonzero, Epoch 3 Probability of Nonzero, Epoch 4 Probability of Nonzero, Epoch 5 105 104 Cou 103 102 Par 10 100 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='5 ProbabilityofNonzero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='Epoch6 ProbabilityofNonzero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='Epoch7 Probability of Nonzero, Epoch 8 ProbabilityofNonzero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='Epoch9 ProbabilityofNonzero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='Epoch1022 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' DUERSCH logical activation functions [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' The model was trained with 60, 000 cases and 10 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Figure 4 illustrates the progress of the sparsity sieve on the first training trail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' The trained model achieves over 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='9% zeros upon completion, while retaining 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='9% accuracy on 12, 000 test cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' See Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='3 for additional details on the structure of the network and numerical integration comparisons for various quadratures applied to the mean-field distribution at the start and at the end of training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Viewed from the context of tensors, mean-field distributions are equivalent to rank-1 functions [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' We can compare a rank-r Canonical Polyadic (CP) approximation, a rank-r tensor, to a mean-field mixture as Xj ≈ r � k=1 λk d−1 � m=0 A(m) jmk p(θ) ≈ r � k=1 q(k) d−1 � i=0 q(θi | k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Although this work does not explore higher-rank variational distributions, the re- sulting mean-field mixtures would also be scalable and feasible to integrate, only multiplying the integration cost by the rank, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' the number of mixture components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Although the analysis in Section 4 only examines Gaussian mean-field distribu- tions and Dirac-Gauss mixtures, a related approach may be suitable for Laplacian mean-field distributions or perhaps even Dirac-Laplace mixtures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' One difficulty, how- ever, is that the quadratic basis functions, (θ − ν)2, would have to be replaced by absolute value functions, |θ − ν|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Unfortunately, this would require an adaptive basis, rather than just algebraically manipulating a fixed basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' It may be possible to improve sparsity further by taking correlated parameter structures into account by referencing the computational graph in a manor similar to the method by Jantre, Bhattacharya, and Maiti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' That said, a key benefit of this work is its generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Many deep learning architectures do not adhere to an elementary chain of matrix multiplications and elementwise activation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' For example, the computational graph dependencies in transformer architectures [44] may span multiple layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' If, however, the probability of a parameter zero can be efficiently tied to such dependencies, it would be possible to control entire swaths of parameters at once and improve computational efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Summary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' This work began by investigating stochastic blocked mean-field quadratures in order to ensure numerical integration would retain high accuracy within specific parameter blocks, while also ensuring many samples converge to a tensor-product cubature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' By considering quasirandom sequences to speed up con- vergence, we arrived at quadrature sequences composed of antithetic evaluation pairs from the cross-polytope sequence in the Hadamard basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' In d dimensions, this method exactly integrates approximately 1 4d2 multivariate quadratic basis functions using only 4 function evaluations, which gives the highest exactness efficiency for all the methods tested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='3 shows how every doubling of the number of evaluations increases exactness to include half of the remaining quadratic basis functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' We then examined the optimal structure of variational distributions that can be written as an exponential of a linear combination of variational basis functions, which ties efficient numerical integration to a fixed-point iteration for variational updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Adjusting the quadrature sequences to act on gradients allows us to approximate Hessian diagonals for Gaussian mean-field distributions, as well as reconstruct the probabilities of specific parameters realizing to zero in Dirac-Gauss mixtures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Finally, a practical sparsifying methodology was devised to overcome several op- timization challenges for the complicated loss structures that typify deep learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' AN EFFICIENT QUADRATURE SEQUENCE AND SPARSIFYING METHODOLOGY 23 Numerical experiments demonstrate the ability to achieve strong sparsity while re- taining high validation accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' This work opens new approaches to reduce both storage and operating energy requirements for trained machine learning models by allowing dense matrix multi- plications and tensor contractions to be replaced with more efficient sparse versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' With regard to previous work on logical activation functions, sparsity allows us to suppress logical complexity of predictions during training in pursuit of better gener- alization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' My sincere thanks to Erin Acquesta, Tommie Catanach, Jaideep Ray, and Cosmin Safta for providing early feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Tommie suggested an exponential annealing schedule and log-likelihood integration experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Jaideep noted that by limiting the diameter of evaluation nodes, these quadrature sequences may provide an additional benefit to prediction models that cannot support large parameter perturbations, such as physics models that cannot accept unphysical states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' By probing these challenges from many perspectives, these conversations provide an important means to deepen understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Numerical Integration Experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' This set of experiments, Figures 5 to 7, shows the typical range of integration errors for several numerical approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' The first approach is pure Monte Carlo integration via sampling the mean-field distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' The second and third approaches are quasi-Monte Carlo, translating the set of samples to match the mean (second) and also scaling to match both the mean and the variance (third).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' The fourth approach demonstrates stochastic blocked mean-field quadratures with a block size of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' This requires 3 sigma points, given by the simplex vertices from Algorithm A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='1, within in each 2D block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Within each block, the evaluation nodes are then permuted uniformly at random and concatenated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' The fifth approach shows antithetic pairs of cross-polytope vertices in the Hada- mard basis, Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Signed errors are stored and sorted for each integration method from 5000 trials to obtain the 90% confidence intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Symmetric Distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' The first two mean-field distributions are Gaussian and Laplacian, Figure 5 and Figure 6, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Both are both symmetric Algorithm A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='1 Simplex Polytope Sigma Points Input: d is the number of dimensions in which the desired sigma points are embedded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Output: X is d × d + 1 matrix of evaluation nodes and corresponding weights, w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' 1: function (X, w) = simplex quadrature(d) 2: r = √ d 3: X = 0d×d+1 4: for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' , d do 5: Xii = r and Xij = −r d+1−i for j = i + 1, i + 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' , d + 1 6: r ← r √ (d+1−i)2−1 d+1−i 7: end for 8: Return X and shared weight, w = 1 d+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' 9: end function 24 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' DUERSCH Quadrature Error for Mean-Field Gaussian Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Selected quadrature errors for, q(θi) ≡ �d i=0 N(θi | 0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Each plot shows the quadra- ture error for products of orthonormal polynomials, ϕd(·), where d indicates the degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Row 1: Univariate basis functions show how quasi-Monte Carlo methods achieve exactness on first and second degree polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' These methods do not necessarily reduce error on higher-order basis functions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' compare to Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' The stochastic 2-block mean-field quadrature does not correctly integrate 3rd-order basis functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' The cross-polytope sequence is exact for these functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Row 2: Quasi-Monte Carlo methods offer little improvement to these multivariate quadratic in- tegrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Stochastic blocking retains exactness for quadratics within each block, column 1, but not between blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Finally, the cross-polytope sequence shows the exactness periodicity we expect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Row 3: Stochastic blocking appears to reduces the error for these 3rd-order functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' The cross- polytope sequence produces Gauss points, roots of ϕ2(θ0), causing all products to vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Row 4: The cross-polytope sequence generates odd pairs of evaluations for each odd basis function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' The product of an odd number of such evaluations remains odd, thus correctly summing to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' This is why both quasi-Monte Carlo methods integrate to zero with two samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' and demonstrate higher-order exactness for the cross-polytope quadrature sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' The univariate basis functions in Row 1 show how quasi-Monte Carlo methods achieve exactness on first and second degree polynomials by transforming samples to match leading moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' These methods may also reduce error on higher-order basis functions, but not always.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' See Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' The stochastic blocked mean-field quadrature does not correctly integrate 3rd-order basis functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' In contrast, the cross-polytope sequence in the Hadamard basis, always produces a pair of Gauss-points in each dimension, thus integrating all univariate 3rd-order polynomials exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' The multivariate quadratics in Row 2 show that even variance-matched quasi- 1 p1(0o) P2(00) P3(0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='. Monte Carlo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='. Mean-MatchedQuasi-MonteCarlo ---X---.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='Variance-MatchedQuasi-MonteCarlo 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='5 ----.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='. Stochastic 2-Block Mean-Field .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='. Cross-Polytope Vertex Sequence Error 0 ***+***** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='5 1 4 8 12 16 20 24 28 32 4 8 12 20 24 28 32 4 8 12 16 20 24 28 32 4 8 12 20 24 28 32 P1(00)p1(01) P1(00)P1(02) P1(00)P1(04) P1(00)p1(07) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='5 众众 0 A 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='5 X 4 8 12 16 20 24 28 32 4 8 12 16 20 24 28 32 4 8 12 16 20 24 28 32 4 8 12 16 20 24 28 32 2(0)1(01) 2(00)P1(02) P2(00)1(04) (10) 1()3) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='5 Error F ned Sign 国 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='5 4 8 12 16 20 24 28 32 4 8 12 16 20 24 28 32 4 8 12 16 20 24 28 32 4 8 12 16 20 24 28 32 P1(00)P1(01)1(02) P1(00)91(01)91(04) 1(00)P1(03)01(04) P1(00)1(01)1(07) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='5 ror S 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='5 4 8 12 16 20 24 28 32 4 8 12 16 20 24 28 32 4 8 12 16 20 24 28 32 4 8 12 16 20 24 28 32 Function Evaluations Function Evaluations Function Evaluations Function EvaluationsAN EFFICIENT QUADRATURE SEQUENCE AND SPARSIFYING METHODOLOGY 25 Quadrature Error for Mean-Field Laplace Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Selected quadrature errors for, q(θi) ≡ �d i=0 Laplace(θi | 0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' The critical property of the mean-field distribution that determines the characteristics of these error plots is symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Since Laplace distributions are symmetric, we observe the same structures as the Gaussian case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' The only notable difference is the scale of errors for some of the stochastic methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' For example, errors associated with ϕ3(θ0) (top-right) are significantly smaller in this case, as are the errors for the multivariate cubics in row 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' The cross-polytope sequence quadratures are exact for all the same cases as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Monte Carlo may not significantly improve mixed second-order integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Stochastic blocking retains exactness for quadratics within each block, column 1, but not be- tween blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Finally, the cross-polytope sequence shows the exactness periodicity we expect, based on the leading mismatched bit between each pair of parameter in- dices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' For example, ϕ1(θ0)ϕ1(θ7) has the same exactness periodicity as ϕ1(θ0)ϕ1(θ1) because the bit strings, 000 and 111, differ in the leading bit, just as 000 and 001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' In Row 3, the stochastic mean-field quadrature reduces error for these 3rd-order functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Since the cross-polytope sequence produces Gauss points in each dimension, the zeros of ϕ2(θ0), all of these products correctly vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' The cross-polytope sequence performs well in Row 4 because it generates odd evaluation pairs for each odd function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Since the product of an odd number of such evaluations is still odd, the average sums to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' This also explains why both moment- matching methods integrate to zero when they only contain two samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' 1 p1(0o) P2(00) P3(0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='. Monte Carlo .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Mean-Matched Quasi-Monte Carlo .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='---X---.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Variance-Matched Quasi-Monte Carlo 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='5 ----.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='. Stochastic 2-Block Mean-Field --.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Cross-Polytope Vertex Sequence Error 0 XXXX 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='5 1 4 8 12 20 24 28 32 4 8 12 20 24 28 32 4 8 12 16 20 24 28 32 4 8 12 20 24 28 32 P1(00)p1(01) P1(00)P1(02) P1(00)P1(04) P1(00)p1(07) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='5 众众 ** 0 A 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='5 4 8 12 16 20 24 28 32 4 8 12 16 20 24 28 32 4 8 12 16 20 24 28 32 4 8 12 16 20 24 28 32 2(0)1(01) 2(00)P1(02) 2(00)P1(04) 2(00)P1(07) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='5 1 4 8 12 16 20 24 28 32 4 8 12 16 20 24 28 32 4 8 12 16 20 24 28 32 4 8 12 16 20 24 28 32 P1(00)P1(01)1(02) 1(00)91(01)91(04) 1(00)P1(03)1(04) P1(00)1(01)1(07) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='5 ror S 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='5 4 8 12 16 20 24 28 32 4 8 12 16 20 24 28 32 4 8 12 16 20 24 28 32 4 8 12 16 20 24 28 32 Function Evaluations Function Evaluations Function Evaluations Function Evaluations26 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' DUERSCH Quadrature Error for Mean-Field Dirac-Gauss Mixture Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Selected quadrature errors for, q(θi) ≡ �d i=0 � 1 2 δ(θi) + 1 2 N(θi | 2, 1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Row 1: Note the conspicuous errors for the few-sample variance-matched quasi-Monte Carlo cases with ϕ2(θ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Since the cross-polytope sequence does not produce Gaussian quadratures in each coordinate, we no longer obtain 3rd-order exactness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Row 2: The cross-polytope sequence still operates as designed to integrate multivariate quadratics with the same periodicity as the previous cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Row 3: The cross-polytope sequence is no longer always exact for these cases because it no longer evaluates at roots of ϕ2(θi), but we still obtain the same exactness periodicity as Row 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Row 4: The cross-polytope sequence is still exact for these cases for the same reason as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Dirac-Gauss Mixture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' The third mean-field distribution tested is a spike and slab, Figure 7, which is not symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' As a consequence of the asymmetry, it is not possible for the cross-polytope sequence to generate Gaussian quadrature pairs that also have equal-weights, the essential property that allowed the cross-polytope sequence to generate 3rd-order cubatures earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' We can still construct equal-weight quadratures for this purpose, but they are only 2nd-order in each dimension, thus only becoming second-order cubatures with the exactness periodicity of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' The errors in Row 1 for the few-sample variance-matched quasi-Monte Carlo cases with ϕ2(θ0) occur because each factor distribution contains finite probability mass at θi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' With only a few samples, a specific coordinate is often zero for all samples, meaning it is not possible to match the sample variance to the distribution variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Only being able to match the mean causes these results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Again, since the cross- polytope sequence does not produce Gaussian quadratures in each coordinate, we no 1 p1(0o) p2(0) (p3(00) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='. Monte Carlo .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Mean-Matched Quasi-Monte Carlo .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='--X---.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Variance-MatchedQuasi-MonteCarlo 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='5 ---.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='Stochastic 2-Block Mean-Field .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='. Cross-Polytope Vertex Sequence Error 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='5 1 4 8 12 16 20 24 28 32 4 8 12 16 20 24 28 32 4 8 12 16 20 24 28 32 4 8 12 20 24 28 32 P1(00)p1(01) P1(00)P1(02) (0) (0) 1 (10) () ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='5 众众 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='5 X X 4 8 12 16 20 24 28 32 4 8 12 16 20 24 28 32 4 8 12 16 20 24 28 32 4 8 12 16 20 24 28 32 2(0)1(01) p2(00)p1(02) P2(00)1(04) 2(00)P1(07) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='5 Error 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='5 4 8 12 16 20 24 28 32 4 8 12 16 20 24 28 32 4 8 12 16 20 24 28 32 4 8 12 16 20 24 28 32 P1(00)1(01)1(02) P1(00)1(01)P1(04) 1(00)P1(03)1(04) P1(00)1(01)1(07) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' ror 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='5 4 8 12 16 20 24 28 32 4 8 12 16 20 24 28 32 4 8 12 16 20 24 28 32 4 8 12 16 20 24 28 32 Function Evaluations Function Evaluations Function Evaluations Function EvaluationsAN EFFICIENT QUADRATURE SEQUENCE AND SPARSIFYING METHODOLOGY 27 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Left: The initial, untrained, mean-field distribution exhibits high loss, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' negative log-likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Numerical integration with cross-polytope vertices in the Hadamard basis produces slightly tighter confidence intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Right: At the end of training, the model has converged to a sparse Gaussian mean-field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Both the stochastic 2-block simplex sigma points and the cross polytope sequence produce significantly tighter confidence intervals on the test data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' longer obtain cubic exactness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Row 2 shows that the cross-polytope sequence still operates as designed to inte- grate multivariate quadratics with the same periodicity as the previous cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Row 3, however, shows that the cross-polytope sequence is no longer always exact for these cases, since it no longer evaluates roots of ϕ2(θi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Instead, we revert to the same ex- actness periodicity as seen in Row 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' In Row 4, the same reasoning for odd products of odd function evaluations still holds, so these are still exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' MNIST CNN Negative Log-Likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' This simple network uses ternary logical activation functions, which were designed to forge a relationship be- tween parameter sparsity and logical complexity [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' The specific layer structure is shown in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' At the beginning of training, we have a Dirac-Gauss mixture to support inference of a sparsity pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' At the end of the final epoch, the model has converged to a full realization of a specific sparsity pattern, leaving the mean- field distribution as a Gaussian in each nonzero parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Both the initial and final mean-field integrals are shown on the left and right, respectively, of Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' These results use a fixed sequence of test cases and the 90% confidence intervals are taken by sorting outcomes from 3000 trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' For each trial, the random number generator is seeded with the trial index before constructing each quadrature sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='Table 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='Convolutional Neural Network with Ternary Logic Activations ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='Layer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='Channels In ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='Channels Out ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='Kernel ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='Stride ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='Parameters ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='Convolution ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='2 × 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='48 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='Ternary ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='32 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='Convolution ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='48 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='2 × 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='768 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='Ternary ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='48 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='128 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='Convolution ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='192 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='2 × 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='12288 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='Ternary ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='192 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='64 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='512 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='Convolution ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='64 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='192 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='2 × 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='49152 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='Ternary ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='192 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='64 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='512 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='Convolution ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='64 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='192 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='2 × 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='49152 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='Ternary ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='192 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='64 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='512 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='Linear ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='64 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='640 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='SoftMax ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='We see that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' initially,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' the cross-polytope sequence in the Hadamard basis provides a marginal improvement on the integrals as we average over several data samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' At MNIST CNN, Initial Dirac-Gauss Mean-Field Final Sparse Gaussian Mean-Field 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='16 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Monte Carlo 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Mean-Matched Quasi-Monte Carlo Loss 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='14 X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='. :Variance-Matched Quasi-Monte Carlo Stochastic 2-Block Mean-Field Cross-Polytope Vertex Sequence 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='1 for 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='3 Intervals 果果贝果果 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='08 [] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='25 Confidence 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='06 0 0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='04 %06 88881 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='02 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='15 0 4 8 12 16 20 24 28 32 4 8 12 16 20 24 28 32 Test Data Count Test Data Count28 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' DUERSCH the end of training, however, both the 2-blocked simplex sigma points and the cross- polytope sequence produce much tighter integral bounds than the other approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Note that the third test case on the left has an atypically large loss, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' a poor prediction, that demonstrates how the average loss integrals respond to typical, albeit intermittent, perturbations to the average integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Since all permutations are equally likely, we have EP1,P2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=',Pnb f(θ(1)) = 1 nqnb nq � q1=1 nq � q2=1 · · nq � qnb=1 nb � b=1 f (b)(θ(qb) b ) = � 1 nq nq � q1=1 f (1)(θ(q1) 1 ) � � 1 nq nq � q2=1 f (2)(θ(q2) 2 ) � · · � � 1 nq nq � qnb=1 f (nb)(θ (qnb) nb ) � � = �� dθ1 q(θ1)f (1)(θ1) � �� dθ2 q(θ2)f (2)(θ2) � · · �� dθnb q(θnb)f (nb)(θnb) � = � dθ q(θ)f(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Since this holds for all evaluation nodes, the average yields the same result□ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' pi1 xor pi2 = � nb xor j=1 [bitj(i1) ∧ bitj(q)] � xor � nb xor j=1 [bitj(i2) ∧ bitj(q)] � = nb xor j=1 [(bitj(i1) ∧ bitj(q)) xor (bitj(i2) ∧ bitj(q))] = nb xor j=1 [(bitj(i1) xor bitj(i2)) ∧ bitj(q)] = nb xor j=1 [xj ∧ bitj(q)]□ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' This result easily follows from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' As q increases, the relative parity, pi1 xor pi2, can only switch when at least one bit, bitj(q), at a position j ≥ b flips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNE_T4oBgHgl3EQf8RxN/content/2301.08374v1.pdf'} +page_content=' Thus, starting at q = z2b, 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a/oNFRT4oBgHgl3EQfbzf_/content/tmp_files/2301.13562v1.pdf.txt b/oNFRT4oBgHgl3EQfbzf_/content/tmp_files/2301.13562v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..46bc282c7c16c8ec3d835bf8fd3f48f0c25c3c50 --- /dev/null +++ b/oNFRT4oBgHgl3EQfbzf_/content/tmp_files/2301.13562v1.pdf.txt @@ -0,0 +1,1742 @@ +SciPost Physics +Submission +MCNET-23-01, IPPP/23/04 +Unweighting multijet event generation using +factorisation-aware neural networks +T. Janßen1, D. Maˆıtre2, S. Schumann1, F. Siegert3, H. Truong2 +1 Institut f¨ur Theoretische Physik, Georg-August-Universit¨at G¨ottingen, G¨ottingen, +Germany +2 Institute for Particle Physics Phenomenology, Department of Physics, Durham +University, United Kingdom +3 Institut f¨ur Kern- und Teilchenphysik, TU Dresden, Dresden, Germany +February 1, 2023 +Abstract +In this article we combine the recently proposed method for factorisation-aware matrix +element surrogates [1] with the unbiased unweighting algorithm of [2]. +We show that +employing a sophisticated neural network emulation of QCD multijet matrix elements +based on Catani–Seymour dipole factorisation can lead to a drastic acceleration of un- +weighted event generation. We train neural networks for a selection of partonic channels +contributing at the tree-level to Z + 4, 5 jets and t¯t + 3, 4 jets production at the LHC +which necessitates a generalisation of the dipole emulation model to include initial state +partons as well as massive final state quarks. +We also present first steps towards the +emulation of colour-sampled amplitudes. We incorporate these emulations as fast and +accurate surrogates in a two-stage rejection sampling algorithm within the SHERPA Monte +Carlo that yields unbiased unweighted events suitable for phenomenological analyses and +post-processing in experimental workflows, e.g. as input to a time-consuming detector +simulation. For the computational cost of unweighted events we achieve a reduction by +factors between 16 and 350 for the considered channels. +1 +arXiv:2301.13562v1 [hep-ph] 31 Jan 2023 + +SciPost Physics +Submission +Contents +1 +Introduction +2 +2 +Improved matrix element emulation using neural networks +4 +2.1 +Neural networks based on dipoles +4 +2.2 +Extension to initial-state and massive partons +8 +2.3 +Colour-sampled matrix elements +10 +3 +Event unweighting utilising matrix element surrogates +12 +3.1 +Two-stage unweighting method +12 +3.2 +Performance analysis +14 +4 +Implementation and application to LHC processes +16 +4.1 +Implementation in the SHERPA framework +16 +4.2 +Results for LHC multijet production processes +17 +5 +Conclusions +22 +A Auxiliary weight distributions +24 +References +25 +1 +Introduction +Physics simulations for current and future high-energy accelerator experiments pose a se- +vere computational challenge not only due to the complexity of the studied signatures and +the demand for higher theoretical accuracy, but also owing to the sheer number of simu- +lated events needed to match the enormous collider luminosities. This has sparked a wide +range of algorithmic developments to accelerate key elements of the simulation tool chain +and to improve their computational efficiency and thus reduce their resource requirements. +Machine learning based methods play a prominent role in these developments [3]. +Traditionally, the largest fraction of resources has been spent on the complex simulation +of the detector response to collision final states, while the generation of the collision events +constituted only O(10% − 20%) of the budget. With many recent activities reducing the +computational footprint of detector simulations, the event generation speed has become a +more and more important area to facilitate the full exploitation of future collider data. +This situation is amplified with the upcoming increased luminosity at the LHC and +its focus turning to more complex processes. With the advent of matching and merging +techniques theoretical predictions of increased precision have become accessible for these +high multiplicity final states. But as the computational cost increases strongly with the +multiplicity the resources needed for the theoretical description of processes of interest +has surged, see for example [4]. Besides conventional approaches for improving the perfor- +mance of Monte Carlo methods in event generators, more recently also machine learning +methods are explored [5]. +2 + +SciPost Physics +Submission +A particular challenge is related to the generation of the hard scattering component +that forms the core of the event evolution, thereby representing the parton-level truth +signal and/or background hypotheses in physics analyses [6, 7]. +In Monte Carlo event +generators the hard process is assumed to be factorised from parton showers in the initial +and final state as well as non-perturbative phenomena such as hadronisation and the +underlying event. Algorithmically the generation of hard scattering events corresponds to +the evaluation, i.e. the stochastic sampling, of the phase space integral over the squared +transition matrix element of the considered process at a given order in perturbation theory. +There has been renewed interest in improving phase space integration and sampling +techniques, mostly based on neural networks [8–14], using a variety of methods, but also +Nested Sampling [15], and mixed-kernel Markov chain algorithms [16] have been investi- +gated. Broadly speaking these approaches share the goal of adjusting a sampling distribu- +tion as closely as possible to the true target distribution, i.e. the actual transition matrix +element. In the case of a traditional Monte Carlo integration this will typically result in +events with weights of ideally small spread. However, for a resource efficient generation +process for physics analyses, in particular due to very time-consuming components such +as a detector simulation, events with (largely) unit weight are desirable. This is typically +solved via von Neumann rejection sampling, i.e. an accept–reject procedure for weighted +event samples. However, even with advanced adaptive sampling techniques in particu- +lar for multi-particle final states the efficiency of the unweighting procedure can be quite +small, resulting in the repeated trial evaluation of the scattering matrix element that ul- +timately get rejected. For LHC key processes such as the multijet-associated production +of gauge bosons this might result in O(105) evaluations of the computationally expensive +matrix element for a single unit-weight event [17]. +This suggests a complementary opportunity for saving resources, namely the usage of a +fast and accurate surrogate for the trial weights. A corresponding two-stage unweighting +procedure that fully corrects for the potential mismatch between the surrogate weight +and the actual value of the full matrix element has recently been presented in Ref. [2]. +To demonstrate the algorithm, a rather simple neural network designed to replicate the +weight of partonic events, represented by their external momenta was used. For tree-level +contributions to Z/W + 4 jets and t¯t + 3 jets production at the LHC significant gains +have been observed, however, for less complex partonic channels ordinary unweighting +could not be outperformed. Over the last few years more sophisticated matrix element +surrogates based on neural networks have been developed [1,18,19], addressing tree-level +and one-loop amplitudes. In particular, Ref. [1] presented a method to emulate scattering +matrix elements employing the factorisation properties of QCD amplitudes in the soft- +and collinear limits. +In this work we explore the potential of a combination of the approaches in Refs. [2] +and [1] in unweighted event generation for multijet production processes at the LHC. +To this end we generalise the method presented in Ref. [1] to the case of colour-charged +initial states and massive final-state partons. +We also explore, for the first time, the +emulation of colour-sampled QCD amplitudes in the colour-flow decomposition. With an +implementation in the SHERPA event generator framework [20, 21] we benchmark tree- +level contributions to Z + 4, 5 jets and t¯t + 3, 4 jets production. The paper is structured +as follows: In Sec. 2 we review the dipole emulation model of Ref. [1] and present our +new developments to address hadronic collisions and massive final-state partons. In Sec. 3 +we review the unweighting procedure worked out in Ref. [2]. In Sec. 4 we discuss the +implementation of both algorithms in the SHERPA framework, and present our results +obtained for selected partonic channels contributing to pp → Z+4, 5 jets and pp → t¯t+3, 4 +jets. +3 + +SciPost Physics +Submission +2 +Improved matrix element emulation using neural networks +To demonstrate the accelerated surrogate unweighting algorithm in Ref. [2] the authors +used only a simple neural network as model for the event weights. It was clear from the +study that a bottleneck in this procedure was the accuracy of the event weight approxima- +tions coming from the surrogate model, leading to low efficiencies in the second unweighting +step. In this work we consider replacing that simple model with the factorisation-aware +neural network model introduced in Ref. [1], which has been shown to exhibit more ac- +curate predictions on a per-point basis compared to existing methods. Below we review +the construction of this model and detail the necessary extensions to facilitate multijet +production processes at the LHC. +2.1 +Neural networks based on dipoles +In this section we briefly review the framework from Ref. [1], where an ansatz for matrix +elements based on the factorisation properties of QCD matrix elements in their soft and +collinear limits was used. This factorisation can be depicted as +|Mn+1|2 → |Mn|2 ⊗ Vijk , +(1) +where the matrix element in the (n + 1)-body phase space reduces to a matrix element +in the momentum mapped n-body phase space multiplied by a singular factor, Vijk. For +single infrared limits, Vijk contains all the singularity structure of the matrix element. +In the dipole formalism these singular factors are encapsulated in the Catani–Seymour +dipoles Dijk [22]. This factorisation property of matrix elements leads to the form of our +ansatz, which is inspired by the dipole factorisation formula. It is given by +|Mn+1|2 ≃ +� +{ijk} +CijkDijk , +(2) +where i, j, and k denote the three partons involved in the dipole function. Instead of +fitting the matrix element directly where there are divergences in the infrared regions of +phase space, we let the neural network fit the coefficients Cijk as a function of phase +space. These coefficients are more well-behaved than the full matrix element in the soft +and collinear limits as the singular behaviour is described by the dipole functions. By +combining these well-behaved coefficients with the analytically known dipoles, we produce +an approximation for the matrix element. +This enables the fitting of matrix elements +across the entire sampled phase space with a single neural network. Whilst the model +predicts the Cijk coefficients, it should be noted that these are not entirely meaningful +by themselves. Only once they are combined with the corresponding dipoles do we get +the approximation of the matrix element. It is this approximation of the matrix element +which should be seen as the model prediction and which appears in the loss function. +In singly unresolved limits, only relevant dipoles are large and so constrain the cor- +responding Cijk in the fit. +Outside of these limits all dipoles are of similar order of +magnitude and the ansatz in Eq. (2) is more under-constrained. In these regions of phase +space, the excellent fitting capabilities of neural networks are leveraged to interpolate +the non-singular matrix elements. The accuracy achieved in this approach is due to the +fact that the coefficients being fit by the network, for single soft or collinear kinematics, +are free of divergences. This facet of the emulation model makes it particularly apt for +the case of multijet production processes where matrix elements are plagued with many +well-understood divergent structures. +4 + +SciPost Physics +Submission +Table 1: Hyperparameters of the neural network and their values. +Parameter +Value +Hidden layers +4 +Nodes in hidden layers +128 +Activation function +swish [25,26] +Weight initialiser +Glorot uniform [27] +Loss function +MSE +Batch size +512 +Optimiser +ADAM [28] +Initial learning rate +10−3 +Callbacks +EarlyStopping, ReduceLROnPlateau +In this work we employ networks of similar size and complexity to those in Ref. [2], +namely, we use KERAS [23] and TENSORFLOW [24] to build a neural network (NN) model +with four hidden layers, each consisting of 128 nodes. These hidden layers use the swish +activation function [25,26] and their weights are initialised according to the Glorot Uniform +distribution [27]. +The swish activation function has a similar shape to the more well-known ReLU ac- +tivation function, but it is a smooth continuous function allowing small negative values, +instead of thresholding them to 0 like ReLU does. We find that in practice swish outper- +forms ReLU in all our trained models. We also find that instead of using a linear activation +function in the output layer, using a swish activation function is more performant. +The NN is fitted to the data generated from SHERPA (see Section 4.2 for more informa- +tion on the generation of data) by minimising the loss function encoding the discrepancy +between the prediction made with the neural network to the true matrix element provided +by SHERPA. We use the mean squared error (MSE) as the loss function, with the training +optimised using ADAM [28] with an initial learning rate of 10−3. +The learning rate is +reduced when the validation loss shows no improvement for 30 epochs of training by using +the ReduceLROnPlateau callback, and EarlyStopping is used to terminate training when +there is no improvement in the validation loss after 60 epochs. A summary of the neural +network hyperparameters is given in Tab. 1 for reference. +As inputs to our generalised network model we feed: the 4-momenta of all initial- and +final-state particles, the phase-space mapping variables corresponding to the dipoles in +the ansatz, denoted as yijk1, and the kinematic invariants sij for all pairs of particles in +the process considered. Note that the yijk and the sij input variables are not independent +from the external momenta. To aid the neural network training process, we pre-process +the yijk, such that the shape and widths of their distributions are similar for the different +dipole configurations (we explicitly state the new dipoles included in the model in Sec. 2.2). +1The initial-state phase-space mapping variables are referred to as xijk in [22, 29] but we will refer to +all phase-space mappings as yijk for brevity. +5 + +SciPost Physics +Submission +This amounts to +yijk → +� +� +� +� +� +� +� +� +� +log(yijk) +if massive FI dipole , +log(1 − yijk) +if massless FI, IF, or II dipole , +log(yijk) +otherwise (massless FF dipole) . +(3) +We note that since we do not have massive partons in the initial-state we only require +the corresponding transformation for massive FI dipoles. The kinematic invariants are +also transformed with the logarithm sij → log(sij) as they can span many orders of +magnitude. It should be noted that the particles involved in the dipole functions and +mapping variables denoted by the subscript ijk are the colour-charged particles in the +initial- and final-state. Non colour-charged particles, for example, electrons and positrons, +do not appear in the dipoles, but nevertheless their momenta and kinematic invariants are +fed into the network as inputs such that it learns of their dependence. All of these inputs +are standardised to zero mean and unit variance, with the 4-momenta being standardised +along each component. +To use them more effectively in the loss function, we pre-process the matrix elements +as +|Mn+1|2 → arsinh +� +|Mn+1|2 +Spred +� +, +(4) +and standardise to zero mean and unit variance. Spred is the prediction scale taken to be +the minimum matrix element value found in our training set. This transformation aids +the neural network in training by reducing the span of the target distribution. +The output nodes of our neural network correspond not to the matrix elements directly, +but instead to the dipole coefficients. The raw outputs, denoted by cijk, are transformed +to the coefficients appearing in Eq. (2), Cijk, via the transformation +Cijk = Scoef × sinh (cijk) +(5) +where Scoef is the coefficient scale, taken to be Spred/Sdipole. Sdipole is the representative +value of a dipole, which we take to be the median of all dipoles in our training set. +The neural network prediction is made by using Eq. (2) to combine the predicted Cijk +coefficients with the corresponding dipoles Dijk. In order to compare with the scaled target +matrix elements, we have to transform the neural network predicted matrix element with +Eq. (4) with the same Spred. We can then compare the matrix element as predicted by +the neural network, with the truth value, as given by SHERPA, in the MSE loss function. +A diagram illustrating the NN emulator architecture is given in Fig. 1. +In Ref. [1], the neural network predictions were given by the average over an ensemble +of 20 independent replicas trained on different shuffled subsets of the training set and +with different initial random seeds for model weight initialisation. Here we take a similar +approach by training a set of 10 replica models, however, for predictions we select the +model with the lowest validation loss. We stress that this is not a special choice as all +individual replica models converge to a similar point. As an illustrative example, we plot +in Fig. 2 the loss curves for the partonic channel gg → e−e+ggd ¯d, which is a leading- +order contribution to Z + 4 jets production at the LHC. We observe convergence across +all replicas with training terminating at similar values of the MSE. +The reasoning behind ensembling a prediction is to reduce the effects of stochasticity of +the training process, to reduce random model weight initialisation, and to reduce variance +in the prediction. In this work we strive for a balance of accuracy and speed, meaning it +is advantageous to use a single model to make predictions. The reasoning is as follows. +6 + +SciPost Physics +Submission +Figure 1: A simplified sketch of our neural network emulator showing inputs, hidden +layers, and outputs Cijk. +0 +100 +200 +300 +400 +500 +600 +700 +800 +Epochs +10−3 +10−2 +10−1 +100 +MSE loss +gg → e−e+ggd ¯d +Training loss +Validation loss +Figure 2: Training and validation loss for 10 replica models, for the gg → e−e+ggd ¯d +channel, shown as solid lines. The epochs at which training is terminated are illustrated +as the solid circles. We depict the training and validation loss of the selected model in +dashed horizontal lines. +In an ensemble of models where replicas are trained on different subsets of the same +training data, there is overlapping information learnt by the individual models. This leads +to diminishing returns in predictive accuracy, meaning that whilst evaluation time grows +linearly with the number of replicas, accuracy does not. We have therefore observed a +single model to be the most performant configuration. It is important to stress that this +does not mean that one model cannot be sufficiently accurate, as we will demonstrate in +Sec. 4.2. +This decision to use only a single NN for predictions also guided our choice of num- +ber of nodes in the hidden layers. With 128 nodes we reach a balance of having enough +parameters to model the matrix elements whilst reducing the effects of overfitting. De- +creasing the number of nodes in the hidden layers to create a more compact NN has little +effect on the evaluation time for a single network when we use the ONNX Runtime [30] +for evaluation, there would only be loss in accuracy which represents a decrease in overall +unweighting efficiency. +7 + +SciPost Physics +Submission +Dij,k +i +j +k +�ij +pi +pj +pk +(a) FF dipole +Da +ij +i +j +a +�ij +pi +pj +pa +(b) FI dipole +Dai +k +a +i +k +�ai +pa +pi +pk +(c) IF dipole +Dai,b +a +i +b +�ai +pa +pi +pb +(d) II dipole +Figure 3: Schematic diagrams of the four classes of Catani–Seymour dipoles. The +dipoles are named according to whether the emitter and spectator are in the initial +(upper indices) or final state (lower indices). Each dipole consists of a composite particle +(denoted by tilde) that decays into two partons, and a spectator that recoils to conserve +momentum. The grey blob represents the hard scattering process, with incoming and +outgoing lines representing initial- and final-state partons, respectively. The black circle +represents the splitting function within the dipole function which contains the divergent +behaviour. +2.2 +Extension to initial-state and massive partons +In Ref. [1], the authors considered jet production processes initiated via electron–positron +annihilation where only final-state QCD radiation occurs, meaning the set of dipoles built +into the emulation model were of the FF (final-state radiator, final-state spectator) kind. +Furthermore, the model was restricted to the production of massless QCD partons. +In this work we consider the extension to hadronic initial states, which is relatively +straightforward: we need to account for the additional radiation that comes from the +colour-charged initial-state particles. To this end, we add the initial-state dipoles to the +ansatz, namely, we add the IF (initial-state radiator, final-state spectator), FI (final- +state radiator, initial-state spectator) and II (initial-state radiator, initial-state spectator) +splitting configurations. This means that the emitter i, and spectator k, in the ansatz can +now be in the initial-state. Illustrations for the complete set of dipoles now included in +the model are shown in Fig. 3. +To showcase the extension to massless initial state dipoles we consider the emulation of +tree-level matrix elements for the partonic channels gg → e−e+ggd ¯d and gg → e−e+gggd ¯d, +which are leading order contributions to Z + 4 jets and Z + 5 jets production at the LHC, +respectively. As validation of the emulation accuracy of the NN model for this extension +to initial states, we examine the ability of the model to predict matrix elements across +the sampled phase space, but in particular for the case of soft and collinear kinematics, +where QCD matrix elements are strongly enhanced. We plot in Fig. 4 a 2d histogram +of the truth-to-prediction ratio, |M|2 +true/|M|2 +pred, against the true value, |M|2 +true, for 1M +gg → e−e+ggd ¯d test events with standard cuts as described in Sec. 4.2. Along the sides, +we plot the marginal distributions of the matrix element (top) and the ratio (right). The +8 + +SciPost Physics +Submission +10−3 +10−1 +101 +Fraction of points [%] +10−47 +10−39 +10−31 +10−23 +10−15 +10−7 +|M|2 +true +10−8 +10−6 +10−4 +10−2 +100 +102 +104 +106 +108 +|M|2 +true / |M|2 +pred +gg → e−e+ggd ¯d (colour-summed) +N = 1M events +10−2 +101 +Fraction of points [%] +100 +101 +102 +103 +104 +105 +Number of points in bin +Figure 4: 2d histogram showing the distribution of truth-to-prediction ratios of the +matrix element against the value of the true matrix element for the Z +4j process gg → +e−e+ggd ¯d. Along the axes, we plot the marginal distributions of the matrix element +(top), and the truth-to-prediction ratio (right). High population bins are illustrated as +yellow, with low population bins, down to single points, are depicted in purple. +results illustrate that the ratio depicting model accuracy is centred around the ideal value +of 1, with a steep drop off. This applies to the bulk of the events, as depicted by yellow +coloured bins, tightly constrained to a narrow band. The purple coloured bins represent +low population bins, or single points, which shows that the tails of the ratio distribution +are primarily seen for smaller matrix element weights. Furthermore, the model accuracy +remains high for the largest values of the matrix element, signalling that the infrared +behaviour is well controlled. This is a key property of the factorisation-aware model. The +emulation performance for the Z + 5 jets process is presented in Sec. 4.2. +An additional extension we study in this article is the inclusion of massive dipoles to +our ansatz. This allows us to examine QCD processes with massive partons which is of +particular importance for top-quark pair production in association with jets. We include +the massive FF, FI, and IF dipoles from Ref. [29] into the emulation model. The massive +dipoles are generalisations of the massless dipoles, meaning in principle it would be possible +to remove the massless dipoles from the ansatz. However, in practice, we only include the +minimal set of necessary dipoles for a given partonic channel and so the inclusion of +the massless dipoles reduces overall computational cost due to their relatively simpler +expressions. With the massive dipoles implemented, our model contains the complete set +of dipoles and is in principle able to take advantage of the factorisation-aware model for +9 + +SciPost Physics +Submission +10−3 +10−1 +101 +Fraction of points [%] +10−43 +10−36 +10−29 +10−22 +10−15 +10−8 +|M|2 +true +10−8 +10−6 +10−4 +10−2 +100 +102 +104 +106 +108 +|M|2 +true / |M|2 +pred +gg → t¯tggg (colour-summed) +N = 1M events +10−2 +101 +Fraction of points [%] +100 +101 +102 +103 +104 +105 +Number of points in bin +Figure 5: 2d histogram showing the distribution of truth-to-prediction ratios of the +matrix element against the value of the true matrix element for the t¯t + 3j process +gg → t¯tggg. Along the axes, we plot the marginal distributions of the matrix element +(top), and the truth-to-prediction ratio (right). +arbitrary processes involving QCD-enhanced behaviour at tree-level. +In order to showcase the extension to massive dipoles, we consider emulating tree-level +matrix elements of three partonic channels: gg → t¯tggg, and u¯u → t¯tgd ¯d, contributing to +leading order t¯t+3 jets production, and ug → t¯tgggu which is a leading order contribution +to t¯t+4 jets production. To validate the inclusion of these massive dipoles into the model, +we show in Fig. 5 the deviation similar to Fig. 4 but for 1M tree-level events of gg → t¯tggg +in proton–proton collisions at √s = 13 TeV, with cuts described in Sec. 4.2. We again +observe the narrow yellow band, indicating that the bulk of the test events are accurately +predicted, with the outliers corresponding to smaller matrix element values. The infrared +behaviour is well captured by the model as can be seen by the narrow head for the largest +matrix element values. For emulation performance of channels not described here, we refer +the reader to Sec. 4.2 and App. A. +2.3 +Colour-sampled matrix elements +The discussion so far has been focused on the emulation of colour-summed matrix elements +as it was the case for Ref. [1]. In this work we take the first steps towards emulating colour- +sampled matrix elements, such as those obtained from the COMIX generator [31,32]. +Based on the colour-flow decomposition of QCD amplitudes [33,34], for each event, the +generator samples a momentum configuration and a valid colour assignment, i.e. colour +10 + +SciPost Physics +Submission +10−2 +100 +Fraction of points [%] +10−44 +10−37 +10−30 +10−23 +10−16 +10−9 +|M|2 +true +10−8 +10−6 +10−4 +10−2 +100 +102 +104 +106 +108 +|M|2 +true / |M|2 +pred +gg → t¯tggg (colour-sampled) +N = 1M events +10−2 +101 +Fraction of points [%] +100 +101 +102 +103 +104 +Number of points in bin +Figure 6: Truth-to-prediction ratio for colour-sampled gg → t¯tggg matrix elements +against the colour-ordered partial amplitudes, |M|2. Marginal distributions are plotted +for the matrix elements (top) and ratios (right). +indices. The colour assignment thereby is represented by a vector of integers, C, where +entries in the vector, ci ∈ {1, 2, 3}, denote the colour assigned to a colour-charged parton +in the process. Gluons have two colour indices corresponding to colour and anti-colour, +whereas quarks/anti-quarks carry only one index. +We add this vector of colour assignments as an additional input to the NN to include the +colour-sampled information from the generator. We one-hot encode the colour assignments +such that colours are represented by 3-element vectors, e.g. R = [0, 0, 1], G = [0, 1, 0], and +B = [1, 0, 0], as the integer representation of colour assignments is not useful to the NN. +Given the actual colour of a parton is ambiguous, the matrix element should be invari- +ant to any cyclic permutation of the specific colour assigned to a given quark or gluon. To +give an example, the three permutations C1 = [R, B, B, G, R], C2 = [G, R, R, B, G], and +C3 = [B, G, G, R, B] of a five colour assignment would lead to the same matrix element +weight. To aid the NN in learning this behaviour, we take the three permutations and du- +plicate the other model inputs such that the training data is enlarged by a factor of three. +This did not cause us to run into any computational bottlenecks in terms of memory or +time taken to train the models. Note that this duplication of data is not required when +making predictions. +The rest of the inputs to the NN model remain identical. We study to what extent +a naive approach of using the same dipole functions, which are most suitable for colour- +summed matrix elements, works for the case of colour-sampled matrix elements. In the +11 + +SciPost Physics +Submission +future, a more promising approach might be the application of coloured dipole terms +directly. Their form has already been derived [35] and implemented for the dipole sub- +traction in the COMIX event generator [31] but is not implemented in our NN-based model +yet. +To illustrate the emulation accuracy of colour-sampled matrix elements, here denoted +|M|2, we plot the truth-to-prediction ratio in Fig. 6 for the gg → t¯tggg channel. While +we again observe the property of well-behaved predictions for the larger matrix elements, +evidently, the ratio distribution is much wider than in the colour-summed case. +This +decrease in accuracy directly translates to a lower expected gain factor when using this +emulator as a surrogate model for event unweighting. This is discussed further in Sec. 4.2 +where we elaborate on specific reasons for this decrease in accuracy and present possible +future endeavours. +3 +Event unweighting utilising matrix element surrogates +The unweighting of hard-scattering parton-level event samples constitutes an important +step in the simulation of scattering events. The obtained unit-weight events then get passed +on to subsequent evolution stages, including QCD parton showers, hadronisation, and +possibly a detector simulation. However, the unweighting, based on rejection sampling, +can pose a severe computational challenge, in particular when the evaluation time of the +matrix element is long and the efficiency of the unweighting is rather low. To address +this challenge Ref. [2] proposed a novel two-stage rejection sampling algorithm based +on fast surrogates that we briefly review in this section. We furthermore generalise the +performance measures to the case where the surrogate replaces the matrix element only, +rather than its combination with the phase space weight as was the case in Ref. [2]. +3.1 +Two-stage unweighting method +The Monte Carlo method provides a numerical procedure to estimate integrals, e.g. par- +tonic cross sections in high energy physics. When the integrand is non-trivial we use im- +portance sampling to reduce the variance of the integral estimate. For a positive-definite +target function f : Ω ⊂ Rd → [0, ∞) defined over the unit hypercube Ω = [0, 1]d and a +probability density function g the Monte Carlo estimate of the integral +I = +� +Ω +f(u′) du′ +(6) +is given by +I ≈ 1 +N +N +� +i=1 +f(ui) +g(ui) = ⟨w⟩g +(7) +with the pointwise event weight wi = f(ui)/g(ui). A suitable g can reduce the variance +of the integral estimate and thereby increase the efficiency of the numerical integration. +Finding such a function g is a difficult task, though, as one needs a way to efficiently draw +samples from it. For multimodal target functions it is attractive to use a multi-channel +approach, where g is defined by a mixture distribution. The weights of the channels can +then be adapted automatically [36]. VEGAS [37] is an algorithm to automatically construct +a sampling distribution g by optimising the bin widths of a piecewise-constant function. +It can also be used to remap a given g or even the individual channels of a multi-channel +distribution [38]. +12 + +SciPost Physics +Submission +Besides the total integral we are typically interested in differential distributions of +the points ui, i.e. histograms of physical observables. Monte Carlo sampling produces +weighted events so every entry in a histogram comes with a weight. Variance reduction +methods like importance sampling also reduce the spread of weights but only a perfect +sampler results in strictly uniform weights. A large weight spread is problematic when the +samples are to be post-processed by detector simulations, as these are very expensive in +terms of computation time per event. It is inefficient to apply them to events that yield +a minuscule contribution to the total cross section. The alternative is to first impose a +rejection sampling step to extract unit-weight samples. This converts a sample of Ntrials +weighted events into a set of N ≤ Ntrials unweighted events by randomly accepting or +rejecting every weighted event with the acceptance probability w/wmax where wmax is +the maximal event weight. Even though the information of the rejected events is lost the +overall efficiency can be significantly increased when detector simulation is more expensive +than event generation. +A convenient measure for the performance of a Monte Carlo event generator is the +unweighting efficiency ϵ of the rejection sampling step, defined as +ϵ := +N +Ntrials . +(8) +For a large number of trial events it can be estimated by +ϵ ≈ ⟨w⟩ +wmax +, +(9) +where ⟨w⟩ is the mean of the Ntrials weights in the event sample. The average number of +target function evaluations needed to get one accepted event is then given by 1/ϵ. Similar +to how the uncertainty on the integral estimate can be diminished by variance reduction +methods, the unweighting efficiency can be increased by optimising the sampling density +g for smaller wmax. +There is another way of reducing the computational footprint especially if the target +function takes a long time to evaluate and has a rather low unweighting efficiency. This +is typically the case for high multiplicity scattering processes. The enormous growth in +the number of contributing Feynman diagrams makes high multiplicity matrix elements +increasingly expensive. At the same time, the high dimensionality of phase space renders +it difficult to find a sampling density g that is well adapted to the target everywhere +in the integration volume. Consequently, the unweighting efficiency typically decreases +with increasing multiplicity, see for example [17]. In this situation one can reduce the +overall event generation time through replacing the expensive matrix element by a fast +and accurate surrogate. The inaccuracy inevitably introduced in this procedure can be +fully corrected for in a second unweighting step, resulting in an unbiased method [2]. An +outline of the algorithm is given in Alg. 1. In addition, a more extensive explanation +follows below. +We begin by generating a weighted trial event in the conventional way. +In a first +unweighting step we then compare the surrogate weight s to the weight maximum wmax +and accept the event with probability s/wmax. For an event that gets rejected at this point +we only had to evaluate the cheap surrogate. If the event gets accepted, however, we need +to evaluate the true weight w and attach a correction weight x = w/s to the event. In a +second unweighting step, the event has an acceptance probability of x/xmax. Like wmax, +xmax has to be predetermined. When the surrogate yields an accurate approximation of +the true weight, a large proportion of events gets accepted in the second unweighting step. +We note that the algorithm can easily be extended to the case of not strictly positive event +weights as shown in [2]. +13 + +SciPost Physics +Submission +Algorithm 1: Two-stage rejection-sampling unweighting algorithm using an +event-wise weight estimate. +while true do +generate phase-space point u; +calculate approximate event weight s; +generate uniform random number R1 ∈ [0, 1); +# first unweighting step +if s > R1 · wmax then +calculate exact event weight w; +determine ratio x = w/s; +generate uniform random number R2 ∈ [0, 1); +# second unweighting step +if x > R2 · xmax then +return u and �w = max(1, s/wmax) · max(1, x/xmax) +end +end +end +Alg. 1 contains a crucial detail regarding the weight maxima, namely that even after +unweighting events can end up with weights ˜w > 1 if s is larger than wmax or if x is +larger than xmax. +If the true maxima were used, this could never happen. +However, +given finite-sized samples an exact determination of wmax is realistically not possible. It +is often not even desirable since a small number of points with large weights can induce +a prohibitively small unweighting efficiency without contributing significantly to the total +integral. It can therefore be useful to work with a deliberately reduced maximum, provided +the rare mismatches are corrected for by event weights. +The resulting events will be +partially unweighted since there can be some events that overshoot the maximum. These +will receive an overweight ˜w = w/wmax > 1. Hereinafter, we adopt the approach used in +SHERPA for finding the reduced maximum. The aim is that the remaining overweights do +not contribute more than a fixed proportion to the integral. We set this share to 0.1 %. +This can be achieved by taking the sorted weights of a sample of weighted points and +finding the weight that cuts off the desired quantile. In SHERPA this is done automatically +during the integration phase. +We point out that using a reduced maximum is a fully +unbiased technique commonly used in event generators. +It is especially helpful when +weight surrogates are used since the limited approximation quality of the surrogate can +lead to particularly large outliers. +3.2 +Performance analysis +To fairly evaluate the performance gain of the two-stage unweighting algorithm shown in +Alg. 1 we take the average time it takes to generate a single (partially) unweighted event +and compare it to the time it would take to generate the statistical equivalent using the +standard unweighting procedure. We call the ratio between the two the effective gain +factor feff: +feff := Tstandard +Tsurrogate +. +(10) +14 + +SciPost Physics +Submission +In order to separate the actual unweighting from program initialisation and other aspects +of event generation, we break the calculation down to the relevant ingredients: +feff = +Ntrials +full +· +� +⟨tME⟩ + ⟨tPS⟩ +� +Ntrials +1st,surr · +� +⟨tsurr⟩ + ⟨tPS⟩ +� ++ Ntrials +2nd,surr · ⟨tME⟩ +(11) += +1 +⟨tsurr⟩+⟨tPS⟩ +⟨tME⟩+⟨tPS⟩ · +ϵfull +ϵ1st,surrϵ2nd,surr + +⟨tME⟩ +⟨tME⟩+⟨tPS⟩ · +ϵfull +ϵ2nd,surr +. +(12) +The average evaluation times of the full matrix element weight, the phase space weight +and the matrix element surrogate, respectively, are denoted as ⟨tME⟩, ⟨tPS⟩ and ⟨tsurr⟩. By +Ntrials +step we denote the number of trials in the respective unweighting step. The unweighting +efficiencies are defined as +ϵfull := +N +Ntrials +full +, +ϵ1st,surr := +Ntrials +2nd,surr +Ntrials +1st,surr +and +ϵ2nd,surr := +N +Ntrials +2nd,surr +. +(13) +It should be noted that events rejected due to phase space constraints do not affect the un- +weighting efficiencies since the selection cuts can be applied solely based on the kinematics +without having to evaluate the matrix element. +From Eq. (12) it is clear that an important requirement for significant gains are short +evaluation times for the surrogate in comparison to the full matrix element, i.e. ⟨tsurr⟩ ≪ +⟨tME⟩. +Furthermore, even with a fast and accurate surrogate gains are only possible +when the original unweighting efficiency ϵfull is small enough. Therefore, the surrogate +unweighting method is of limited use when the sampling density is very well adapted to +the target. For suitable processes it will thus be important to find a good balance between +fast evaluation and high accuracy of the surrogate. +The efficiency ϵfull can be estimated by +ϵfull ≈ ⟨w⟩ +wmax +(14) +from the weights w generated during an initial integration run, i.e. after adapting the phase +space generator. For wmax we use the reduced value as described in Sec. 3.1. Analogously, +we estimate ϵ1st,surr by +ϵ1st,surr ≈ +⟨s⟩ +wmax +(15) +using the same weight maximum and the surrogate weights s determined for the events in +the test dataset. Since the deviations of the surrogate should average out, one can expect +the values of ϵfull and ϵ1st,surr to be close. The second unweighting efficiency ϵ2nd,surr can +be estimated by +ϵ2nd,surr ≈ ⟨x⟩ +xmax +(16) +using the values x = w/s determined for the events in the test dataset. The reduced +maximum xmax can be calculated analogously to wmax with the restriction that we have +to weight the values of x by their corresponding values of s to take into account the +acceptance probability in the first unweighting step. +To determine the times ⟨tME⟩, ⟨tPS⟩ and ⟨tsurr⟩ we repeat the calculation of the +full/surrogate matrix element and phase space weights for a number of events from the test +dataset. Depending on the complexity of the process we need between 10 and 10 000 events +for a reliable time estimate. Note, the value of ⟨tsurr⟩ includes the time for preprocessing +the inputs and post-processing the outputs of the surrogate model. +15 + +SciPost Physics +Submission +4 +Implementation and application to LHC processes +In this section we present the application of the dipole model emulation of QCD matrix +elements in the unweighting of event samples for high-multiplicity scattering processes at +the LHC, i.e. Z + 4, 5 jets and t¯t + 3, 4 jets production at the LHC. Results presented in +Ref. [2] were based on a simplified neural network surrogate, however, also included an +approximation for the phase space weight. We will here contrast the results obtained before +to the sophisticated dipole model surrogate and also comment on the challenges when using +colour-sampled QCD amplitudes. We furthermore briefly describe an implementation in +the workflow of the SHERPA framework [20,21]. Note that we here only need to consider the +generation of the hard process partons, as this is factorised from the generation of initial- +and final-state parton showers as well as non-perturbative phases such as hadronisation +and the underlying event [6]. +Furthermore we note that systematic variations of the +hard event related to alternative PDF sets, or modifications in the scale choices can be +evaluated on-the-fly for unweighted events, represented by variational weights, see for +example [39,40]. +4.1 +Implementation in the SHERPA framework +The two-stage unweighting algorithm described in Sec. 3.1 has been implemented in +SHERPA [2]. The framework provides two built-in tree-level matrix element generators: +AMEGIC [41] and COMIX [31]. We use AMEGIC to evaluate colour-summed matrix ele- +ments and COMIX for colour-sampled ones. +To adapt the integrator to the integrand +SHERPA runs an initial optimisation phase. This is followed by an integration phase in +which the optimised integrator is used to calculate the total cross section of the process. +From the event weights produced in this phase the value of wmax is determined, based on +the 0.1 % maximum reduction method introduced in Sec. 3.1. We take 2M events from the +integration phase as a training dataset by saving the momenta, matrix element and phase +space weights, and, when using colour sampling, colour assignments. From the training +dataset we use 800k events for training the model, 200k for validation during the training +and 1M for testing the performance afterwards. We train the dipole model described in +Sec. 2 using KERAS [23] with the TENSORFLOW [24] backend and save it in the ONNX +format [42]. The 1M events from the test dataset are used to determine the value of xmax. +After the training of the surrogate model has been completed successfully, the de- +termination of the surrogate matrix element value during event generation with SHERPA +proceeds as follows. At the point where normally the matrix element would be calculated +with AMEGIC or COMIX, we use the momenta of the current trial event to determine the +additional inputs yijk and sij. Along with the momenta these are then fed into the model +which we evaluate on a single CPU core using the C++ API of the ONNX Runtime pack- +age [30]. We find that ONNX Runtime evaluates the model several times faster than the +header-only library frugally-deep [43], which was used in [2]. It is important to note that +this introduces an additional dependency on a software library. We would, however like +to emphasise that our method does not depend on the code with which the surrogate is +evaluated. This affects only the evaluation time. It would even be possible to create an +interface through which any suitable tool could be used for this purpose. The model evalu- +ation yields the dipole coefficients Cijk which are then combined with the Catani–Seymour +dipoles Dijk according to Eq. (2). To determine the relevant dipoles we use a custom imple- +mentation, although there already exists an implementation of Catani–Seymour dipoles +in SHERPA (used in the automated construction of infrared subtraction terms for NLO +QCD and EW calculations [44,45]) which could in principle also be employed for the case +considered here. +16 + +SciPost Physics +Submission +4.2 +Results for LHC multijet production processes +To study the performance of the method we consider various partonic multijet processes. +We thereby follow the validation and benchmark strategies outlined in Ref. [2], considering +Z+jets and t¯t+jets production in proton–proton collisions at √s = 13 TeV. In particular +we present results for Z +{4, 5} jets and t¯t+{3, 4} jets final states, thereby extending our +previous study by one multiplicity. Jets get reconstructed with the anti-kt algorithm [46] +with R = 0.4. As parton density functions we use the NNPDF-3.0 NNLO set [47]. +Z+jets +We examine the partonic channels gg → e−e+ggd ¯d and gg → e−e+gggd ¯d at the tree- +level that represent leading-order contributions to Z + 4 jets and Z + 5 jets production +at the LHC. Correspondingly, using the four-momenta as inputs for the surrogate model +we have parameter spaces with 32 and 36 dimensions. These get supplemented by the +corresponding dipole mapping variables and kinematic invariants, see Sec. 2.2. Cuts are +implemented to constrain the fiducial phase space. A dilepton invariant mass me−e+ > +66 GeV and four, respectively, five jets with pT,j > 20 GeV are enforced. Identical cuts are +used for the training and the prediction. +As a first assessment of the quality of the surrogate we show in Fig. 7a the distribution +of the ratio between the true event weight w and the surrogate event weight s for 1M test +events for the exemplar channel gg → e−e+gggd ¯d. The corresponding plot for the process +with the lower multiplicity is shown in App. A. We compare the results of the dipole +model with the naive model from Ref. [2]. We point out that the naive model learns the +entire event weight, while the dipole model learns only the matrix element weight. For +the representation in Fig. 7a, the approximated matrix element weight of the dipole model +was therefore multiplied by the true phase space weight. While a perfect model would +reproduce the true weight exactly, such that the ratio would be one for all events, our +surrogates show deviations. In both cases the distribution is peaked at one and falls off +rather symmetrically towards higher and lower values. For the dipole model the peak is +more pronounced and has a steep slope towards the tails of the distribution. This indicates +that for the bulk of the events the dipole model produces results that are much closer to +the true values than the ones from the naive model. While the naive model seems to tend +to generate an excessive number of large weights, i.e. s > w, both models generate a small +number of outliers with s ≪ w, reaching values for x = w/s of up to 107. We also indicate +the points where the values of xmax lie to show which parts of the distributions are cut +off in the partial unweighting. The dipole model achieves a much smaller xmax than the +naive model, 3.6 compared to 84.8. +In Tab. 2 we summarise the evaluation times of the full and dipole-model surrogate +weights, the efficiencies of the single- and two-stage unweighting, the maximum xmax for +the second unweighting step and, finally, the effective gain factor feff. The evaluation of the +surrogate is found to be orders of magnitude faster than the full matrix element calculation +with AMEGIC. In the 4-jet case it is more than 300, and in the 5-jet case more than 80.000 +times as fast. The evaluation of the phase space weights is fast in comparison to the full +matrix element. However, it is of order, or even larger than ⟨tsurr⟩. We find that the +additional complexity when increasing the multiplicity from four to five jets increases the +matrix element evaluation time by a factor of 300 and reduces the unweighting efficiency +by a factor of 20. Nevertheless, ⟨tsurr⟩ grows only by a factor less than two, while the +approximation accuracy, reflected by xmax and ϵ2nd,surr, remains very similar. We obtain +the values xmax = 2.6 and ϵ2nd,surr = 0.39 in the 4-jet case compared to xmax = 3.6 and +ϵ2nd,surr = 0.29 for five jets. The effective gain factors yield 16 and 269, respectively. +17 + +SciPost Physics +Submission +Table 2: Performance measures for partonic channels contributing to Z + {4, 5} jets +production at the LHC. +SHERPA default +with dipole-model surrogate +Process +tME[ms] +tPS[ms] +ϵfull +tsurr[ms] +xmax +ϵ1st,surr +ϵ2nd,surr +feff +gg → e−e+ggd ¯d +54 +0.40 +1.411 % +0.14 +2.6 +1.418 % +39 % +16 +gg → e−e+gggd ¯d +16 216 +5.70 +0.076 % +0.20 +3.6 +0.085 % +29 % +269 +10−6 +10−3 +100 +103 +106 +w/s +100 +101 +102 +103 +104 +105 +106 +dN/d(w/s) +gg → e−e+gggd ¯d +N = 1M events +xnaive +max += 84.8 +xdipole +max += 3.6 +naive +dipole +(a) Channel gg → e−e+gggd ¯d +(Z + 5 jets). +10−6 +10−3 +100 +103 +106 +w/s +100 +101 +102 +103 +104 +105 +106 +dN/d(w/s) +u¯u → t¯tgd ¯d +N = 1M events +xnaive +max += 132.8 +xdipole +max += 1.5 +naive +dipole +(b) Channel u¯u → t¯tgd ¯d +(t¯t + 3 jets). +Figure 7: Ratio distributions of exact weights and their surrogate for the factorisation- +aware emulation of the matrix-element weight (dipole) and the combined matrix-element +and phase-space weight from Ref. [2] (naive). +t¯t+jets +As contributions to the processes t¯t + 3 jets and t¯t + 4 jets in hadronic collisions we +here consider three partonic channels with varying number of external gluons, namely +u¯u → t¯tgd ¯d, gg → t¯tggg and ug → t¯tgggu. In contrast to the previous examples these are +pure QCD processes featuring massive coloured particles. Even though the final states +contain one particle fewer than the Z+jets channels, these processes still pose a severe +computational challenge. +The direct coupling of gluons to the top quarks leads to a +significant proliferation of Feynman diagrams in their jet-associated production. The input +space dimensionalities are now 28 and 32, respectively. For the processes contributing to +t¯t+3 jets we require three anti-kt jets with pT,j > 20 GeV. The fiducial phase space of the +t¯t+4 jets channel is constrained by requiring four jets with staggered transverse-momentum +cuts, namely pT,1 > 100 GeV, pT,2 > 50 GeV, pT,3 > 40 GeV and pT,4 > 20 GeV. We do +not impose phase space restrictions on the external top quarks, that we treat as on-shell +in the matrix element calculation, i.e. p2 +t = p2¯t = m2 +t with mt = 173.4 GeV. +In Fig. 7b we show the ratio distributions of the true event weights and their surrogates +for the dipole model and the naive model using the example of the partonic channel +u¯u → t¯tgd ¯d. Note that the corresponding distributions for the other channels are shown +in App. A. In comparison to Fig. 7a it can be seen that the distribution of the naive model +is wider while the one of the dipole model is even narrower in this example. Moreover, it +has visibly fewer outliers. This is also reflected in the values of xmax, where the excellent +result of 1.5 for the dipole model is two orders of magnitude smaller than the one for the +18 + +SciPost Physics +Submission +Table 3: Performance measures for partonic channels contributing to t¯t + {3, 4} jets +production at the LHC. +SHERPA default +with dipole-model surrogate +Process +tME[ms] +tPS[ms] +ϵfull +tsurr[ms] +xmax +ϵ1st,surr +ϵ2nd,surr +feff +u¯u → t¯tgd ¯d +5 +0.04 +0.092 % +0.14 +1.5 +0.092 % +69 % +20 +gg → t¯tggg +3262 +0.90 +1.093 % +0.18 +1.4 +1.128 % +69 % +61 +ug → t¯tgggu +51 200 +4.00 +0.153 % +0.24 +1.8 +0.160 % +57 % +354 +0 +50 +100 +150 +200 +250 +300 +350 +effective gain factor feff +naive +dipole +naive +dipole +naive +dipole +naive +dipole +naive +dipole +gg → e−e+ggd ¯d +(Z + 4 jets) +gg → e−e+gggd ¯d +(Z + 5 jets) +u¯u → t¯td ¯dg +(t¯t + 3 jets) +gg → t¯tggg +(t¯t + 3 jets) +ug → t¯tgggu +(t¯t + 4 jets) +2.1 +16 +26 +269 +1.0 +20 +2.8 +61 +11 +354 +→ higher is better +Figure 8: Effective gain factors for different processes. +For comparison the results +obtained using the naive neural network surrogate model from Ref. [2] are shown. Note +that the naive model includes the phase space weight while the dipole model learns the +matrix element weight only. +naive model. +In Tab. 3 we compile the results obtained for the three partonic channels comparing the +ordinary unweighting procedure with the two-stage surrogate technique. Again, we find +significant speedups when using the dipole-model surrogate. For the process ug → t¯tgggu +the surrogate is in fact more than 200.000 times faster than the full matrix element weight +evaluation. For all three examples the surrogate gives accurate approximations leading to +values of xmax between 1.4 and 1.8. The gain factors feff lie between 20 for the process +u¯u → t¯tgd ¯d and 354 for ug → t¯tgggu. +We compare the results for the effective gain factors for all five example processes +in Fig. 8. +For comparison we also include the results obtained using the simpler NN +surrogate from Ref. [2] that were not contained in the tables. +The values differ from +the original publication because the definition of the renormalisation and factorisation +scales has changed from a momenta-dependent one as used in Ref. [2] to a fixed value +as used in Ref. [1]. This change leads to a slightly simpler learning problem and thus to +slightly better performance. It can be seen that the dipole model achieves much larger +19 + +SciPost Physics +Submission +100k 200k +500k +1M +training data size +2 +4 +6 +8 +10 +xmax +gg → e−e+ggd ¯d +gg → e−e+gggd ¯d +u¯u → t¯tgd ¯d +gg → t¯tggg +ug → t¯tgggu +(a) absolute +100k 200k +500k +1M +training data size +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +xmax +max(xmax) +gg → e−e+ggd ¯d +gg → e−e+gggd ¯d +u¯u → t¯tgd ¯d +gg → t¯tggg +ug → t¯tgggu +(b) relative +Figure 9: Influence of the training data size on the value of xmax. +gain factors. This can be attributed to the fact that the dipole model approximates the +matrix elements much better because it already knows the relevant dipole structures for +QCD emissions that dominate the multijet processes considered here. Furthermore, it is +found that the respective highest multiplicity channels of the two process groups yield +the largest gain factors. Adding an additional external particle causes the complexity of +the calculation of the matrix element to grow significantly. This leads to a considerably +increased evaluation time tME for the full weight, while the time tsurr for the surrogate +changes only insignificantly. The impressive performance of the dipole surrogate model +facilitates high gains even for those channels where the naive model from Ref. [2] led to +minor gains only. +The influence of the training dataset size +In Fig. 9 we show how the value of xmax depends on the event sample size used to train +the surrogate model for the different example processes. The number of training events is +varied between 105 and 106. A hierarchy can be identified: the models with the highest, +i.e. worst, values of xmax gain the most from additional training data. For the process +gg → e−e+gggd ¯d for example the resulting xmax is more than halved by going from 105 to +106 events. The processes with smaller xmax in comparison benefit less. For the process +gg → t¯tggg the gain is only 23 %. These observations carry over to Fig. 10 where the +dependence of feff on the training dataset size is shown. According to Eq. (12) we have +feff ∝ ϵ2nd,surr and according to Eq. (14) we have ϵ2nd,surr ∝ 1/xmax. Therefore feff is +inversely proportional to xmax. The largest improvement can again be seen for the process +gg → e−e+gggd ¯d where the value of feff increases by 125 % when going from 105 to 106 +events. Likewise, the smallest improvement relates to the process gg → t¯tggg where the +increase is only 22 %. +Results for colour-sampled amplitudes +The above examples are based on matrix elements with an explicit sum over the SU(3) +colour configurations of the involved partons. Using Monte Carlo integration techniques for +phase space sampling, and possibly partonic flavours, a further option arises: just like the +20 + +SciPost Physics +Submission +100k 200k +500k +1M +training data size +0 +100 +200 +300 +400 +500 +feff +gg → e−e+ggd ¯d +gg → e−e+gggd ¯d +u¯u → t¯tgd ¯d +gg → t¯tggg +ug → t¯tgggu +(a) absolute +100k200k +500k +1M +training data size +0.2 +0.4 +0.6 +0.8 +1.0 +feff +max(feff) +gg → e−e+ggd ¯d +gg → e−e+gggd ¯d +u¯u → t¯tgd ¯d +gg → t¯tggg +ug → t¯tgggu +(b) relative +Figure 10: Influence of the training data size on the value of feff. +kinematic variables, we can also sample the colour assignments for the external partons. It +can be shown [48] that colour sampling has a superior scaling behaviour compared to colour +summation and therefore becomes much faster for large parton multiplicities. This holds +even though colour sampling needs more points to reach a certain target precision. With +6−8 colour charged legs our examples already feature quite high dimensional colour spaces. +It is thus worthwhile to test the performance of our method for colour sampled matrix +elements. +As a benchmark we use SHERPA with its built-in matrix element generator +COMIX that implements colour sampling based on the colour-flow decomposition of QCD +amplitudes [31,32]. To keep things simple, we use a naive approach and employ basically +the same surrogate model as before with the colour configuration as an additional input, +see Sec. 2.3. While our model ansatz Eq. (2) is averaged over the colours, the neural +network can try to learn the colour structure and encode it in the coefficients. As discussed +in Sec. 2.3, an improved approach could use a new set of dipoles with explicit colour +assignment in the future. +We trained the dipole model on the processes gg → e−e+ggd ¯d and gg → t¯tggg and +found gain factors of 0.23 and 0.26, respectively. The performance is thus worse than using +the standard unweighting when sampling colours. We checked that increasing the size of +the training dataset does not lead to much higher gains. Three effects come into play +here: first, the approximation quality of the model is worse because the complexity of the +emulation problem increases significantly due to the additional colour degrees of freedom. +Secondly, the evaluation time tME for the matrix element is now much shorter because +instead of the whole sum only a single colour point needs to be evaluated. Thirdly, the +evaluation time tPS for the phase space weight is now no longer negligible. With COMIX +it is of the same order of magnitude as tME. This makes it much more difficult to achieve +large gains. +A way to deal with the last two points would be to let the surrogate also approximate +the phase space weight such that +s′ ≈ wME · wPS . +(17) +Let us demonstrate this for the effective gain factor. In the limit of a highly accurate +21 + +SciPost Physics +Submission +surrogate with ϵ1st,surr ≈ ϵfull and ϵ2nd,surr ≈ 1 Eq. (12) becomes: +feff ≈ +1 +⟨tsurr⟩+⟨tPS⟩ +⟨tfull⟩ ++ ⟨tME⟩ +⟨tfull⟩ · ϵfull +. +(18) +Even in the ideal case where ⟨tsurr⟩ → 0 and ϵfull → 0 there is an upper limit given by +feff ≤ ⟨tfull⟩ +⟨tPS⟩ . +(19) +This is unproblematic as long as the evaluation of wPS is cheap compared to wME. If this +is not the case a surrogate that emulates the full weight is beneficial and results in an +effective gain factor of: +f +′ +eff = +1 +⟨t′surr⟩ +⟨tfull⟩ · +ϵfull +ϵ′1st,surrϵ′2nd,surr + +ϵfull +ϵ′2nd,surr +. +(20) +Considering again the limit of a highly accurate surrogate leads to +f +′ +eff ≈ +1 +⟨t′surr⟩ +⟨tfull⟩ + ϵfull +. +(21) +The largest possible gain factor is thus f′max +eff += ϵfull−1. This corresponds to the same +acceptance rate as without surrogate but with zero evaluation time. +As was done in +Ref. [2] we adapted the dipole surrogate model to include the phase space weight and +evaluated the performance for the same two processes as above. We find gain factors of +0.02 and 0.22, respectively. Again, we do not achieve any gains compared to the standard +unweighting. In this case the problem is that the neural network gives an even worse +approximation since we include the phase space mapping which already tries to flatten +the structures in the soft and collinear regions. So the model has to deal with a situation +it was originally not designed for. The resulting losses eat up the gain from not having to +calculate wPS for every trial event. +The observations described above open up various options for improvement. One pos- +sibility, as mentioned before, would be to develop a surrogate model with colour-dependent +dipoles, adequately representing amplitudes in a specific colour-flow assignment. In ad- +dition, one could attempt to explicitly incorporate knowledge about the employed phase +space mappings. +5 +Conclusions +We presented a case study of using a fast and accurate neural network emulation model +for scattering matrix elements in the context of unweighted event generation for multijet +processes. To this end we have generalised the model originally presented in Ref. [1], based +on Catani–Seymour dipole factorisation, to account also for initial-state emissions and +massive final-state partons. When considering QCD multijet processes this factorisation- +aware model – using the parton four-momenta, dipole variables and kinematic invariants +as inputs – provides very precise estimates for the squared transition amplitudes. This +has been showcased for a selection of partonic channels contributing at the tree-level to +hadronic Z + 4, 5 jets and t¯t + 3, 4 jets production. +We then considered the trained networks in the ONNX format as fast surrogates for +the full squared matrix elements in a two-stage rejection algorithm, originally presented +22 + +SciPost Physics +Submission +in Ref. [2], in the SHERPA framework. This enables the production of unbiased samples +of unweighted events that reproduce the exact target distribution, i.e. the true squared +matrix element of the considered scattering process. Given a fast and accurate surrogate +model, the effective gains are largest when two conditions are met: (i) the unweighting +efficiency of the phase space integrator is rather low, and, (ii) the matrix element is time +consuming to evaluate. For example, for the channels gg → e−e+gggd ¯d and ug → t¯tgggu in +proton–proton collisions at √s = 13 TeV, featuring default unweighting efficiencies for the +AMEGIC integrator of 0.08% and 0.153%, we found gain factors of 269 and 354, respectively, +when using our dipole-model surrogate. Accordingly, the computational resources needed +to generate a given number of unweighted events get reduced by more than two orders +of magnitude. At the same time, the overheads for training the surrogate network model +are very modest, given that events from the compulsory integration phase prior to the +generation process can be used for that purpose. +The underlying workflow for colour-summed squared matrix elements should be easily +adaptable also for other matrix element providers and usage in experimental computing +frameworks, given that in contrast to the original treatment from Ref. [2] we only employ +the emulation of the matrix element expression and no longer include the generator specific +phase space weight in the first-stage approximation. Furthermore, the ONNX standard +allows one to easily store, transfer and exchange the trained neural networks, offering +much flexibility in the method used to train the model. +Our results are valid for a sequential event generation workflow where events are gen- +erated one after the other on a single CPU core. We expect that the performance can be +further increased by moving to a parallel workflow that generates multiple events at the +same time using parallel hardware. The evaluation of the neural networks, which form +the basis for the surrogate models, can be easily vectorised and benefits in particular from +accelerators such as GPUs. +Our study targeted high-multiplicity tree-level contributions that constitute a severe +computational challenge in state-of-the-art matrix element plus parton shower simula- +tions of multijet production processes [49–54], given that one can typically achieve NLO +QCD accuracy only for somewhat lower multiplicities, see for instance [55]. However, in +particular for the highest multiplicities sampling the colour assignments of the external +partons outperforms their explicit summation. This poses new challenges to emulation +models, given the high-dimensionality of the colour space. We explored naive extensions +towards a suitably adjusted network model, though we were not able to achieve significant +gains using a surrogate based on colour-summed dipoles. This is partly also due to the +reduced evaluation times for partial amplitudes in the colour-flow decomposition. We are +confident that under the same strategy but using colour-stripped dipoles in the surrogate +ansatz and incorporating the phase space weight into the emulation useful gain factors +could be achieved. +Acknowledgements +We are grateful for fruitful discussions with Enrico Bothmann, Stefan H¨oche, and Max +Knobbe. +Funding information +The work of SS and TJ was supported by BMBF (contract +05H21MGCAB) and Deutsche Forschungsgemeinschaft (DFG, German Research Founda- +tion) - project number 456104544. FS’s research was supported by the German Research +Foundation (DFG) under grant No. SI 2009/1-1. DM’s research was supported by STFC +under grant ST/X003167/1. +23 + +SciPost Physics +Submission +A +Auxiliary weight distributions +In this appendix we collect in Figs. 11a–11c auxiliary plots for the emulation accuracy of +our dipole surrogate model (dipole) and the combined neural network surrogate for the +matrix-element and phase-space weight from Ref. [2] (naive) for the remaining partonic +channels. Shown are the ratios of the true weights and the respective surrogates. The two +vertical lines indicate the corresponding maxima based on the 0.1 % maximum reduction +method, see Sec. 3.1. +10−6 +10−3 +100 +103 +106 +w/s +100 +101 +102 +103 +104 +105 +106 +dN/d(w/s) +gg → e−e+ggd ¯d +N = 1M events +xnaive +max += 41.5 +xdipole +max += 2.6 +naive +dipole +(a) Channel gg → e+e−ggd ¯d +(Z + 4 jets). +10−6 +10−3 +100 +103 +106 +w/s +100 +101 +102 +103 +104 +105 +106 +dN/d(w/s) +gg → t¯tggg +N = 1M events +xnaive +max += 41.1 +xdipole +max += 1.5 +naive +dipole +(b) Channel gg → t¯tggg +(t¯t + 3 jets). +10−6 +10−3 +100 +103 +106 +w/s +100 +101 +102 +103 +104 +105 +106 +dN/d(w/s) +ug → t¯tgggu +N = 1M events +xnaive +max += 81.2 +xdipole +max += 1.8 +naive +dipole +(c) Channel ug → t¯tgggu +(t¯t + 4 jets). +Figure 11: Ratio distributions of exact weights and their surrogate for the factorisation- +aware emulation of the matrix-element weight (dipole) and the combined matrix-element +and phase-space weight from Ref. 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Aad et al., “Modelling and computational improvements +to the simulation of single vector-boson plus jet processes for the ATLAS +experiment,” JHEP 08 (2022) 089, arXiv:2112.09588 [hep-ex]. +28 + diff --git a/oNFRT4oBgHgl3EQfbzf_/content/tmp_files/load_file.txt b/oNFRT4oBgHgl3EQfbzf_/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9f59276daace9965ad4013645ae4bf13f3613548 --- /dev/null +++ b/oNFRT4oBgHgl3EQfbzf_/content/tmp_files/load_file.txt @@ -0,0 +1,1117 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf,len=1116 +page_content='SciPost Physics Submission MCNET-23-01, IPPP/23/04 Unweighting multijet event generation using factorisation-aware neural networks T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' Janßen1, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' Maˆıtre2, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' Schumann1, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' Siegert3, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' Truong2 1 Institut f¨ur Theoretische Physik, Georg-August-Universit¨at G¨ottingen, G¨ottingen, Germany 2 Institute for Particle Physics Phenomenology, Department of Physics, Durham University, United Kingdom 3 Institut f¨ur Kern- und Teilchenphysik, TU Dresden, Dresden, Germany February 1, 2023 Abstract In this article we combine the recently proposed method for factorisation-aware matrix element surrogates [1] with the unbiased unweighting algorithm of [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' We show that employing a sophisticated neural network emulation of QCD multijet matrix elements based on Catani–Seymour dipole factorisation can lead to a drastic acceleration of un- weighted event generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' We train neural networks for a selection of partonic channels contributing at the tree-level to Z + 4, 5 jets and t¯t + 3, 4 jets production at the LHC which necessitates a generalisation of the dipole emulation model to include initial state partons as well as massive final state quarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' We also present first steps towards the emulation of colour-sampled amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' We incorporate these emulations as fast and accurate surrogates in a two-stage rejection sampling algorithm within the SHERPA Monte Carlo that yields unbiased unweighted events suitable for phenomenological analyses and post-processing in experimental workflows, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' as input to a time-consuming detector simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' For the computational cost of unweighted events we achieve a reduction by factors between 16 and 350 for the considered channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='13562v1 [hep-ph] 31 Jan 2023 SciPost Physics Submission Contents 1 Introduction 2 2 Improved matrix element emulation using neural networks 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='1 Neural networks based on dipoles 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='2 Extension to initial-state and massive partons 8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='3 Colour-sampled matrix elements 10 3 Event unweighting utilising matrix element surrogates 12 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='1 Two-stage unweighting method 12 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='2 Performance analysis 14 4 Implementation and application to LHC processes 16 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='1 Implementation in the SHERPA framework 16 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='2 Results for LHC multijet production processes 17 5 Conclusions 22 A Auxiliary weight distributions 24 References 25 1 Introduction Physics simulations for current and future high-energy accelerator experiments pose a se- vere computational challenge not only due to the complexity of the studied signatures and the demand for higher theoretical accuracy, but also owing to the sheer number of simu- lated events needed to match the enormous collider luminosities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' This has sparked a wide range of algorithmic developments to accelerate key elements of the simulation tool chain and to improve their computational efficiency and thus reduce their resource requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' Machine learning based methods play a prominent role in these developments [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' Traditionally, the largest fraction of resources has been spent on the complex simulation of the detector response to collision final states, while the generation of the collision events constituted only O(10% − 20%) of the budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' With many recent activities reducing the computational footprint of detector simulations, the event generation speed has become a more and more important area to facilitate the full exploitation of future collider data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' This situation is amplified with the upcoming increased luminosity at the LHC and its focus turning to more complex processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' With the advent of matching and merging techniques theoretical predictions of increased precision have become accessible for these high multiplicity final states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' But as the computational cost increases strongly with the multiplicity the resources needed for the theoretical description of processes of interest has surged, see for example [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' Besides conventional approaches for improving the perfor- mance of Monte Carlo methods in event generators, more recently also machine learning methods are explored [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' 2 SciPost Physics Submission A particular challenge is related to the generation of the hard scattering component that forms the core of the event evolution, thereby representing the parton-level truth signal and/or background hypotheses in physics analyses [6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' In Monte Carlo event generators the hard process is assumed to be factorised from parton showers in the initial and final state as well as non-perturbative phenomena such as hadronisation and the underlying event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' Algorithmically the generation of hard scattering events corresponds to the evaluation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' the stochastic sampling, of the phase space integral over the squared transition matrix element of the considered process at a given order in perturbation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' There has been renewed interest in improving phase space integration and sampling techniques, mostly based on neural networks [8–14], using a variety of methods, but also Nested Sampling [15], and mixed-kernel Markov chain algorithms [16] have been investi- gated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' Broadly speaking these approaches share the goal of adjusting a sampling distribu- tion as closely as possible to the true target distribution, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' the actual transition matrix element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' In the case of a traditional Monte Carlo integration this will typically result in events with weights of ideally small spread.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' However, for a resource efficient generation process for physics analyses, in particular due to very time-consuming components such as a detector simulation, events with (largely) unit weight are desirable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' This is typically solved via von Neumann rejection sampling, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' an accept–reject procedure for weighted event samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' However, even with advanced adaptive sampling techniques in particu- lar for multi-particle final states the efficiency of the unweighting procedure can be quite small, resulting in the repeated trial evaluation of the scattering matrix element that ul- timately get rejected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' For LHC key processes such as the multijet-associated production of gauge bosons this might result in O(105) evaluations of the computationally expensive matrix element for a single unit-weight event [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' This suggests a complementary opportunity for saving resources, namely the usage of a fast and accurate surrogate for the trial weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' A corresponding two-stage unweighting procedure that fully corrects for the potential mismatch between the surrogate weight and the actual value of the full matrix element has recently been presented in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' To demonstrate the algorithm, a rather simple neural network designed to replicate the weight of partonic events, represented by their external momenta was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' For tree-level contributions to Z/W + 4 jets and t¯t + 3 jets production at the LHC significant gains have been observed, however, for less complex partonic channels ordinary unweighting could not be outperformed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' Over the last few years more sophisticated matrix element surrogates based on neural networks have been developed [1,18,19], addressing tree-level and one-loop amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' In particular, Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' [1] presented a method to emulate scattering matrix elements employing the factorisation properties of QCD amplitudes in the soft- and collinear limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' In this work we explore the potential of a combination of the approaches in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' [2] and [1] in unweighted event generation for multijet production processes at the LHC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' To this end we generalise the method presented in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' [1] to the case of colour-charged initial states and massive final-state partons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' We also explore, for the first time, the emulation of colour-sampled QCD amplitudes in the colour-flow decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' With an implementation in the SHERPA event generator framework [20, 21] we benchmark tree- level contributions to Z + 4, 5 jets and t¯t + 3, 4 jets production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' The paper is structured as follows: In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' 2 we review the dipole emulation model of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' [1] and present our new developments to address hadronic collisions and massive final-state partons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' 3 we review the unweighting procedure worked out in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' 4 we discuss the implementation of both algorithms in the SHERPA framework, and present our results obtained for selected partonic channels contributing to pp → Z+4, 5 jets and pp → t¯t+3, 4 jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' 3 SciPost Physics Submission 2 Improved matrix element emulation using neural networks To demonstrate the accelerated surrogate unweighting algorithm in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' [2] the authors used only a simple neural network as model for the event weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' It was clear from the study that a bottleneck in this procedure was the accuracy of the event weight approxima- tions coming from the surrogate model, leading to low efficiencies in the second unweighting step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' In this work we consider replacing that simple model with the factorisation-aware neural network model introduced in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' [1], which has been shown to exhibit more ac- curate predictions on a per-point basis compared to existing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' Below we review the construction of this model and detail the necessary extensions to facilitate multijet production processes at the LHC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='1 Neural networks based on dipoles In this section we briefly review the framework from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' [1], where an ansatz for matrix elements based on the factorisation properties of QCD matrix elements in their soft and collinear limits was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' This factorisation can be depicted as |Mn+1|2 → |Mn|2 ⊗ Vijk , (1) where the matrix element in the (n + 1)-body phase space reduces to a matrix element in the momentum mapped n-body phase space multiplied by a singular factor, Vijk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' For single infrared limits, Vijk contains all the singularity structure of the matrix element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' In the dipole formalism these singular factors are encapsulated in the Catani–Seymour dipoles Dijk [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' This factorisation property of matrix elements leads to the form of our ansatz, which is inspired by the dipole factorisation formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' It is given by |Mn+1|2 ≃ � {ijk} CijkDijk , (2) where i, j, and k denote the three partons involved in the dipole function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' Instead of fitting the matrix element directly where there are divergences in the infrared regions of phase space, we let the neural network fit the coefficients Cijk as a function of phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' These coefficients are more well-behaved than the full matrix element in the soft and collinear limits as the singular behaviour is described by the dipole functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' By combining these well-behaved coefficients with the analytically known dipoles, we produce an approximation for the matrix element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' This enables the fitting of matrix elements across the entire sampled phase space with a single neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' Whilst the model predicts the Cijk coefficients, it should be noted that these are not entirely meaningful by themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' Only once they are combined with the corresponding dipoles do we get the approximation of the matrix element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' It is this approximation of the matrix element which should be seen as the model prediction and which appears in the loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' In singly unresolved limits, only relevant dipoles are large and so constrain the cor- responding Cijk in the fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' Outside of these limits all dipoles are of similar order of magnitude and the ansatz in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' (2) is more under-constrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' In these regions of phase space, the excellent fitting capabilities of neural networks are leveraged to interpolate the non-singular matrix elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' The accuracy achieved in this approach is due to the fact that the coefficients being fit by the network, for single soft or collinear kinematics, are free of divergences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' This facet of the emulation model makes it particularly apt for the case of multijet production processes where matrix elements are plagued with many well-understood divergent structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' 4 SciPost Physics Submission Table 1: Hyperparameters of the neural network and their values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' Parameter Value Hidden layers 4 Nodes in hidden layers 128 Activation function swish [25,26] Weight initialiser Glorot uniform [27] Loss function MSE Batch size 512 Optimiser ADAM [28] Initial learning rate 10−3 Callbacks EarlyStopping, ReduceLROnPlateau In this work we employ networks of similar size and complexity to those in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' [2], namely, we use KERAS [23] and TENSORFLOW [24] to build a neural network (NN) model with four hidden layers, each consisting of 128 nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' These hidden layers use the swish activation function [25,26] and their weights are initialised according to the Glorot Uniform distribution [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' The swish activation function has a similar shape to the more well-known ReLU ac- tivation function, but it is a smooth continuous function allowing small negative values, instead of thresholding them to 0 like ReLU does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' We find that in practice swish outper- forms ReLU in all our trained models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' We also find that instead of using a linear activation function in the output layer, using a swish activation function is more performant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' The NN is fitted to the data generated from SHERPA (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='2 for more informa- tion on the generation of data) by minimising the loss function encoding the discrepancy between the prediction made with the neural network to the true matrix element provided by SHERPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' We use the mean squared error (MSE) as the loss function, with the training optimised using ADAM [28] with an initial learning rate of 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' The learning rate is reduced when the validation loss shows no improvement for 30 epochs of training by using the ReduceLROnPlateau callback, and EarlyStopping is used to terminate training when there is no improvement in the validation loss after 60 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' A summary of the neural network hyperparameters is given in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' 1 for reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' As inputs to our generalised network model we feed: the 4-momenta of all initial- and final-state particles, the phase-space mapping variables corresponding to the dipoles in the ansatz, denoted as yijk1, and the kinematic invariants sij for all pairs of particles in the process considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' Note that the yijk and the sij input variables are not independent from the external momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' To aid the neural network training process, we pre-process the yijk, such that the shape and widths of their distributions are similar for the different dipole configurations (we explicitly state the new dipoles included in the model in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' 1The initial-state phase-space mapping variables are referred to as xijk in [22, 29] but we will refer to all phase-space mappings as yijk for brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' 5 SciPost Physics Submission This amounts to yijk → � � � � � � � � � log(yijk) if massive FI dipole , log(1 − yijk) if massless FI, IF, or II dipole , log(yijk) otherwise (massless FF dipole) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' (3) We note that since we do not have massive partons in the initial-state we only require the corresponding transformation for massive FI dipoles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' The kinematic invariants are also transformed with the logarithm sij → log(sij) as they can span many orders of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' It should be noted that the particles involved in the dipole functions and mapping variables denoted by the subscript ijk are the colour-charged particles in the initial- and final-state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' Non colour-charged particles, for example, electrons and positrons, do not appear in the dipoles, but nevertheless their momenta and kinematic invariants are fed into the network as inputs such that it learns of their dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' All of these inputs are standardised to zero mean and unit variance, with the 4-momenta being standardised along each component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' To use them more effectively in the loss function, we pre-process the matrix elements as |Mn+1|2 → arsinh � |Mn+1|2 Spred � , (4) and standardise to zero mean and unit variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' Spred is the prediction scale taken to be the minimum matrix element value found in our training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' This transformation aids the neural network in training by reducing the span of the target distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' The output nodes of our neural network correspond not to the matrix elements directly, but instead to the dipole coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' The raw outputs, denoted by cijk, are transformed to the coefficients appearing in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' (2), Cijk, via the transformation Cijk = Scoef × sinh (cijk) (5) where Scoef is the coefficient scale, taken to be Spred/Sdipole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' Sdipole is the representative value of a dipole, which we take to be the median of all dipoles in our training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' The neural network prediction is made by using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' (2) to combine the predicted Cijk coefficients with the corresponding dipoles Dijk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' In order to compare with the scaled target matrix elements, we have to transform the neural network predicted matrix element with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' (4) with the same Spred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' We can then compare the matrix element as predicted by the neural network, with the truth value, as given by SHERPA, in the MSE loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' A diagram illustrating the NN emulator architecture is given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' [1], the neural network predictions were given by the average over an ensemble of 20 independent replicas trained on different shuffled subsets of the training set and with different initial random seeds for model weight initialisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' Here we take a similar approach by training a set of 10 replica models, however, for predictions we select the model with the lowest validation loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' We stress that this is not a special choice as all individual replica models converge to a similar point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' As an illustrative example, we plot in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' 2 the loss curves for the partonic channel gg → e−e+ggd ¯d, which is a leading- order contribution to Z + 4 jets production at the LHC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' We observe convergence across all replicas with training terminating at similar values of the MSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' The reasoning behind ensembling a prediction is to reduce the effects of stochasticity of the training process, to reduce random model weight initialisation, and to reduce variance in the prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' In this work we strive for a balance of accuracy and speed, meaning it is advantageous to use a single model to make predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' The reasoning is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' 6 SciPost Physics Submission Figure 1: A simplified sketch of our neural network emulator showing inputs, hidden layers, and outputs Cijk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' 0 100 200 300 400 500 600 700 800 Epochs 10−3 10−2 10−1 100 MSE loss gg → e−e+ggd ¯d Training loss Validation loss Figure 2: Training and validation loss for 10 replica models, for the gg → e−e+ggd ¯d channel, shown as solid lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' The epochs at which training is terminated are illustrated as the solid circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' We depict the training and validation loss of the selected model in dashed horizontal lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' In an ensemble of models where replicas are trained on different subsets of the same training data, there is overlapping information learnt by the individual models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' This leads to diminishing returns in predictive accuracy, meaning that whilst evaluation time grows linearly with the number of replicas, accuracy does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' We have therefore observed a single model to be the most performant configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' It is important to stress that this does not mean that one model cannot be sufficiently accurate, as we will demonstrate in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' This decision to use only a single NN for predictions also guided our choice of num- ber of nodes in the hidden layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' With 128 nodes we reach a balance of having enough parameters to model the matrix elements whilst reducing the effects of overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' De- creasing the number of nodes in the hidden layers to create a more compact NN has little effect on the evaluation time for a single network when we use the ONNX Runtime [30] for evaluation, there would only be loss in accuracy which represents a decrease in overall unweighting efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' 7 SciPost Physics Submission Dij,k i j k �ij pi pj pk (a) FF dipole Da ij i j a �ij pi pj pa (b) FI dipole Dai k a i k �ai pa pi pk (c) IF dipole Dai,b a i b �ai pa pi pb (d) II dipole Figure 3: Schematic diagrams of the four classes of Catani–Seymour dipoles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' The dipoles are named according to whether the emitter and spectator are in the initial (upper indices) or final state (lower indices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' Each dipole consists of a composite particle (denoted by tilde) that decays into two partons, and a spectator that recoils to conserve momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' The grey blob represents the hard scattering process, with incoming and outgoing lines representing initial- and final-state partons, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' The black circle represents the splitting function within the dipole function which contains the divergent behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='2 Extension to initial-state and massive partons In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' [1], the authors considered jet production processes initiated via electron–positron annihilation where only final-state QCD radiation occurs, meaning the set of dipoles built into the emulation model were of the FF (final-state radiator, final-state spectator) kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' Furthermore, the model was restricted to the production of massless QCD partons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' In this work we consider the extension to hadronic initial states, which is relatively straightforward: we need to account for the additional radiation that comes from the colour-charged initial-state particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' To this end, we add the initial-state dipoles to the ansatz, namely, we add the IF (initial-state radiator, final-state spectator), FI (final- state radiator, initial-state spectator) and II (initial-state radiator, initial-state spectator) splitting configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' This means that the emitter i, and spectator k, in the ansatz can now be in the initial-state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' Illustrations for the complete set of dipoles now included in the model are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' To showcase the extension to massless initial state dipoles we consider the emulation of tree-level matrix elements for the partonic channels gg → e−e+ggd ¯d and gg → e−e+gggd ¯d, which are leading order contributions to Z + 4 jets and Z + 5 jets production at the LHC, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' As validation of the emulation accuracy of the NN model for this extension to initial states, we examine the ability of the model to predict matrix elements across the sampled phase space, but in particular for the case of soft and collinear kinematics, where QCD matrix elements are strongly enhanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' We plot in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' 4 a 2d histogram of the truth-to-prediction ratio, |M|2 true/|M|2 pred, against the true value, |M|2 true, for 1M gg → e−e+ggd ¯d test events with standard cuts as described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' Along the sides, we plot the marginal distributions of the matrix element (top) and the ratio (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' The ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='SciPost Physics ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='Submission ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='10−3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='10−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='Fraction of points [%] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='10−47 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='10−39 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='10−31 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='10−23 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='10−15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='10−7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='|M|2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='true ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='10−8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='10−6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='10−4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='10−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='104 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='106 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='108 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='|M|2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='true / |M|2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='pred ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='gg → e−e+ggd ¯d (colour-summed) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='N = 1M events ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='10−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='Fraction of points [%] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='103 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='104 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='105 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='Number of points in bin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='Figure 4: 2d histogram showing the distribution of truth-to-prediction ratios of the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='matrix element against the value of the true matrix element for the Z +4j process gg → ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='e−e+ggd ¯d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' Along the axes, we plot the marginal distributions of the matrix element (top), and the truth-to-prediction ratio (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' High population bins are illustrated as yellow, with low population bins, down to single points, are depicted in purple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' results illustrate that the ratio depicting model accuracy is centred around the ideal value of 1, with a steep drop off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' This applies to the bulk of the events, as depicted by yellow coloured bins, tightly constrained to a narrow band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' The purple coloured bins represent low population bins, or single points, which shows that the tails of the ratio distribution are primarily seen for smaller matrix element weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' Furthermore, the model accuracy remains high for the largest values of the matrix element, signalling that the infrared behaviour is well controlled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' This is a key property of the factorisation-aware model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' The emulation performance for the Z + 5 jets process is presented in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' An additional extension we study in this article is the inclusion of massive dipoles to our ansatz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' This allows us to examine QCD processes with massive partons which is of particular importance for top-quark pair production in association with jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' We include the massive FF, FI, and IF dipoles from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' [29] into the emulation model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' The massive dipoles are generalisations of the massless dipoles, meaning in principle it would be possible to remove the massless dipoles from the ansatz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' However, in practice, we only include the minimal set of necessary dipoles for a given partonic channel and so the inclusion of the massless dipoles reduces overall computational cost due to their relatively simpler expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' With the massive dipoles implemented,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' our model contains the complete set ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='of dipoles and is in principle able to take advantage of the factorisation-aware model for ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='SciPost Physics ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='Submission ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='10−3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='10−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='Fraction of points [%] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='10−43 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='10−36 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='10−29 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='10−22 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='10−15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='10−8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='|M|2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='true ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='10−8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='10−6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='10−4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='10−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='104 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='106 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='108 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='|M|2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='true / |M|2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='pred ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='gg → t¯tggg (colour-summed) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='N = 1M events ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='10−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='Fraction of points [%] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='103 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='104 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='105 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='Number of points in bin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='Figure 5: 2d histogram showing the distribution of truth-to-prediction ratios of the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='matrix element against the value of the true matrix element for the t¯t + 3j process ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='gg → t¯tggg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' Along the axes, we plot the marginal distributions of the matrix element (top), and the truth-to-prediction ratio (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' arbitrary processes involving QCD-enhanced behaviour at tree-level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' In order to showcase the extension to massive dipoles, we consider emulating tree-level matrix elements of three partonic channels: gg → t¯tggg, and u¯u → t¯tgd ¯d, contributing to leading order t¯t+3 jets production, and ug → t¯tgggu which is a leading order contribution to t¯t+4 jets production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' To validate the inclusion of these massive dipoles into the model, we show in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' 5 the deviation similar to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' 4 but for 1M tree-level events of gg → t¯tggg in proton–proton collisions at √s = 13 TeV, with cuts described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' We again observe the narrow yellow band, indicating that the bulk of the test events are accurately predicted, with the outliers corresponding to smaller matrix element values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' The infrared behaviour is well captured by the model as can be seen by the narrow head for the largest matrix element values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' For emulation performance of channels not described here, we refer the reader to Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='2 and App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='3 Colour-sampled matrix elements The discussion so far has been focused on the emulation of colour-summed matrix elements as it was the case for Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' In this work we take the first steps towards emulating colour- sampled matrix elements, such as those obtained from the COMIX generator [31,32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' Based on the colour-flow decomposition of QCD amplitudes [33,34], for each event, the generator samples a momentum configuration and a valid colour assignment, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' colour 10 SciPost Physics Submission 10−2 100 Fraction of points [%] 10−44 10−37 10−30 10−23 10−16 10−9 |M|2 true 10−8 10−6 10−4 10−2 100 102 104 106 108 |M|2 true / |M|2 pred gg → t¯tggg (colour-sampled) N = 1M events 10−2 101 Fraction of points [%] 100 101 102 103 104 Number of points in bin Figure 6: Truth-to-prediction ratio for colour-sampled gg → t¯tggg matrix elements against the colour-ordered partial amplitudes, |M|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' Marginal distributions are plotted for the matrix elements (top) and ratios (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' The colour assignment thereby is represented by a vector of integers, C, where entries in the vector, ci ∈ {1, 2, 3}, denote the colour assigned to a colour-charged parton in the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' Gluons have two colour indices corresponding to colour and anti-colour, whereas quarks/anti-quarks carry only one index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' We add this vector of colour assignments as an additional input to the NN to include the colour-sampled information from the generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' We one-hot encode the colour assignments such that colours are represented by 3-element vectors, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' R = [0, 0, 1], G = [0, 1, 0], and B = [1, 0, 0], as the integer representation of colour assignments is not useful to the NN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' Given the actual colour of a parton is ambiguous, the matrix element should be invari- ant to any cyclic permutation of the specific colour assigned to a given quark or gluon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' To give an example, the three permutations C1 = [R, B, B, G, R], C2 = [G, R, R, B, G], and C3 = [B, G, G, R, B] of a five colour assignment would lead to the same matrix element weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' To aid the NN in learning this behaviour, we take the three permutations and du- plicate the other model inputs such that the training data is enlarged by a factor of three.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' This did not cause us to run into any computational bottlenecks in terms of memory or time taken to train the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' Note that this duplication of data is not required when making predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' The rest of the inputs to the NN model remain identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' We study to what extent a naive approach of using the same dipole functions, which are most suitable for colour- summed matrix elements, works for the case of colour-sampled matrix elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' In the 11 SciPost Physics Submission future, a more promising approach might be the application of coloured dipole terms directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' Their form has already been derived [35] and implemented for the dipole sub- traction in the COMIX event generator [31] but is not implemented in our NN-based model yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' To illustrate the emulation accuracy of colour-sampled matrix elements, here denoted |M|2, we plot the truth-to-prediction ratio in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' 6 for the gg → t¯tggg channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' While we again observe the property of well-behaved predictions for the larger matrix elements, evidently, the ratio distribution is much wider than in the colour-summed case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' This decrease in accuracy directly translates to a lower expected gain factor when using this emulator as a surrogate model for event unweighting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' This is discussed further in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='2 where we elaborate on specific reasons for this decrease in accuracy and present possible future endeavours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' 3 Event unweighting utilising matrix element surrogates The unweighting of hard-scattering parton-level event samples constitutes an important step in the simulation of scattering events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' The obtained unit-weight events then get passed on to subsequent evolution stages, including QCD parton showers, hadronisation, and possibly a detector simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' However, the unweighting, based on rejection sampling, can pose a severe computational challenge, in particular when the evaluation time of the matrix element is long and the efficiency of the unweighting is rather low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' To address this challenge Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' [2] proposed a novel two-stage rejection sampling algorithm based on fast surrogates that we briefly review in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' We furthermore generalise the performance measures to the case where the surrogate replaces the matrix element only, rather than its combination with the phase space weight as was the case in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='1 Two-stage unweighting method The Monte Carlo method provides a numerical procedure to estimate integrals, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' par- tonic cross sections in high energy physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' When the integrand is non-trivial we use im- portance sampling to reduce the variance of the integral estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' For a positive-definite target function f : Ω ⊂ Rd → [0, ∞) defined over the unit hypercube Ω = [0, 1]d and a probability density function g the Monte Carlo estimate of the integral I = � Ω f(u′) du′ (6) is given by I ≈ 1 N N � i=1 f(ui) g(ui) = ⟨w⟩g (7) with the pointwise event weight wi = f(ui)/g(ui).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' A suitable g can reduce the variance of the integral estimate and thereby increase the efficiency of the numerical integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' Finding such a function g is a difficult task, though, as one needs a way to efficiently draw samples from it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' For multimodal target functions it is attractive to use a multi-channel approach, where g is defined by a mixture distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' The weights of the channels can then be adapted automatically [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' VEGAS [37] is an algorithm to automatically construct a sampling distribution g by optimising the bin widths of a piecewise-constant function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' It can also be used to remap a given g or even the individual channels of a multi-channel distribution [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' 12 SciPost Physics Submission Besides the total integral we are typically interested in differential distributions of the points ui, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' histograms of physical observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' Monte Carlo sampling produces weighted events so every entry in a histogram comes with a weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' Variance reduction methods like importance sampling also reduce the spread of weights but only a perfect sampler results in strictly uniform weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' A large weight spread is problematic when the samples are to be post-processed by detector simulations, as these are very expensive in terms of computation time per event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' It is inefficient to apply them to events that yield a minuscule contribution to the total cross section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' The alternative is to first impose a rejection sampling step to extract unit-weight samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' This converts a sample of Ntrials weighted events into a set of N ≤ Ntrials unweighted events by randomly accepting or rejecting every weighted event with the acceptance probability w/wmax where wmax is the maximal event weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' Even though the information of the rejected events is lost the overall efficiency can be significantly increased when detector simulation is more expensive than event generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' A convenient measure for the performance of a Monte Carlo event generator is the unweighting efficiency ϵ of the rejection sampling step, defined as ϵ := N Ntrials .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' (8) For a large number of trial events it can be estimated by ϵ ≈ ⟨w⟩ wmax , (9) where ⟨w⟩ is the mean of the Ntrials weights in the event sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' The average number of target function evaluations needed to get one accepted event is then given by 1/ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' Similar to how the uncertainty on the integral estimate can be diminished by variance reduction methods, the unweighting efficiency can be increased by optimising the sampling density g for smaller wmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' There is another way of reducing the computational footprint especially if the target function takes a long time to evaluate and has a rather low unweighting efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' This is typically the case for high multiplicity scattering processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' The enormous growth in the number of contributing Feynman diagrams makes high multiplicity matrix elements increasingly expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' At the same time, the high dimensionality of phase space renders it difficult to find a sampling density g that is well adapted to the target everywhere in the integration volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' Consequently, the unweighting efficiency typically decreases with increasing multiplicity, see for example [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' In this situation one can reduce the overall event generation time through replacing the expensive matrix element by a fast and accurate surrogate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' The inaccuracy inevitably introduced in this procedure can be fully corrected for in a second unweighting step, resulting in an unbiased method [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' An outline of the algorithm is given in Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' In addition, a more extensive explanation follows below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' We begin by generating a weighted trial event in the conventional way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' In a first unweighting step we then compare the surrogate weight s to the weight maximum wmax and accept the event with probability s/wmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' For an event that gets rejected at this point we only had to evaluate the cheap surrogate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' If the event gets accepted, however, we need to evaluate the true weight w and attach a correction weight x = w/s to the event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' In a second unweighting step, the event has an acceptance probability of x/xmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' Like wmax, xmax has to be predetermined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' When the surrogate yields an accurate approximation of the true weight, a large proportion of events gets accepted in the second unweighting step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' We note that the algorithm can easily be extended to the case of not strictly positive event weights as shown in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' 13 SciPost Physics Submission Algorithm 1: Two-stage rejection-sampling unweighting algorithm using an event-wise weight estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' while true do generate phase-space point u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' calculate approximate event weight s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' generate uniform random number R1 ∈ [0, 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' # first unweighting step if s > R1 · wmax then calculate exact event weight w;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' determine ratio x = w/s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' generate uniform random number R2 ∈ [0, 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' # second unweighting step if x > R2 · xmax then return u and �w = max(1, s/wmax) · max(1, x/xmax) end end end Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' 1 contains a crucial detail regarding the weight maxima, namely that even after unweighting events can end up with weights ˜w > 1 if s is larger than wmax or if x is larger than xmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' If the true maxima were used, this could never happen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' However, given finite-sized samples an exact determination of wmax is realistically not possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' It is often not even desirable since a small number of points with large weights can induce a prohibitively small unweighting efficiency without contributing significantly to the total integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' It can therefore be useful to work with a deliberately reduced maximum, provided the rare mismatches are corrected for by event weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' The resulting events will be partially unweighted since there can be some events that overshoot the maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' These will receive an overweight ˜w = w/wmax > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' Hereinafter, we adopt the approach used in SHERPA for finding the reduced maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' The aim is that the remaining overweights do not contribute more than a fixed proportion to the integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' We set this share to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='1 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' This can be achieved by taking the sorted weights of a sample of weighted points and finding the weight that cuts off the desired quantile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' In SHERPA this is done automatically during the integration phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' We point out that using a reduced maximum is a fully unbiased technique commonly used in event generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' It is especially helpful when weight surrogates are used since the limited approximation quality of the surrogate can lead to particularly large outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='2 Performance analysis To fairly evaluate the performance gain of the two-stage unweighting algorithm shown in Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' 1 we take the average time it takes to generate a single (partially) unweighted event and compare it to the time it would take to generate the statistical equivalent using the standard unweighting procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' We call the ratio between the two the effective gain factor feff: feff := Tstandard Tsurrogate .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' (10) 14 SciPost Physics Submission In order to separate the actual unweighting from program initialisation and other aspects of event generation, we break the calculation down to the relevant ingredients: feff = Ntrials full � ⟨tME⟩ + ⟨tPS⟩ � Ntrials 1st,surr · � ⟨tsurr⟩ + ⟨tPS⟩ � + Ntrials 2nd,surr · ⟨tME⟩ (11) = 1 ⟨tsurr⟩+⟨tPS⟩ ⟨tME⟩+⟨tPS⟩ · ϵfull ϵ1st,surrϵ2nd,surr + ⟨tME⟩ ⟨tME⟩+⟨tPS⟩ · ϵfull ϵ2nd,surr .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' (12) The average evaluation times of the full matrix element weight, the phase space weight and the matrix element surrogate, respectively, are denoted as ⟨tME⟩, ⟨tPS⟩ and ⟨tsurr⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' By Ntrials step we denote the number of trials in the respective unweighting step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' The unweighting efficiencies are defined as ϵfull := N Ntrials full , ϵ1st,surr := Ntrials 2nd,surr Ntrials 1st,surr and ϵ2nd,surr := N Ntrials 2nd,surr .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' (13) It should be noted that events rejected due to phase space constraints do not affect the un- weighting efficiencies since the selection cuts can be applied solely based on the kinematics without having to evaluate the matrix element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' (12) it is clear that an important requirement for significant gains are short evaluation times for the surrogate in comparison to the full matrix element, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' ⟨tsurr⟩ ≪ ⟨tME⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' Furthermore, even with a fast and accurate surrogate gains are only possible when the original unweighting efficiency ϵfull is small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' Therefore, the surrogate unweighting method is of limited use when the sampling density is very well adapted to the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' For suitable processes it will thus be important to find a good balance between fast evaluation and high accuracy of the surrogate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' The efficiency ϵfull can be estimated by ϵfull ≈ ⟨w⟩ wmax (14) from the weights w generated during an initial integration run, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' after adapting the phase space generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' For wmax we use the reduced value as described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' Analogously, we estimate ϵ1st,surr by ϵ1st,surr ≈ ⟨s⟩ wmax (15) using the same weight maximum and the surrogate weights s determined for the events in the test dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' Since the deviations of the surrogate should average out, one can expect the values of ϵfull and ϵ1st,surr to be close.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' The second unweighting efficiency ϵ2nd,surr can be estimated by ϵ2nd,surr ≈ ⟨x⟩ xmax (16) using the values x = w/s determined for the events in the test dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' The reduced maximum xmax can be calculated analogously to wmax with the restriction that we have to weight the values of x by their corresponding values of s to take into account the acceptance probability in the first unweighting step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' To determine the times ⟨tME⟩, ⟨tPS⟩ and ⟨tsurr⟩ we repeat the calculation of the full/surrogate matrix element and phase space weights for a number of events from the test dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' Depending on the complexity of the process we need between 10 and 10 000 events for a reliable time estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' Note, the value of ⟨tsurr⟩ includes the time for preprocessing the inputs and post-processing the outputs of the surrogate model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' 15 SciPost Physics Submission 4 Implementation and application to LHC processes In this section we present the application of the dipole model emulation of QCD matrix elements in the unweighting of event samples for high-multiplicity scattering processes at the LHC, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' Z + 4, 5 jets and t¯t + 3, 4 jets production at the LHC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' Results presented in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' [2] were based on a simplified neural network surrogate, however, also included an approximation for the phase space weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' We will here contrast the results obtained before to the sophisticated dipole model surrogate and also comment on the challenges when using colour-sampled QCD amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' We furthermore briefly describe an implementation in the workflow of the SHERPA framework [20,21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' Note that we here only need to consider the generation of the hard process partons, as this is factorised from the generation of initial- and final-state parton showers as well as non-perturbative phases such as hadronisation and the underlying event [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' Furthermore we note that systematic variations of the hard event related to alternative PDF sets, or modifications in the scale choices can be evaluated on-the-fly for unweighted events, represented by variational weights, see for example [39,40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='1 Implementation in the SHERPA framework The two-stage unweighting algorithm described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='1 has been implemented in SHERPA [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' The framework provides two built-in tree-level matrix element generators: AMEGIC [41] and COMIX [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' We use AMEGIC to evaluate colour-summed matrix ele- ments and COMIX for colour-sampled ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' To adapt the integrator to the integrand SHERPA runs an initial optimisation phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' This is followed by an integration phase in which the optimised integrator is used to calculate the total cross section of the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' From the event weights produced in this phase the value of wmax is determined, based on the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='1 % maximum reduction method introduced in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' We take 2M events from the integration phase as a training dataset by saving the momenta, matrix element and phase space weights, and, when using colour sampling, colour assignments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' From the training dataset we use 800k events for training the model, 200k for validation during the training and 1M for testing the performance afterwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' We train the dipole model described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' 2 using KERAS [23] with the TENSORFLOW [24] backend and save it in the ONNX format [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' The 1M events from the test dataset are used to determine the value of xmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' After the training of the surrogate model has been completed successfully, the de- termination of the surrogate matrix element value during event generation with SHERPA proceeds as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' At the point where normally the matrix element would be calculated with AMEGIC or COMIX, we use the momenta of the current trial event to determine the additional inputs yijk and sij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' Along with the momenta these are then fed into the model which we evaluate on a single CPU core using the C++ API of the ONNX Runtime pack- age [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' We find that ONNX Runtime evaluates the model several times faster than the header-only library frugally-deep [43], which was used in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' It is important to note that this introduces an additional dependency on a software library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' We would, however like to emphasise that our method does not depend on the code with which the surrogate is evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' This affects only the evaluation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' It would even be possible to create an interface through which any suitable tool could be used for this purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' The model evalu- ation yields the dipole coefficients Cijk which are then combined with the Catani–Seymour dipoles Dijk according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' To determine the relevant dipoles we use a custom imple- mentation, although there already exists an implementation of Catani–Seymour dipoles in SHERPA (used in the automated construction of infrared subtraction terms for NLO QCD and EW calculations [44,45]) which could in principle also be employed for the case considered here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' 16 SciPost Physics Submission 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='2 Results for LHC multijet production processes To study the performance of the method we consider various partonic multijet processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' We thereby follow the validation and benchmark strategies outlined in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' [2], considering Z+jets and t¯t+jets production in proton–proton collisions at √s = 13 TeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' In particular we present results for Z +{4, 5} jets and t¯t+{3, 4} jets final states, thereby extending our previous study by one multiplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' Jets get reconstructed with the anti-kt algorithm [46] with R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' As parton density functions we use the NNPDF-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='0 NNLO set [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' Z+jets We examine the partonic channels gg → e−e+ggd ¯d and gg → e−e+gggd ¯d at the tree- level that represent leading-order contributions to Z + 4 jets and Z + 5 jets production at the LHC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' Correspondingly, using the four-momenta as inputs for the surrogate model we have parameter spaces with 32 and 36 dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' These get supplemented by the corresponding dipole mapping variables and kinematic invariants, see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' Cuts are implemented to constrain the fiducial phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' A dilepton invariant mass me−e+ > 66 GeV and four, respectively, five jets with pT,j > 20 GeV are enforced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' Identical cuts are used for the training and the prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' As a first assessment of the quality of the surrogate we show in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' 7a the distribution of the ratio between the true event weight w and the surrogate event weight s for 1M test events for the exemplar channel gg → e−e+gggd ¯d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' The corresponding plot for the process with the lower multiplicity is shown in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' We compare the results of the dipole model with the naive model from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' We point out that the naive model learns the entire event weight, while the dipole model learns only the matrix element weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' For the representation in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' 7a, the approximated matrix element weight of the dipole model was therefore multiplied by the true phase space weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' While a perfect model would reproduce the true weight exactly, such that the ratio would be one for all events, our surrogates show deviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' In both cases the distribution is peaked at one and falls off rather symmetrically towards higher and lower values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' For the dipole model the peak is more pronounced and has a steep slope towards the tails of the distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' This indicates that for the bulk of the events the dipole model produces results that are much closer to the true values than the ones from the naive model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' While the naive model seems to tend to generate an excessive number of large weights, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' s > w, both models generate a small number of outliers with s ≪ w, reaching values for x = w/s of up to 107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' We also indicate the points where the values of xmax lie to show which parts of the distributions are cut off in the partial unweighting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' The dipole model achieves a much smaller xmax than the naive model, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='6 compared to 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' In Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' 2 we summarise the evaluation times of the full and dipole-model surrogate weights, the efficiencies of the single- and two-stage unweighting, the maximum xmax for the second unweighting step and, finally, the effective gain factor feff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' The evaluation of the surrogate is found to be orders of magnitude faster than the full matrix element calculation with AMEGIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' In the 4-jet case it is more than 300, and in the 5-jet case more than 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='000 times as fast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' The evaluation of the phase space weights is fast in comparison to the full matrix element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' However, it is of order, or even larger than ⟨tsurr⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' We find that the additional complexity when increasing the multiplicity from four to five jets increases the matrix element evaluation time by a factor of 300 and reduces the unweighting efficiency by a factor of 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' Nevertheless, ⟨tsurr⟩ grows only by a factor less than two, while the approximation accuracy, reflected by xmax and ϵ2nd,surr, remains very similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' We obtain the values xmax = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='6 and ϵ2nd,surr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='39 in the 4-jet case compared to xmax = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='6 and ϵ2nd,surr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='29 for five jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' The effective gain factors yield 16 and 269, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' 17 SciPost Physics Submission Table 2: Performance measures for partonic channels contributing to Z + {4, 5} jets production at the LHC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' SHERPA default with dipole-model surrogate Process tME[ms] tPS[ms] ϵfull tsurr[ms] xmax ϵ1st,surr ϵ2nd,surr feff gg → e−e+ggd ¯d 54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='40 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='411 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='14 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='418 % 39 % 16 gg → e−e+gggd ¯d 16 216 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='076 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='20 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='085 % 29 % 269 10−6 10−3 100 103 106 w/s 100 101 102 103 104 105 106 dN/d(w/s) gg → e−e+gggd ¯d N = 1M events xnaive max = 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='8 xdipole max = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='6 naive dipole (a) Channel gg → e−e+gggd ¯d (Z + 5 jets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' 10−6 10−3 100 103 106 w/s 100 101 102 103 104 105 106 dN/d(w/s) u¯u → t¯tgd ¯d N = 1M events xnaive max = 132.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='8 xdipole max = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='5 naive dipole (b) Channel u¯u → t¯tgd ¯d (t¯t + 3 jets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' Figure 7: Ratio distributions of exact weights and their surrogate for the factorisation- aware emulation of the matrix-element weight (dipole) and the combined matrix-element and phase-space weight from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' [2] (naive).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' t¯t+jets As contributions to the processes t¯t + 3 jets and t¯t + 4 jets in hadronic collisions we here consider three partonic channels with varying number of external gluons, namely u¯u → t¯tgd ¯d, gg → t¯tggg and ug → t¯tgggu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' In contrast to the previous examples these are pure QCD processes featuring massive coloured particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' Even though the final states contain one particle fewer than the Z+jets channels, these processes still pose a severe computational challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' The direct coupling of gluons to the top quarks leads to a significant proliferation of Feynman diagrams in their jet-associated production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' The input space dimensionalities are now 28 and 32, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' For the processes contributing to t¯t+3 jets we require three anti-kt jets with pT,j > 20 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' The fiducial phase space of the t¯t+4 jets channel is constrained by requiring four jets with staggered transverse-momentum cuts, namely pT,1 > 100 GeV, pT,2 > 50 GeV, pT,3 > 40 GeV and pT,4 > 20 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' We do not impose phase space restrictions on the external top quarks, that we treat as on-shell in the matrix element calculation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' p2 t = p2¯t = m2 t with mt = 173.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='4 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' 7b we show the ratio distributions of the true event weights and their surrogates for the dipole model and the naive model using the example of the partonic channel u¯u → t¯tgd ¯d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' Note that the corresponding distributions for the other channels are shown in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' In comparison to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' 7a it can be seen that the distribution of the naive model is wider while the one of the dipole model is even narrower in this example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' Moreover, it has visibly fewer outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' This is also reflected in the values of xmax, where the excellent result of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='5 for the dipole model is two orders of magnitude smaller than the one for the 18 SciPost Physics Submission Table 3: Performance measures for partonic channels contributing to t¯t + {3, 4} jets production at the LHC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' SHERPA default with dipole-model surrogate Process tME[ms] tPS[ms] ϵfull tsurr[ms] xmax ϵ1st,surr ϵ2nd,surr feff u¯u → t¯tgd ¯d 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='092 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='14 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='092 % 69 % 20 gg → t¯tggg 3262 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='90 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='093 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='18 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='128 % 69 % 61 ug → t¯tgggu 51 200 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='153 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='24 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='160 % 57 % 354 0 50 100 150 200 250 300 350 effective gain factor feff naive dipole naive dipole naive dipole naive dipole naive dipole gg → e−e+ggd ¯d (Z + 4 jets) gg → e−e+gggd ¯d (Z + 5 jets) u¯u → t¯td ¯dg (t¯t + 3 jets) gg → t¯tggg (t¯t + 3 jets) ug → t¯tgggu (t¯t + 4 jets) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='1 16 26 269 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='0 20 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='8 61 11 354 → higher is better Figure 8: Effective gain factors for different processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' For comparison the results obtained using the naive neural network surrogate model from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' [2] are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' Note that the naive model includes the phase space weight while the dipole model learns the matrix element weight only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' naive model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' In Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' 3 we compile the results obtained for the three partonic channels comparing the ordinary unweighting procedure with the two-stage surrogate technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' Again, we find significant speedups when using the dipole-model surrogate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' For the process ug → t¯tgggu the surrogate is in fact more than 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='000 times faster than the full matrix element weight evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' For all three examples the surrogate gives accurate approximations leading to values of xmax between 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='4 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' The gain factors feff lie between 20 for the process u¯u → t¯tgd ¯d and 354 for ug → t¯tgggu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' We compare the results for the effective gain factors for all five example processes in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' For comparison we also include the results obtained using the simpler NN surrogate from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' [2] that were not contained in the tables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' The values differ from the original publication because the definition of the renormalisation and factorisation scales has changed from a momenta-dependent one as used in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' [2] to a fixed value as used in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' This change leads to a slightly simpler learning problem and thus to slightly better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' It can be seen that the dipole model achieves much larger 19 SciPost Physics Submission 100k 200k 500k 1M training data size 2 4 6 8 10 xmax gg → e−e+ggd ¯d gg → e−e+gggd ¯d u¯u → t¯tgd ¯d gg → t¯tggg ug → t¯tgggu (a) absolute 100k 200k 500k 1M training data size 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='0 xmax max(xmax) gg → e−e+ggd ¯d gg → e−e+gggd ¯d u¯u → t¯tgd ¯d gg → t¯tggg ug → t¯tgggu (b) relative Figure 9: Influence of the training data size on the value of xmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' gain factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' This can be attributed to the fact that the dipole model approximates the matrix elements much better because it already knows the relevant dipole structures for QCD emissions that dominate the multijet processes considered here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' Furthermore, it is found that the respective highest multiplicity channels of the two process groups yield the largest gain factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' Adding an additional external particle causes the complexity of the calculation of the matrix element to grow significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' This leads to a considerably increased evaluation time tME for the full weight, while the time tsurr for the surrogate changes only insignificantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' The impressive performance of the dipole surrogate model facilitates high gains even for those channels where the naive model from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' [2] led to minor gains only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' The influence of the training dataset size In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' 9 we show how the value of xmax depends on the event sample size used to train the surrogate model for the different example processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' The number of training events is varied between 105 and 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' A hierarchy can be identified: the models with the highest, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' worst, values of xmax gain the most from additional training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' For the process gg → e−e+gggd ¯d for example the resulting xmax is more than halved by going from 105 to 106 events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' The processes with smaller xmax in comparison benefit less.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' For the process gg → t¯tggg the gain is only 23 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' These observations carry over to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' 10 where the dependence of feff on the training dataset size is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' According to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' (12) we have feff ∝ ϵ2nd,surr and according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' (14) we have ϵ2nd,surr ∝ 1/xmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' Therefore feff is inversely proportional to xmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' The largest improvement can again be seen for the process gg → e−e+gggd ¯d where the value of feff increases by 125 % when going from 105 to 106 events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' Likewise, the smallest improvement relates to the process gg → t¯tggg where the increase is only 22 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' Results for colour-sampled amplitudes The above examples are based on matrix elements with an explicit sum over the SU(3) colour configurations of the involved partons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' Using Monte Carlo integration techniques for phase space sampling, and possibly partonic flavours, a further option arises: just like the 20 SciPost Physics Submission 100k 200k 500k 1M training data size 0 100 200 300 400 500 feff gg → e−e+ggd ¯d gg → e−e+gggd ¯d u¯u → t¯tgd ¯d gg → t¯tggg ug → t¯tgggu (a) absolute 100k200k 500k 1M training data size 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='0 feff max(feff) gg → e−e+ggd ¯d gg → e−e+gggd ¯d u¯u → t¯tgd ¯d gg → t¯tggg ug → t¯tgggu (b) relative Figure 10: Influence of the training data size on the value of feff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' kinematic variables, we can also sample the colour assignments for the external partons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' It can be shown [48] that colour sampling has a superior scaling behaviour compared to colour summation and therefore becomes much faster for large parton multiplicities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' This holds even though colour sampling needs more points to reach a certain target precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' With 6−8 colour charged legs our examples already feature quite high dimensional colour spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' It is thus worthwhile to test the performance of our method for colour sampled matrix elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' As a benchmark we use SHERPA with its built-in matrix element generator COMIX that implements colour sampling based on the colour-flow decomposition of QCD amplitudes [31,32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' To keep things simple, we use a naive approach and employ basically the same surrogate model as before with the colour configuration as an additional input, see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' While our model ansatz Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' (2) is averaged over the colours, the neural network can try to learn the colour structure and encode it in the coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' As discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='3, an improved approach could use a new set of dipoles with explicit colour assignment in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' We trained the dipole model on the processes gg → e−e+ggd ¯d and gg → t¯tggg and found gain factors of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='23 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='26, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' The performance is thus worse than using the standard unweighting when sampling colours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' We checked that increasing the size of the training dataset does not lead to much higher gains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' Three effects come into play here: first, the approximation quality of the model is worse because the complexity of the emulation problem increases significantly due to the additional colour degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' Secondly, the evaluation time tME for the matrix element is now much shorter because instead of the whole sum only a single colour point needs to be evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' Thirdly, the evaluation time tPS for the phase space weight is now no longer negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' With COMIX it is of the same order of magnitude as tME.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' This makes it much more difficult to achieve large gains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' A way to deal with the last two points would be to let the surrogate also approximate the phase space weight such that s′ ≈ wME · wPS .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' (17) Let us demonstrate this for the effective gain factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' In the limit of a highly accurate 21 SciPost Physics Submission surrogate with ϵ1st,surr ≈ ϵfull and ϵ2nd,surr ≈ 1 Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' (12) becomes: feff ≈ 1 ⟨tsurr⟩+⟨tPS⟩ ⟨tfull⟩ + ⟨tME⟩ ⟨tfull⟩ · ϵfull .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' (18) Even in the ideal case where ⟨tsurr⟩ → 0 and ϵfull → 0 there is an upper limit given by feff ≤ ⟨tfull⟩ ⟨tPS⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' (19) This is unproblematic as long as the evaluation of wPS is cheap compared to wME.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' If this is not the case a surrogate that emulates the full weight is beneficial and results in an effective gain factor of: f ′ eff = 1 ⟨t′surr⟩ ⟨tfull⟩ · ϵfull ϵ′1st,surrϵ′2nd,surr + ϵfull ϵ′2nd,surr .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' (20) Considering again the limit of a highly accurate surrogate leads to f ′ eff ≈ 1 ⟨t′surr⟩ ⟨tfull⟩ + ϵfull .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' (21) The largest possible gain factor is thus f′max eff = ϵfull−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' This corresponds to the same acceptance rate as without surrogate but with zero evaluation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' As was done in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' [2] we adapted the dipole surrogate model to include the phase space weight and evaluated the performance for the same two processes as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' We find gain factors of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='02 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='22, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' Again, we do not achieve any gains compared to the standard unweighting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' In this case the problem is that the neural network gives an even worse approximation since we include the phase space mapping which already tries to flatten the structures in the soft and collinear regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' So the model has to deal with a situation it was originally not designed for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' The resulting losses eat up the gain from not having to calculate wPS for every trial event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' The observations described above open up various options for improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' One pos- sibility, as mentioned before, would be to develop a surrogate model with colour-dependent dipoles, adequately representing amplitudes in a specific colour-flow assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' In ad- dition, one could attempt to explicitly incorporate knowledge about the employed phase space mappings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' 5 Conclusions We presented a case study of using a fast and accurate neural network emulation model for scattering matrix elements in the context of unweighted event generation for multijet processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' To this end we have generalised the model originally presented in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' [1], based on Catani–Seymour dipole factorisation, to account also for initial-state emissions and massive final-state partons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' When considering QCD multijet processes this factorisation- aware model – using the parton four-momenta, dipole variables and kinematic invariants as inputs – provides very precise estimates for the squared transition amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' This has been showcased for a selection of partonic channels contributing at the tree-level to hadronic Z + 4, 5 jets and t¯t + 3, 4 jets production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' We then considered the trained networks in the ONNX format as fast surrogates for the full squared matrix elements in a two-stage rejection algorithm, originally presented 22 SciPost Physics Submission in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' [2], in the SHERPA framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' This enables the production of unbiased samples of unweighted events that reproduce the exact target distribution, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' the true squared matrix element of the considered scattering process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' Given a fast and accurate surrogate model, the effective gains are largest when two conditions are met: (i) the unweighting efficiency of the phase space integrator is rather low, and, (ii) the matrix element is time consuming to evaluate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' For example, for the channels gg → e−e+gggd ¯d and ug → t¯tgggu in proton–proton collisions at √s = 13 TeV, featuring default unweighting efficiencies for the AMEGIC integrator of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='08% and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='153%, we found gain factors of 269 and 354, respectively, when using our dipole-model surrogate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' Accordingly, the computational resources needed to generate a given number of unweighted events get reduced by more than two orders of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' At the same time, the overheads for training the surrogate network model are very modest, given that events from the compulsory integration phase prior to the generation process can be used for that purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' The underlying workflow for colour-summed squared matrix elements should be easily adaptable also for other matrix element providers and usage in experimental computing frameworks, given that in contrast to the original treatment from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' [2] we only employ the emulation of the matrix element expression and no longer include the generator specific phase space weight in the first-stage approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' Furthermore, the ONNX standard allows one to easily store, transfer and exchange the trained neural networks, offering much flexibility in the method used to train the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' Our results are valid for a sequential event generation workflow where events are gen- erated one after the other on a single CPU core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' We expect that the performance can be further increased by moving to a parallel workflow that generates multiple events at the same time using parallel hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' The evaluation of the neural networks, which form the basis for the surrogate models, can be easily vectorised and benefits in particular from accelerators such as GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' Our study targeted high-multiplicity tree-level contributions that constitute a severe computational challenge in state-of-the-art matrix element plus parton shower simula- tions of multijet production processes [49–54], given that one can typically achieve NLO QCD accuracy only for somewhat lower multiplicities, see for instance [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' However, in particular for the highest multiplicities sampling the colour assignments of the external partons outperforms their explicit summation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' This poses new challenges to emulation models, given the high-dimensionality of the colour space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' We explored naive extensions towards a suitably adjusted network model, though we were not able to achieve significant gains using a surrogate based on colour-summed dipoles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' This is partly also due to the reduced evaluation times for partial amplitudes in the colour-flow decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' We are confident that under the same strategy but using colour-stripped dipoles in the surrogate ansatz and incorporating the phase space weight into the emulation useful gain factors could be achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' Acknowledgements We are grateful for fruitful discussions with Enrico Bothmann, Stefan H¨oche, and Max Knobbe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' Funding information The work of SS and TJ was supported by BMBF (contract 05H21MGCAB) and Deutsche Forschungsgemeinschaft (DFG, German Research Founda- tion) - project number 456104544.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' FS’s research was supported by the German Research Foundation (DFG) under grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' SI 2009/1-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' DM’s research was supported by STFC under grant ST/X003167/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' 23 SciPost Physics Submission A Auxiliary weight distributions In this appendix we collect in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' 11a–11c auxiliary plots for the emulation accuracy of our dipole surrogate model (dipole) and the combined neural network surrogate for the matrix-element and phase-space weight from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' [2] (naive) for the remaining partonic channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' Shown are the ratios of the true weights and the respective surrogates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' The two vertical lines indicate the corresponding maxima based on the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='1 % maximum reduction method, see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' 10−6 10−3 100 103 106 w/s 100 101 102 103 104 105 106 dN/d(w/s) gg → e−e+ggd ¯d N = 1M events xnaive max = 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='5 xdipole max = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='6 naive dipole (a) Channel gg → e+e−ggd ¯d (Z + 4 jets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' 10−6 10−3 100 103 106 w/s 100 101 102 103 104 105 106 dN/d(w/s) gg → t¯tggg N = 1M events xnaive max = 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='1 xdipole max = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='5 naive dipole (b) Channel gg → t¯tggg (t¯t + 3 jets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' 10−6 10−3 100 103 106 w/s 100 101 102 103 104 105 106 dN/d(w/s) ug → t¯tgggu N = 1M events xnaive max = 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='2 xdipole max = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content='8 naive dipole (c) Channel ug → t¯tgggu (t¯t + 4 jets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFRT4oBgHgl3EQfbzf_/content/2301.13562v1.pdf'} +page_content=' Figure 11: Ratio distributions of exact weights and their surrogate for the factorisation- aware emulation of the 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0000000000000000000000000000000000000000..a8dcc559614adf7b74a31c815f4beda3cd4b5c14 --- /dev/null +++ b/odAzT4oBgHgl3EQfAPrQ/content/2301.00924v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:4a1450ef2c1e1239fe8c14f73bcb750081ec71069315e0b567ad89400c2a15e8 +size 1493328 diff --git a/p9AyT4oBgHgl3EQfZPfT/content/tmp_files/2301.00221v1.pdf.txt b/p9AyT4oBgHgl3EQfZPfT/content/tmp_files/2301.00221v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..53ebf48ce93ee3788d8bfca71913ddde455e2adb --- /dev/null +++ b/p9AyT4oBgHgl3EQfZPfT/content/tmp_files/2301.00221v1.pdf.txt @@ -0,0 +1,741 @@ +arXiv:2301.00221v1 [cs.DL] 31 Dec 2022 +The Role of Author Identities in Peer Review +Nihar B. Shah +Carnegie Mellon University +nihars@cs.cmu.edu +Abstract +There is widespread debate on whether to anonymize author identities in peer review. +The key +argument for anonymization is to mitigate bias, whereas arguments against anonymization posit various +uses of author identities in the review process. The Innovations in Theoretical Computer Science (ITCS) +2023 conference adopted a middle ground by initially anonymizing the author identities from reviewers, +revealing them after the reviewer had submitted their initial reviews, and allowing the reviewer to change +their review subsequently. We present an analysis of the reviews pertaining to the identification and use +of author identities. Our key findings are: (I) A majority of reviewers self-report not knowing and being +unable to guess the authors’ identities for the papers they were reviewing. (II) After the initial submission +of reviews, only 7% of reviews changed their overall merit score and 3.8% changed their self-reported +reviewer expertise. +(III) Even among reviews that changed, there was little correlation between the +change in overall merit and the rank of the authors’ affiliations (Kendall tau b correlation coefficient = +0.09). We also conducted an anonymous survey to obtain opinions from reviewers and authors. The main +findings from the 200 survey responses are: (i) A vast majority of participants favor anonymizing author +identities in some form. (ii) The “middle-ground” initiative of ITCS 2023 was appreciated. (iii) Detecting +conflicts of interest is a challenge that needs to be addressed if author identities are anonymized. Overall, +these findings support anonymization of author identities in some form (e.g., as was done in ITCS 2023), +as long as there is a robust and efficient way to check conflicts of interest. +1 +Introduction +There is a lot of debate in various research communities on whether to anonymize author identities or +not. The primary argument for anonymization is that it mitigates bias in the review process. Indeed, a +number of experiments in computer science [TZH17; MS21] and in other fields [Oki+16; Bla91; Ros+06; +Gar+94; FFS94] have found evidence that reviews are biased if author identities are visible to the reviewers. +The most consistent finding across these studies is that reviews are biased favorably towards authors from +higher-ranked affiliations or authors who are famous. +Traditionally, peer review in the field of theoretical computer science does not anonymize author identities. +This topic has been hotly debated in the field with various opinions and discussions [For18; Bar18; Ven18; +Rei18; Mit18]. Some of the stated reasons against anonymization pertain to using author identities to gain +more confidence in hard-to-very mathematical proofs or wacky ideas, and allocate reviewers’ limited time +and effort accordingly. +The review process of the ITCS 2023 conference thus took a middle-ground approach. Author identities +were initially hidden from reviewers. +Once the reviewer submitted their review, they could see author +identities. Reviewers were subsequently allowed to modify their review if they wished. The rationale behind +this approach was that it could mitigate bias at least in the initial impression, and later allow reviewers to +use author identities in a manner that has been posited in aforementioned discussions. Note that the authors +were free to upload their (non anonymous) papers to preprint servers, social media, etc. +In this paper we address three key questions pertaining to author identities in the review process: +• Identification: Given that author identities were initially hidden, how many reviewers could identify the +authors? +1 + +– We address this question by asking reviewers to self-report their knowledge of author identities +when they submit their review (Section 3.1). +• Usage: Given that author identities were made visible to reviewers after they submitted initial reviews, +how did the reviewers subsequently use the author identities to modify their reviews? +– We address this question by analyzing the change in the reviews after they are initially submitted +(Section 3.2). +• Opinions: What opinions or prespectives do participants have regarding anonymization of author iden- +tities? +– We address this question via an anonymous survey of the participants in the ITCS 2023 review +process (Section 4). +We envisage these results to be useful in designing author-anonymization policies in an evidence-based +fashion. +2 +Related work +A number of studies investigate whether revealing identities of authors to reviewers results in biases in the +reviews. Perhaps the most well-known such experiment in computer science involves a randomized controlled +trial at the WSDM 2017 conference [TZH17] in data mining. This experiment found strong biases favoring +authors who were famous or from top-ranked institutions. (See [SSS19] for some concerns regarding the +experimental design and methods employed in [TZH17].) The study [MS21] conducted an observational +study that found biases favoring authors from top-ranked affiliations in the ICLR conference in machine +learning. +A number of studies in other fields also find biases favoring authors who are famous or from +top-ranked institutions [Oki+16; Bla91; Ros+06; Gar+94; FFS94]. +The studies [War08; MHR13] survey researchers about their perceptions of the peer-review process. The +surveys reveal that researchers across a number of fields prefer anonymizing author identities. In the adjoining +field of information theory where anonymization is currently not the norm, a survey of the members of the +community [SD22] shows significant support for trying out anonymization. +The problem of measuring possible bias in peer review has also spurred research on designing and evalu- +ating the methods to do so. The aforementioned work [SSS19] provides an experimental design framework, +statistical tests, and associated theoretical guarantees for conducting such a measurement in a randomized +controlled trial. The paper [MS21] uses difference-in-difference techniques to estimate bias exploiting the fact +that the ICLR conference changed from not anonymizing to anonymizing author names in a certain year. +In addition, the paper [MS21] also develops methods to test for biases based on the review text rather than +just the review scores. The pair of papers [MD06; Sno06] debate the methods of computing the bias, with +different metrics leading to different conclusions. Finally, the paper [Jec+22] focuses on the tradeoff between +running such a controlled experiment in the peer-review process and the efficiency of the peer-review process. +Even when author identities are not included in the submissions, there are ways in which they may +be deciphered. +First, a reviewer may have seen the paper outside of the review process such as on a +preprint server, on social media, or in a talk. For instance, a little over half of the papers submitted to +the NeurIPS 2019 conference were posted on arXiv, and among these papers, 21% were seen by at least +one reviewer [Bey+19]. Second, some reviewers may actively search for their assigned papers online. For +instance, the study [Ras+22b] found that over a third of the reviewers in two top conferences searched for +their assigned paper online. Third, the reviewer may be able to guess the identity of the authors based on +the content of the paper. The study [Le +18] asked reviewers in three conferences which anonymized author +identities to guess the authors of papers. A total of 70%-86% of the reviews did not provide any guess (which +could mean that the reviewer did not have any idea of the identity or that the reviewer chose to just not +answer the question). Among the subset of reviews which contained a guess, 72%-85% guessed at least one +author correctly. Some other works [HJP03; CUD19; MS20] investigate the possibility of identifying authors +based on content by designing machine learning algorithms to predict the authors based on the paper’s +2 + +content. In our experiment, we also investigate this aspect in ITCS 2023 by asking reviewers to self-report +whether they can identify the authors. +Opening up author identities to reviewers near the end of the review process has been employed before +in other computer science conferences [Hic12]. There, the author identities are usually revealed right before +the program committee meeting. Their stated reasons for this approach are that fully anonymous review +might penalize a paper for failing to cite a certain prior work but this prior work may be unpublished work +by the same authors, and anonymous review may not discover certain conflicts which the reviewer may know +about. Our middle ground approach is also motivated by the other often stated use cases of author identities +(Section 1). Most importantly, a key objective of this approach in ITCS 2023 was to understand the use of +author identities by reviewers to enable subsequent policy-makers to design policies using this evidence. +Questions about bias might arise more when decisions can be subjective, for instance, relying on criteria +such as predictions of potential impact or novelty or interestingness to rank papers. +There are various +experiments in computer science and elsewhere that have found high disagreement among reviewers in +terms of which paper is better [OTD07; Fog+12; LC14; Pie+17; Bey+21], and more recently, experiments +that have also found high disagreements among co-authors about their jointly authored papers [Ras+22a]. +Consequently, there have been various debates about the low acceptance rates and “selectivity” in computer +science conferences [For09; Par16; Var17]. +Our work focuses on biases associated with author identities. In addition to this, there are a variety of +other issues in peer review. See [Sha22] for details. These issues lead to a variety of challenging theoretical +problems which can lead to considerable practical impact if solved, but are not well explored in theoretical +computer science. +3 +Main results 1: Analysis of reviews +In this section, we analyze the reviews to understand their dependence on author identities. We begin with a +brief overview and statistics of the review process at ITCS 2023, and then present our analysis of the change +in the reviews after the reviewer could see the author identities.1 +3.1 +Overview and basic statistics of the review process +The ITCS 2023 conference received a total of 235 paper submissions. The set of reviewers comprised a +program committee (PC) and external reviewers. +The 40-member program committee were ultimately +responsible for the acceptance/rejection decisions on all papers. The program committee also included the +program chair of the conference. The members of the program committee could review papers themselves +or ask experts not in the program committee, called external reviewers, to review. In ITCS 2023, there were +40 members in the program committee who did a total of 405 reviews. There were 315 external reviewers +who did 361 reviews. +The conference asked authors to anonymize their identities on the submitted paper. The conference +policy did not prohibit authors from posting their (non anonymous) papers online or on social media or give +talks. Reviewers were not shown the authors’ identities until they submitted a review. After a reviewer +submitted their review, they could see the authors’ identities for that paper, and had the option to modify +their review in any manner they chose. The reviewers were told of this policy beforehand, and were also told +that the (changes in the) reviews will be analyzed. +Reviewers were asked to fill out following form when submitting their review for any paper: +• Overall merit: 5. Strong accept, 4. Accept, 3. Weak accept, 2. Weak reject, 1. Reject +• Reviewer expertise: 4. Expert, 3. Knowledgeable, 2. Some familiarity, 1. No familiarity +• Paper summary +• Comments for authors +1See https://aspredicted.org/ng77i.pdf for the preregistration of this analysis. +3 + +1 2 3 4 5 +0 +20 +40 +9 +283129 +3 +% Reviews +(a) Overall merit +1 2 3 4 +0 +20 +40 +6 +31 +44 +19 +% Reviews +(b) Reviewer expertise +0 +500 +1000 +1500 +2000 +2500 +3000 +0 +20 +40 +# Reviews +(c) Review length (number of words) +Figure 1: Basic statistics of the reviews in ITCS 2023. +Know +Guess +No idea +0 +20 +40 +60 +80 +100 +20 +13 +67 +% Reviews +(a) Program committee members +Know +Guess +No idea +0 +20 +40 +60 +80 +100 +25 +21 +54 +% Reviews +(b) External reviewers +Figure 2: Distribution of reviewers’ self reports about their knowledge of author identities. +• Comments for PC +• Author identity: Please indicate your knowledge of author identities. [Visible to program chairs. Not +visible to authors at any time.] +A. I know the author identities (e.g., saw a talk or on arxiv etc.) +B. I have a reasonable guess for author identities +C. I have no idea about author identities +In Figure 1, we plot the basic statistics pertaining to the reviews – distribution of the overall merit scores, +the self-reported reviewer expertise, and a histogram of the review length. Note that the reviewers could +edit their reviews after they submit an initial version, and these statistics pertain to the final state of each +review. The mean length of the reviews was 480 words and median was 425 words. +We now come to the question about knowledge of author identities. It is important to note that this +question was asked at the time the reviewer was submitting the review. This allows their answer to be based +on a careful reading of the paper, and not just the title or abstract or a brief glance at the paper. In Figure 2 +we plot the distribution of reviewers’ answers to this question. We observe among both program committee +members and external reviewers, a majority of respondents report having no idea of the identities +of the authors. +3.2 +Change in reviews after initial submission +Recall that the reviewers could not see the identities of the authors of the paper initially, but the identities +were made visible to them after they submitted a review for the paper. The reviewers could then edit their +reviews, and in this section, we analyze the changes to the reviews after initial submission. +4 + +Author identity +Change in overall merit +Change in reviewer expertise +Program +Know +1/20 = 5.0% +1/20 = 5.0% +committee +Guess +1/11 = 9.1% +0/11 = 0% +member +No idea +4/59 = 6.8% +1/59 = 1.7% +External +Know +5/90 = 5.6% +6/90 = 6.7% +reviewer +Guess +6/76 = 7.9% +4/76 = 5.2% +No Idea +15/195 = 7.7% +5/195 = 2.6% +Total +32/451 = 7.1% +17/451 = 3.8% +Table 1: Number and percentage of reviews that changed after initial submission. +Before delving into the details, we must address one confounder. Once a program committee member +submitted their review, they could also see all other reviews submitted for that paper. Thus any change +in their review may be due to other reviews, and such an influence of other reviews is well documented in +the literature [Tep+19]. Consequently, for program committee members, we restrict attention to only those +reviews that were the first to be submitted for the paper. In other words, in this analysis, we only consider +a review by a program committee member if there was no other review submitted for that paper when this +review was first submitted. The external reviewers could not see other reviews at any time, and hence we +consider all external reviews. This leaves us with 90 reviews by program committee members, along with +the 361 reviews by external reviewers for our analysis. In the remainder of this section, we restrict attention +to these 451 reviews. +We tabulate the percentage and number of reviews that changed in Table 1. Observe that the number +of reviews that changed the overall merit and reviewer expertise is very small. Also observe that among +the reviews that changed, a sizable fraction indicated that they knew the author identities, which suggests +that either these reviews are changed for reasons not having to do with author identities, or that initially +hiding the identities from the paper had some effect despite the reviewers’ knowledge of author identities +from elsewhere. Among the 32 reviews that changed overall merit, the mean change was −0.4 and median +change was −1, and among the 17 reviews that changed reviewer expertise, the mean change was +0.47 and +median was +1. +We now delve deeper into the changes in overall merit and reviewer expertise, where we compute the +correlation between these changes and the rank of the affiliations of the authors of the paper. We compute +the rank of any affiliation based on csrankings.org (2012 to 2022, Theory → Algorithms and Complexity). +Affiliations that were not listed on csrankings (such as industry labs) where ranked by the program chair. +Then for any paper, we take the best rank among all affiliations associated with the authors of that paper. +Restricting attention to the 32 reviews where the overall merit was changed after initial submission, we +compute the Kendall tau b correlation between the change in overall merit and the rank of the associated +paper. A positive correlation means that the overall merit for a paper with a better rank increased after the +initial submission. We find that even after restricting to only the reviews that changed, the correlation is +very weak, with a correlation coefficient of 0.09. +Next we look at the 17 reviews where the reviewer expertise was changed after initial submission. In +these 17 reviews, we wish to check if the self-reported reviewer expertise increased if the paper was from a +better-ranked affiliation and the reviewer gave a high overall merit score, or if the paper was from a lower- +ranked affiliation and the reviewer gave a low overall merit score. In what follows, this would be implied +by a positive correlation coefficient and the opposite by a negative correlation coefficient. We compute the +Kendall tau b correlation between the affiliation-rank of the paper and the quantity (updated expertise - +old expertise)*(1 if overall merit=accept and -1 if overall merit=reject). We find that the coefficient is 0.46, +meaning that restricting attention to these reviews we find a moderate effect, although we emphasize that +this pertains only to a small number of reviews which actually changed. +Finally, in 5 of the 315 reviews by external reviewers, the reviewer changed their answer about knowledge +of author identities after initial submission. In two of these cases, they changed from ‘no idea’ to ‘know’, one +changed from ‘know’ to ‘no idea’, one from ‘guess’ to ‘no idea’, and one from ‘no idea’ to ‘guess’. None of +the 90 reviews by program committee members changed the answer to the question on knowledge of author +identities. +5 + +4 +Main results 2: Survey +We conducted an anonymous, optional survey among the participants of the ITCS 2023 conference. The +survey was sent out via email on November 2, 2022 (soon after the paper acceptance decisions were an- +nounced) and was open till November 15, 2022. We first present the contents of the survey as shown to the +participants, and then present aggregated responses. +4.1 +Questionnaire presented to participants +In what follows, we present the survey questionnaire verbatim as presented to the participants. +Our field has long debated whether to anonymize authors in the review process. The main reported benefit +of anonymizing authors is that of reducing (biased) dependence on author identities in the review, as has +been found in a number of experiments in other fields. A number of cons are also often discussed: +• Challenges in conflict-of-interest detection +• Reviewers want to use author identities for some part of their review (e.g., using track record to gain +confidence in whacky ideas or complex proofs) +• Reduces ambiguity w.r.t. prior literature (e.g., can know that a preprint was by the same authors) +• Imperfectness of anonymization. +ITCS 2023 adopted a middle ground where reviewers initially did not see author identities, but could +see after submitting their initial reviews, and could then modify their reviews. We would love to know your +thoughts on this debate of anonymizing authors. When answering, please assume that conflict-of-interest +detection is taken care of even if authors are anonymized (as done in many other conferences that adopt +author anonymization). +I participated in ITCS 2023 as: +□ PC member +□ External reviewer +□ Author +Preferences regarding anonymizing authors:2 +□ I prefer anonymizing author identities through the entire review process +□ I prefer anonymizing author identities until the time for PC meetings/discussions +□ I prefer anonymizing author identities until the reviewer submits their initial review +□ I prefer not anonymizing author identities +□ Other (please specify) +Please share the reasons for your preferences, as well as any other comments/experiences/opinions on this +topic: +4.2 +Analysis of survey responses +We received 200 responses to the questionnaire. In this section, we summarize and analyze these responses. +2The ordering of the first four options was randomized to be either this order or reverse order. +6 + +0 +20 +40 +60 +80 +77 +29 +58 +52 +12 +# Responses +Anonymizing through entire review process +Anonymizing until PC meetings/discussions +Anonymizing until reviewer submits initial review +Not anonymizing author identities +Other +(a) All responses (200 responses) +0 +20 +40 +60 +80 +6 +1 +4 +9 +3 +# Responses +(b) Program committee (PC) members +(19 responses) +0 +20 +40 +60 +80 +22 +16 +36 +17 +4 +(c) External reviewers +(84 responses) +0 +20 +40 +60 +80 +49 +12 +18 +26 +5 +(d) Authors excluding external re- +viewers and PC (97 responses) +Figure 3: Answers given by respondents to the prompt “Preferences regarding anonymizing authors.” +4.2.1 +Quantitative analysis +We begin with an analysis of the responses to the quantitative question on the respondents’ preferences about +anonymizing authors. We report the results in Figure 3.3 The key takeaway is that there is considerable +support for anonymizing authors in at least some part of the review process, particularly from participants +outside the program committee. +Notice in Figure 3a that twelve respondents chose the option ‘other.’ These respondents also provided +comments alongside this choice. The three PC members who selected ‘other’ also indicated support for +anonymization in some form, such as anonymization to reviewers but not to program committee members. +Five of the twelve respondents who selected ‘other’ and did not select any other option. Three of these five +were supportive of anonymizing authors in peer review, but were concerned about challenges in detecting +conflicts of interest (CoI). In absence of other ways for CoI detection, they supported anonymizing except +for CoI detection. One of them strongly supported anonymization at least for external reviewers. One other +respondent did not have an opinion. The remaining seven respondents who selected ‘other’ also selected +other options alongside, and are already counted in Figure 3a. We discuss their text comments below along +with all other general comments made by respondents. +4.2.2 +Free-text comments +We finally discuss the free-text comments provided by respondents. A total of 103 respondents provided +free-text comments, and in what follows we summarize these comments. We saw in Figure 3 that there +were disagreements among respondents on the best policies for anonymizing author identities, and these +disagreements are also reflected in the free-text comments. In our summarization below, we will focus on +the content of the comments rather than the pure opinion about which policies to use, which were already +captured in Section 4.2.1. +We begin with comments that were common to a large number of respondents. +3The participants were allowed to choose more than one option. Here we report the total counts in terms of the number +of respondents who chose any option (hence the sum of all options in Figure 3 is greater than 200). An alternative approach +would be as follows: If a participant chooses k options, then count each chosen option as 1 +k . The result from this alternative +approach is qualitatively similar to the current results, and hence we omit it for brevity. +7 + +• Respondents opine that revealing author identities can bias the reviews. For example, some representa- +tive comments are: +– “There’s ample evidence now that our judgements can be biased by aspects of individuals’ identities, +despite our best intentions otherwise. Anonymizing author identities should help with that.” +– “This should be done to avoid bias. This also welcomes new authors to the field.” +• Respondents appreciated the initiative taken by ITCS 2023 in adopting a middle-ground approach. +Some representative comments are: +– “I think that hiding author identities helps well-intentioned reviewers reduce their bias. Having +the author identities available later in the process allows one to deal with other issues (concurrent +results, overlap in authorship with previous papers, etc.) more easily. ” +– “I was somewhat surprised when I saw the author list, which I think is a good indication that it +probably was good to withhold the information until after I submitted the review.” +– “I think the middle ground policy we used this time is MUCH better than other more/less anony- +mous policies. It worked really well for me!!” +• Respondents complained about problems in avoiding conflicts-of-interest when assigning external re- +viewers. Some representative comments are: +– “It’s challenging to use subreviewers when author identities are anonymous.” +– “I generally support author anonymity, but I had a bit of a tricky situation with a CoI as a result +this time.” +We now discuss the remaining comments, each of which was given by one or few respondents. We have +broadly classified these comments in terms of their implications regarding author anonymization. +• Problems with anonymization / benefits of knowing author identities: +– Many authors upload their papers on preprint servers like arXiv or elsewhere, thereby resulting in +imperfect anonymization. +– Anonymization makes rejecting a paper of a rival easier: A reviewer can identify the paper and +reject, no one can blame the reviewer since it is “anonymous”. +– Some situations don’t yet have suitable policies for anonymization, e.g., submitting a code repository +on Github. +– In a small community (like ITCS), reviewers can easily guess the identities of authors. +– Even if anonymized, some biasing information about authors may be leaked by the English. +– Introducing anonymization signals mistrust in the integrity of our community and inevitably pushes +people into being less trustworthy. +– Authors may strongly criticize their own past work to bolster the current submission. +If the +submission is anonymized, reviewers will not know that it is the same set of authors. +– It can be challenging for authors to ensure that they do not inadvertently identify themselves. +– Once it is realized that anonymization does not realize its intended goals, it may then lead to +undesirable policies such as preprint publication and talk embargoes. +– Anonymization would create weird incentives around posting papers on preprint servers, where +authors might delay posting papers to stay anonymous or post earlier to make their identities +public, depending on what they think would benefit them. +– The best hedge against bias and bad reviews is a community standard of collegiality, openness, +fairness, and self control. Such an atmosphere can only flourish in an environment where people +are trusted and everything is above board. +– Some authors have a history of making errors or submitting bad papers, so knowing author identities +can help assess correctness concerns. +8 + +– Author identities can be used to assess incremental contributions: followup works by the same set +of authors should be subject to higher bars than followup works by new authors. +– Papers by student authors should have lower bars, or in some cases can be shepherded instead of +outright rejection. +– It is very hard to assess quality in real time, and author reputation can serve as a good distinguisher. +– Authors’ track record provides additional confidence to reviewers, for instance, if they are not +entirely familiar with the paper or if there are very long proofs. +– Author identities tell the reviewer how to read the paper stylistically. +– Reviewers should know who the authors are eventually, so that they can understand whom they +should credit if they happen to learn something during the review process that they later (even +unintentionally) end up using. +– If there are multiple closely related papers on a subject, it is important to understand to what +extent these are distinct groups of authors in each combination or just closely related papers by +very similar authorship teams. +• Benefits of anonymizing: +– Anonymization welcomes new authors to the field. +– “I trusted my evaluation more when I did it without knowing who the authors were.” +– Anonymization helps bring new ideas to the conference. +– Ideas and proofs should talk for themselves, and authors should explain new ideas well. Author +name should not be a substitute. +– It will force reviewers to read the paper carefully rather than judging the quality based on authors. +– It is for the papers in the grey area that author identity will bias decisions, and these papers are +precisely the ones for which the discussion is critical. +– Dependence on author identities to review makes our community unwelcoming and closed to others. +– Revealing author identities may make the reviewer more careful if the reviewer knows that the +identities will be known at the discussion phase. +• Benefits of middle ground: +– Prevents initial bias but then helps reap benefits of not anonymizing, thereby achieving both goals. +– People get curious and will look up the paper, but they will be patient enough to wait until after +the first review if they know they will get a chance to then see and potentially update their review. +– Does not do any harm (presuming the initial review is also saved and the PC can see the changes +made) and is good as a backup option. +– Anonymity until PC meetings will ensure that the reviewers are not biased. Revealing identities +during discussions can rule out confusion regarding originality of the work. +– People work hard to cultivate a good reputation for high quality impactful research. This middle- +ground mitigates implicit discrimination and that explicit discrimination is hopefully not too wide- +spread, and anyhow very hard to deal with. +– Allows those committee members who want to ignore authorship to do so when writing their reviews, +while not constraining others from using it. +• Problems with middle ground: +– Puts back the bias into the process. +– If a reviewer has initially given a negative review to famous authors, after finding the reviewer +identities, a reviewer may be tempted to change the review. +9 + +• Suggestions: +– The program committee members should be able to see the name in the conference management +platform but not on the paper. +– Ask the reviewers to give two scores, one on the quality of the paper/result/novelty and another on +confidence in the correctness/technical part. Among them, only the “correctness confidence score” +can be modified after seeing the authors’ names after submitting an initial review. +– It should be the reviewer’s choice as to whether they want author identities anonymized or not. +– The elimination of author names from the front page is a simple change to the paper. +– Prohibiting arXiv and other external dissemination is not good, but there is no harm in anonymizing +within the review process. +– Can have a two-tiered PC where the senior PC are aware of the author identities but do not actively +participate in reviewing, but helps to resolve any issues with conflicts of interest. +– In this year’s data with the middle-ground approach, it may be possible to test the bias in the +process after initial review submission. Find the correlation between author’s fame or affiliation +ranking and the acceptance decisions, conditioned on the initial review scores.4 +– In the middle ground, PCs should be able to see any change in the reviews once the reviewer sees +the author identities. +• Other comments: +– Anonymization of author identities is already adopted in most subfields of computer science. +– The real problem is that the heavy emphasis on conference publications which can only be refereed +incompletely, and far too few papers end up going through a rigorous refereeing process by a journal. +– We should as a community be more careful about who we pick as reviewers and make sure there is +some diversity in that process as well. It would be good to gather statistics on the demographics +of reviewers. Perhaps a reminder that statistics will be gathered at the end of the process would +be enough to improve things. +5 +Discussion and limitations +Our results suggest support for some form of anonymizing authors in peer review. Doing so can help mitigate +biases pertaining to initial impressions about authors. In the literature, there are some stated benefits of not +anonymizing authors, such as using author identities to gain confidence in a proof or wacky idea, and also to +be circumspect of authors who have a history of submitting low-quality work. While these stated benefits +are debated in the survey responses, if a venue wishes to realize them, the approach followed by ITCS 2023 +may allow them to do so. Although this may result in bias appearing in the process after revelation of author +identities, in ITCS 2023, we did not find much change in the overall merit scores after author identities were +revealed. (Note that we were unable to check changes in review text or eventual discussions, leaving open the +possibility of author identities playing a role there.) It is also sometimes stated that anonymizing authors +is not useful as author identities may be guessed from the contents of the paper. However, we found that a +majority of reviewers were unable to guess the authors’ identities. +4This is certainly an interesting suggestion. However, it may lead to violation of false alarm requirements. To see this, +consider a simplified scenario where papers are either from influential or non-influential authors, and where papers by influential +authors actually have a higher “true” quality. Suppose the true quality of papers by influential authors is 4 out of 5, and that +of papers by non-influential reviewers is 2 out of 5. The review process is noisy, and suppose the mean score given to any paper +is its true score+1 with probability 0.5 and true score-1 otherwise. Finally, in this model, the discussion process simply serves +to denoise the scores, thereby accepting papers with a higher true score. There is thus no bias in the entire process. Now, +the data will comprise a set of papers – some from influential authors and others from non-influential authors – each of which +has a mean reviewer score of 3. However, the acceptance rates for papers with more influential authors will be higher, thereby +leading such an analysis to falsely conclude existence of a bias. Designing statistical tests which can detect this effect with a +guaranteed control on false alarm is an interesting question for future work. +10 + +Another stated benefit of knowing author identities is the ease of checking conflicts of interest. This +was indeed a challenge at ITCS 2023, where several participants in the survey complained about problems +in ensuring that papers are assigned to external reviewers who do not have conflicts of interest with the +paper. Thus an anonymization of authors should be accompanied by an efficient and rigorous process to +check conflicts of interest. One way to do so is to have an automated system to check for co-authorship and +affiliation conflicts, as is done in machine learning conferences. An alternative “manual” option is to have a +small set of volunteers who can check conflicts for any external reviewer that a program committee member +wishes to invite for reviewing. A third possible solution, which leads to some reduction in anonymity, is to +make author identities anonymous only to external reviewers but not program committee members. +Note that importantly, the conference did not impose any restrictions on authors regarding posting +their (non anonymous) papers elsewhere such as on preprint servers. Such policy choices are widely de- +bated [Ras+22b] in some other research communities, where arguments in favor of such restrictions point to +ensuring more thorough anonymization, whereas arguments against such restrictions include allowing free +and open dissemination of research and furthermore of unintentionally biasing against the very people that it +is supposed to protect [Sha22, Chapter 7]. The policies of ITCS 2023, as well as the analysis and discussion +in the present paper pertain to the absence of such restrictions. +This study also has several limitations that we discuss below. +• As briefly mentioned above, we did not have access to any changes in the text of the reviews.5 Author +identities may also have played a role in the discussions between program committee members but we +do not have logs of these discussions to draw any inference. +• One manner in which reviewers may use author identities is to gain confidence in their evaluation. We +attempt to measure this, however, the review questionnaire asked reviewers to self report their expertise. +The confidence and expertise of any reviewer may be related, but not identical. For instance, a self- +report of expertise may pertain to the topic of the paper whereas a self-report of confidence may pertain +to their evaluation of the paper. +• The survey we conducted was anonymous, and hence comes with the usual caveats associated with +anonymous surveys such as possible selection biases. We do stratify the responses by the respondents self- +reported role in ITCS 2023 (program committee member, external reviewer, or author) but investigation +of selection biases or stratification along other attributes is not possible. +• The experiment was announced to reviewers prior to them starting to review, and they were told that +the logs of the review revisions will be analyzed. It is thus possible that this information may have +changed the reviewer’s behavior (Hawthorne effect). +As for the last point above, we think that future conferences adopting such policies of partial anonymiza- +tion should consider imparting transparency with respect to dependence on author identities, in which the +program chairs, or the program committee, or even the authors can see the revisions of any review. +All in all, we hope that this investigation will lead to more discussion and evidence-based policy design +for an improved peer-review process. +Acknowledgments +We sincerely thank the ITCS 2023 program chair Yael Kalai for trying out this middle ground approach to +author anonymization, for facilitating the analysis, and for valuable inputs throughout the analysis. We also +thank Pravesh Kothari for very helpful discussions. We are grateful to all the program committee members +and reviewers of ITCS 2023 for their efforts in the review process, and all the respondents of our survey for +sharing their opinions and suggestions. This work was reviewed and approved by the CMU Institutional +Review Board (IRB). +5Our preregistration did include analysis of the change of text. However, we could not access the logs of the review text +from the HotCRP conference management platform on which the peer-review process was conducted. +11 + +References +[Bar18] +B. Barak and other commentators. On double blind reviews in theory conferences. en. Windows On Theory blog +https://windowsontheory.org/2018/01/11/on-double-blind-reviews-in-theory-conferences/. 2018. +[Bey+19] +A. Beygelzimer, E. Fox, F. d’Alch´e Buc, and H. Larochelle. What we learned from NeurIPS 2019 data. 2019. +[Bey+21] +A. Beygelzimer, Y. Dauphin, P. Liang, and J. Wortman Vaughan. 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Leopold. “Single-blind vs double-blind peer review in the setting of +author prestige”. In: Jama (2016). +[OTD07] +M. Obrecht, K. Tibelius, and G. D’Aloisio. “Examining the value added by committee discussion in the review of +applications for research awards”. In: Research Evaluation (2007). +[Par16] +B. Parhami. “Low acceptance rates of conference papers considered harmful”. In: Computer (2016). +[Pie+17] +E. Pier, J. Raclaw, A. Kaatz, M. Brauer, M. Carnes, M. Nathan, and C. Ford. “Your comments are meaner than +your score: score calibration talk influences intra-and inter-panel variability during scientific grant peer review”. +In: Research Evaluation (2017). +[Ras+22a] +C. Rastogi, I. Stelmakh, A. Beygelzimer, Y. N. Dauphin, P. Liang, J. W. Vaughan, Z. Xue, H. Daum´e III, E. Pierson, +and N. B. Shah. “How do Authors’ Perceptions of their Papers Compare with Co-authors’ Perceptions and Peer- +review Decisions?” In: arXiv:2211.12966. 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In: Publishing Research Consortium (2008). +13 + diff --git a/p9AyT4oBgHgl3EQfZPfT/content/tmp_files/load_file.txt b/p9AyT4oBgHgl3EQfZPfT/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d7b773ed72550132a2a0480c1a888299403e2bc4 --- /dev/null +++ b/p9AyT4oBgHgl3EQfZPfT/content/tmp_files/load_file.txt @@ -0,0 +1,733 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf,len=732 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='00221v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='DL] 31 Dec 2022 The Role of Author Identities in Peer Review Nihar B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Shah Carnegie Mellon University nihars@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='cmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='edu Abstract There is widespread debate on whether to anonymize author identities in peer review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' The key argument for anonymization is to mitigate bias, whereas arguments against anonymization posit various uses of author identities in the review process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' The Innovations in Theoretical Computer Science (ITCS) 2023 conference adopted a middle ground by initially anonymizing the author identities from reviewers, revealing them after the reviewer had submitted their initial reviews, and allowing the reviewer to change their review subsequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' We present an analysis of the reviews pertaining to the identification and use of author identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Our key findings are: (I) A majority of reviewers self-report not knowing and being unable to guess the authors’ identities for the papers they were reviewing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' (II) After the initial submission of reviews, only 7% of reviews changed their overall merit score and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='8% changed their self-reported reviewer expertise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' (III) Even among reviews that changed, there was little correlation between the change in overall merit and the rank of the authors’ affiliations (Kendall tau b correlation coefficient = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='09).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' We also conducted an anonymous survey to obtain opinions from reviewers and authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' The main findings from the 200 survey responses are: (i) A vast majority of participants favor anonymizing author identities in some form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' (ii) The “middle-ground” initiative of ITCS 2023 was appreciated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' (iii) Detecting conflicts of interest is a challenge that needs to be addressed if author identities are anonymized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Overall, these findings support anonymization of author identities in some form (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=', as was done in ITCS 2023), as long as there is a robust and efficient way to check conflicts of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' 1 Introduction There is a lot of debate in various research communities on whether to anonymize author identities or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' The primary argument for anonymization is that it mitigates bias in the review process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Indeed, a number of experiments in computer science [TZH17;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' MS21] and in other fields [Oki+16;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Bla91;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Ros+06;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Gar+94;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' FFS94] have found evidence that reviews are biased if author identities are visible to the reviewers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' The most consistent finding across these studies is that reviews are biased favorably towards authors from higher-ranked affiliations or authors who are famous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Traditionally, peer review in the field of theoretical computer science does not anonymize author identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' This topic has been hotly debated in the field with various opinions and discussions [For18;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Bar18;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Ven18;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Rei18;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Mit18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Some of the stated reasons against anonymization pertain to using author identities to gain more confidence in hard-to-very mathematical proofs or wacky ideas, and allocate reviewers’ limited time and effort accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' The review process of the ITCS 2023 conference thus took a middle-ground approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Author identities were initially hidden from reviewers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Once the reviewer submitted their review, they could see author identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Reviewers were subsequently allowed to modify their review if they wished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' The rationale behind this approach was that it could mitigate bias at least in the initial impression, and later allow reviewers to use author identities in a manner that has been posited in aforementioned discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Note that the authors were free to upload their (non anonymous) papers to preprint servers, social media, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' In this paper we address three key questions pertaining to author identities in the review process: Identification: Given that author identities were initially hidden, how many reviewers could identify the authors?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' 1 – We address this question by asking reviewers to self-report their knowledge of author identities when they submit their review (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Usage: Given that author identities were made visible to reviewers after they submitted initial reviews, how did the reviewers subsequently use the author identities to modify their reviews?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' – We address this question by analyzing the change in the reviews after they are initially submitted (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Opinions: What opinions or prespectives do participants have regarding anonymization of author iden- tities?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' – We address this question via an anonymous survey of the participants in the ITCS 2023 review process (Section 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' We envisage these results to be useful in designing author-anonymization policies in an evidence-based fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' 2 Related work A number of studies investigate whether revealing identities of authors to reviewers results in biases in the reviews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Perhaps the most well-known such experiment in computer science involves a randomized controlled trial at the WSDM 2017 conference [TZH17] in data mining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' This experiment found strong biases favoring authors who were famous or from top-ranked institutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' (See [SSS19] for some concerns regarding the experimental design and methods employed in [TZH17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=') The study [MS21] conducted an observational study that found biases favoring authors from top-ranked affiliations in the ICLR conference in machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' A number of studies in other fields also find biases favoring authors who are famous or from top-ranked institutions [Oki+16;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Bla91;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Ros+06;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Gar+94;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' FFS94].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' The studies [War08;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' MHR13] survey researchers about their perceptions of the peer-review process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' The surveys reveal that researchers across a number of fields prefer anonymizing author identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' In the adjoining field of information theory where anonymization is currently not the norm, a survey of the members of the community [SD22] shows significant support for trying out anonymization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' The problem of measuring possible bias in peer review has also spurred research on designing and evalu- ating the methods to do so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' The aforementioned work [SSS19] provides an experimental design framework, statistical tests, and associated theoretical guarantees for conducting such a measurement in a randomized controlled trial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' The paper [MS21] uses difference-in-difference techniques to estimate bias exploiting the fact that the ICLR conference changed from not anonymizing to anonymizing author names in a certain year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' In addition, the paper [MS21] also develops methods to test for biases based on the review text rather than just the review scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' The pair of papers [MD06;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Sno06] debate the methods of computing the bias, with different metrics leading to different conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Finally, the paper [Jec+22] focuses on the tradeoff between running such a controlled experiment in the peer-review process and the efficiency of the peer-review process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Even when author identities are not included in the submissions, there are ways in which they may be deciphered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' First, a reviewer may have seen the paper outside of the review process such as on a preprint server, on social media, or in a talk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' For instance, a little over half of the papers submitted to the NeurIPS 2019 conference were posted on arXiv, and among these papers, 21% were seen by at least one reviewer [Bey+19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Second, some reviewers may actively search for their assigned papers online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' For instance, the study [Ras+22b] found that over a third of the reviewers in two top conferences searched for their assigned paper online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Third, the reviewer may be able to guess the identity of the authors based on the content of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' The study [Le +18] asked reviewers in three conferences which anonymized author identities to guess the authors of papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' A total of 70%-86% of the reviews did not provide any guess (which could mean that the reviewer did not have any idea of the identity or that the reviewer chose to just not answer the question).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Among the subset of reviews which contained a guess, 72%-85% guessed at least one author correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Some other works [HJP03;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' CUD19;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' MS20] investigate the possibility of identifying authors based on content by designing machine learning algorithms to predict the authors based on the paper’s 2 content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' In our experiment, we also investigate this aspect in ITCS 2023 by asking reviewers to self-report whether they can identify the authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Opening up author identities to reviewers near the end of the review process has been employed before in other computer science conferences [Hic12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' There, the author identities are usually revealed right before the program committee meeting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Their stated reasons for this approach are that fully anonymous review might penalize a paper for failing to cite a certain prior work but this prior work may be unpublished work by the same authors, and anonymous review may not discover certain conflicts which the reviewer may know about.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Our middle ground approach is also motivated by the other often stated use cases of author identities (Section 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Most importantly, a key objective of this approach in ITCS 2023 was to understand the use of author identities by reviewers to enable subsequent policy-makers to design policies using this evidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Questions about bias might arise more when decisions can be subjective, for instance, relying on criteria such as predictions of potential impact or novelty or interestingness to rank papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' There are various experiments in computer science and elsewhere that have found high disagreement among reviewers in terms of which paper is better [OTD07;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Fog+12;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' LC14;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Pie+17;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Bey+21], and more recently, experiments that have also found high disagreements among co-authors about their jointly authored papers [Ras+22a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Consequently, there have been various debates about the low acceptance rates and “selectivity” in computer science conferences [For09;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Par16;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Var17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Our work focuses on biases associated with author identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' In addition to this, there are a variety of other issues in peer review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' See [Sha22] for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' These issues lead to a variety of challenging theoretical problems which can lead to considerable practical impact if solved, but are not well explored in theoretical computer science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' 3 Main results 1: Analysis of reviews In this section, we analyze the reviews to understand their dependence on author identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' We begin with a brief overview and statistics of the review process at ITCS 2023, and then present our analysis of the change in the reviews after the reviewer could see the author identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='1 Overview and basic statistics of the review process The ITCS 2023 conference received a total of 235 paper submissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' The set of reviewers comprised a program committee (PC) and external reviewers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' The 40-member program committee were ultimately responsible for the acceptance/rejection decisions on all papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' The program committee also included the program chair of the conference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' The members of the program committee could review papers themselves or ask experts not in the program committee, called external reviewers, to review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' In ITCS 2023, there were 40 members in the program committee who did a total of 405 reviews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' There were 315 external reviewers who did 361 reviews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' The conference asked authors to anonymize their identities on the submitted paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' The conference policy did not prohibit authors from posting their (non anonymous) papers online or on social media or give talks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Reviewers were not shown the authors’ identities until they submitted a review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' After a reviewer submitted their review, they could see the authors’ identities for that paper, and had the option to modify their review in any manner they chose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' The reviewers were told of this policy beforehand, and were also told that the (changes in the) reviews will be analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Reviewers were asked to fill out following form when submitting their review for any paper: Overall merit: 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Strong accept, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Accept, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Weak accept, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Weak reject, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Reject Reviewer expertise: 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Expert, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Knowledgeable, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Some familiarity, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' No familiarity Paper summary Comments for authors 1See https://aspredicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='org/ng77i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='pdf for the preregistration of this analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' 3 1 2 3 4 5 0 20 40 9 283129 3 % Reviews (a) Overall merit 1 2 3 4 0 20 40 6 31 44 19 % Reviews (b) Reviewer expertise 0 500 1000 1500 2000 2500 3000 0 20 40 # Reviews (c) Review length (number of words) Figure 1: Basic statistics of the reviews in ITCS 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Know Guess No idea 0 20 40 60 80 100 20 13 67 % Reviews (a) Program committee members Know Guess No idea 0 20 40 60 80 100 25 21 54 % Reviews (b) External reviewers Figure 2: Distribution of reviewers’ self reports about their knowledge of author identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Comments for PC Author identity: Please indicate your knowledge of author identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' [Visible to program chairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Not visible to authors at any time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' I know the author identities (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=', saw a talk or on arxiv etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=') B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' I have a reasonable guess for author identities C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' I have no idea about author identities In Figure 1, we plot the basic statistics pertaining to the reviews – distribution of the overall merit scores, the self-reported reviewer expertise, and a histogram of the review length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Note that the reviewers could edit their reviews after they submit an initial version, and these statistics pertain to the final state of each review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' The mean length of the reviews was 480 words and median was 425 words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' We now come to the question about knowledge of author identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' It is important to note that this question was asked at the time the reviewer was submitting the review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' This allows their answer to be based on a careful reading of the paper, and not just the title or abstract or a brief glance at the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' In Figure 2 we plot the distribution of reviewers’ answers to this question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' We observe among both program committee members and external reviewers, a majority of respondents report having no idea of the identities of the authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='2 Change in reviews after initial submission Recall that the reviewers could not see the identities of the authors of the paper initially, but the identities were made visible to them after they submitted a review for the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' The reviewers could then edit their reviews, and in this section, we analyze the changes to the reviews after initial submission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' 4 Author identity Change in overall merit Change in reviewer expertise Program Know 1/20 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='0% 1/20 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='0% committee Guess 1/11 = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='1% 0/11 = 0% member No idea 4/59 = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='8% 1/59 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='7% External Know 5/90 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='6% 6/90 = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='7% reviewer Guess 6/76 = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='9% 4/76 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='2% No Idea 15/195 = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='7% 5/195 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='6% Total 32/451 = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='1% 17/451 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='8% Table 1: Number and percentage of reviews that changed after initial submission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Before delving into the details, we must address one confounder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Once a program committee member submitted their review, they could also see all other reviews submitted for that paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Thus any change in their review may be due to other reviews, and such an influence of other reviews is well documented in the literature [Tep+19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Consequently, for program committee members, we restrict attention to only those reviews that were the first to be submitted for the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' In other words, in this analysis, we only consider a review by a program committee member if there was no other review submitted for that paper when this review was first submitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' The external reviewers could not see other reviews at any time, and hence we consider all external reviews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' This leaves us with 90 reviews by program committee members, along with the 361 reviews by external reviewers for our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' In the remainder of this section, we restrict attention to these 451 reviews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' We tabulate the percentage and number of reviews that changed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Observe that the number of reviews that changed the overall merit and reviewer expertise is very small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Also observe that among the reviews that changed, a sizable fraction indicated that they knew the author identities, which suggests that either these reviews are changed for reasons not having to do with author identities, or that initially hiding the identities from the paper had some effect despite the reviewers’ knowledge of author identities from elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Among the 32 reviews that changed overall merit, the mean change was −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='4 and median change was −1, and among the 17 reviews that changed reviewer expertise, the mean change was +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='47 and median was +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' We now delve deeper into the changes in overall merit and reviewer expertise, where we compute the correlation between these changes and the rank of the affiliations of the authors of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' We compute the rank of any affiliation based on csrankings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='org (2012 to 2022, Theory → Algorithms and Complexity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Affiliations that were not listed on csrankings (such as industry labs) where ranked by the program chair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Then for any paper, we take the best rank among all affiliations associated with the authors of that paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Restricting attention to the 32 reviews where the overall merit was changed after initial submission, we compute the Kendall tau b correlation between the change in overall merit and the rank of the associated paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' A positive correlation means that the overall merit for a paper with a better rank increased after the initial submission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' We find that even after restricting to only the reviews that changed, the correlation is very weak, with a correlation coefficient of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Next we look at the 17 reviews where the reviewer expertise was changed after initial submission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' In these 17 reviews, we wish to check if the self-reported reviewer expertise increased if the paper was from a better-ranked affiliation and the reviewer gave a high overall merit score, or if the paper was from a lower- ranked affiliation and the reviewer gave a low overall merit score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' In what follows, this would be implied by a positive correlation coefficient and the opposite by a negative correlation coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' We compute the Kendall tau b correlation between the affiliation-rank of the paper and the quantity (updated expertise - old expertise)*(1 if overall merit=accept and -1 if overall merit=reject).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' We find that the coefficient is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='46, meaning that restricting attention to these reviews we find a moderate effect, although we emphasize that this pertains only to a small number of reviews which actually changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Finally, in 5 of the 315 reviews by external reviewers, the reviewer changed their answer about knowledge of author identities after initial submission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' In two of these cases, they changed from ‘no idea’ to ‘know’, one changed from ‘know’ to ‘no idea’, one from ‘guess’ to ‘no idea’, and one from ‘no idea’ to ‘guess’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' None of the 90 reviews by program committee members changed the answer to the question on knowledge of author identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' 5 4 Main results 2: Survey We conducted an anonymous, optional survey among the participants of the ITCS 2023 conference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' The survey was sent out via email on November 2, 2022 (soon after the paper acceptance decisions were an- nounced) and was open till November 15, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' We first present the contents of the survey as shown to the participants, and then present aggregated responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='1 Questionnaire presented to participants In what follows, we present the survey questionnaire verbatim as presented to the participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Our field has long debated whether to anonymize authors in the review process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' The main reported benefit of anonymizing authors is that of reducing (biased) dependence on author identities in the review, as has been found in a number of experiments in other fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' A number of cons are also often discussed: Challenges in conflict-of-interest detection Reviewers want to use author identities for some part of their review (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=', using track record to gain confidence in whacky ideas or complex proofs) Reduces ambiguity w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' prior literature (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=', can know that a preprint was by the same authors) Imperfectness of anonymization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' ITCS 2023 adopted a middle ground where reviewers initially did not see author identities, but could see after submitting their initial reviews, and could then modify their reviews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' We would love to know your thoughts on this debate of anonymizing authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' When answering, please assume that conflict-of-interest detection is taken care of even if authors are anonymized (as done in many other conferences that adopt author anonymization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' I participated in ITCS 2023 as: □ PC member □ External reviewer □ Author Preferences regarding anonymizing authors:2 □ I prefer anonymizing author identities through the entire review process □ I prefer anonymizing author identities until the time for PC meetings/discussions □ I prefer anonymizing author identities until the reviewer submits their initial review □ I prefer not anonymizing author identities □ Other (please specify) Please share the reasons for your preferences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' as well as any other comments/experiences/opinions on this topic: 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='2 Analysis of survey responses We received 200 responses to the questionnaire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' In this section, we summarize and analyze these responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' 2The ordering of the first four options was randomized to be either this order or reverse order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='77 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='29 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='58 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='52 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='# Responses ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='Anonymizing through entire review process ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='Anonymizing until PC meetings/discussions ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='Anonymizing until reviewer submits initial review ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='Not anonymizing author identities ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='Other ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='(a) All responses (200 responses) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='# Responses ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='(b) Program committee (PC) members ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='(19 responses) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='22 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='36 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='(c) External reviewers ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='(84 responses) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='49 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='26 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='(d) Authors excluding external re- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='viewers and PC (97 responses) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='Figure 3: Answers given by respondents to the prompt “Preferences regarding anonymizing authors.”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='1 Quantitative analysis We begin with an analysis of the responses to the quantitative question on the respondents’ preferences about anonymizing authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' We report the results in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='3 The key takeaway is that there is considerable support for anonymizing authors in at least some part of the review process, particularly from participants outside the program committee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Notice in Figure 3a that twelve respondents chose the option ‘other.’ These respondents also provided comments alongside this choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' The three PC members who selected ‘other’ also indicated support for anonymization in some form, such as anonymization to reviewers but not to program committee members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Five of the twelve respondents who selected ‘other’ and did not select any other option.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Three of these five were supportive of anonymizing authors in peer review, but were concerned about challenges in detecting conflicts of interest (CoI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' In absence of other ways for CoI detection, they supported anonymizing except for CoI detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' One of them strongly supported anonymization at least for external reviewers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' One other respondent did not have an opinion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' The remaining seven respondents who selected ‘other’ also selected other options alongside, and are already counted in Figure 3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' We discuss their text comments below along with all other general comments made by respondents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='2 Free-text comments We finally discuss the free-text comments provided by respondents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' A total of 103 respondents provided free-text comments, and in what follows we summarize these comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' We saw in Figure 3 that there were disagreements among respondents on the best policies for anonymizing author identities, and these disagreements are also reflected in the free-text comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' In our summarization below, we will focus on the content of the comments rather than the pure opinion about which policies to use, which were already captured in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' We begin with comments that were common to a large number of respondents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' 3The participants were allowed to choose more than one option.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Here we report the total counts in terms of the number of respondents who chose any option (hence the sum of all options in Figure 3 is greater than 200).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' An alternative approach would be as follows: If a participant chooses k options, then count each chosen option as 1 k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' The result from this alternative approach is qualitatively similar to the current results, and hence we omit it for brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' 7 Respondents opine that revealing author identities can bias the reviews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' For example, some representa- tive comments are: – “There’s ample evidence now that our judgements can be biased by aspects of individuals’ identities, despite our best intentions otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Anonymizing author identities should help with that.” – “This should be done to avoid bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' This also welcomes new authors to the field.” Respondents appreciated the initiative taken by ITCS 2023 in adopting a middle-ground approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Some representative comments are: – “I think that hiding author identities helps well-intentioned reviewers reduce their bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Having the author identities available later in the process allows one to deal with other issues (concurrent results, overlap in authorship with previous papers, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=') more easily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' ” – “I was somewhat surprised when I saw the author list, which I think is a good indication that it probably was good to withhold the information until after I submitted the review.” – “I think the middle ground policy we used this time is MUCH better than other more/less anony- mous policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' It worked really well for me!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' !”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Respondents complained about problems in avoiding conflicts-of-interest when assigning external re- viewers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Some representative comments are: – “It’s challenging to use subreviewers when author identities are anonymous.” – “I generally support author anonymity, but I had a bit of a tricky situation with a CoI as a result this time.” We now discuss the remaining comments, each of which was given by one or few respondents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' We have broadly classified these comments in terms of their implications regarding author anonymization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Problems with anonymization / benefits of knowing author identities: – Many authors upload their papers on preprint servers like arXiv or elsewhere, thereby resulting in imperfect anonymization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' – Anonymization makes rejecting a paper of a rival easier: A reviewer can identify the paper and reject, no one can blame the reviewer since it is “anonymous”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' – Some situations don’t yet have suitable policies for anonymization, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=', submitting a code repository on Github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' – In a small community (like ITCS), reviewers can easily guess the identities of authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' – Even if anonymized, some biasing information about authors may be leaked by the English.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' – Introducing anonymization signals mistrust in the integrity of our community and inevitably pushes people into being less trustworthy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' – Authors may strongly criticize their own past work to bolster the current submission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' If the submission is anonymized, reviewers will not know that it is the same set of authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' – It can be challenging for authors to ensure that they do not inadvertently identify themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' – Once it is realized that anonymization does not realize its intended goals, it may then lead to undesirable policies such as preprint publication and talk embargoes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' – Anonymization would create weird incentives around posting papers on preprint servers, where authors might delay posting papers to stay anonymous or post earlier to make their identities public, depending on what they think would benefit them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' – The best hedge against bias and bad reviews is a community standard of collegiality, openness, fairness, and self control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Such an atmosphere can only flourish in an environment where people are trusted and everything is above board.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' – Some authors have a history of making errors or submitting bad papers, so knowing author identities can help assess correctness concerns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' 8 – Author identities can be used to assess incremental contributions: followup works by the same set of authors should be subject to higher bars than followup works by new authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' – Papers by student authors should have lower bars, or in some cases can be shepherded instead of outright rejection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' – It is very hard to assess quality in real time, and author reputation can serve as a good distinguisher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' – Authors’ track record provides additional confidence to reviewers, for instance, if they are not entirely familiar with the paper or if there are very long proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' – Author identities tell the reviewer how to read the paper stylistically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' – Reviewers should know who the authors are eventually, so that they can understand whom they should credit if they happen to learn something during the review process that they later (even unintentionally) end up using.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' – If there are multiple closely related papers on a subject, it is important to understand to what extent these are distinct groups of authors in each combination or just closely related papers by very similar authorship teams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Benefits of anonymizing: – Anonymization welcomes new authors to the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' – “I trusted my evaluation more when I did it without knowing who the authors were.” – Anonymization helps bring new ideas to the conference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' – Ideas and proofs should talk for themselves, and authors should explain new ideas well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Author name should not be a substitute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' – It will force reviewers to read the paper carefully rather than judging the quality based on authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' – It is for the papers in the grey area that author identity will bias decisions, and these papers are precisely the ones for which the discussion is critical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' – Dependence on author identities to review makes our community unwelcoming and closed to others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' – Revealing author identities may make the reviewer more careful if the reviewer knows that the identities will be known at the discussion phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Benefits of middle ground: – Prevents initial bias but then helps reap benefits of not anonymizing, thereby achieving both goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' – People get curious and will look up the paper, but they will be patient enough to wait until after the first review if they know they will get a chance to then see and potentially update their review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' – Does not do any harm (presuming the initial review is also saved and the PC can see the changes made) and is good as a backup option.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' – Anonymity until PC meetings will ensure that the reviewers are not biased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Revealing identities during discussions can rule out confusion regarding originality of the work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' – People work hard to cultivate a good reputation for high quality impactful research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' This middle- ground mitigates implicit discrimination and that explicit discrimination is hopefully not too wide- spread, and anyhow very hard to deal with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' – Allows those committee members who want to ignore authorship to do so when writing their reviews, while not constraining others from using it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Problems with middle ground: – Puts back the bias into the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' – If a reviewer has initially given a negative review to famous authors, after finding the reviewer identities, a reviewer may be tempted to change the review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' 9 Suggestions: – The program committee members should be able to see the name in the conference management platform but not on the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' – Ask the reviewers to give two scores, one on the quality of the paper/result/novelty and another on confidence in the correctness/technical part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Among them, only the “correctness confidence score” can be modified after seeing the authors’ names after submitting an initial review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' – It should be the reviewer’s choice as to whether they want author identities anonymized or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' – The elimination of author names from the front page is a simple change to the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' – Prohibiting arXiv and other external dissemination is not good, but there is no harm in anonymizing within the review process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' – Can have a two-tiered PC where the senior PC are aware of the author identities but do not actively participate in reviewing, but helps to resolve any issues with conflicts of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' – In this year’s data with the middle-ground approach, it may be possible to test the bias in the process after initial review submission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Find the correlation between author’s fame or affiliation ranking and the acceptance decisions, conditioned on the initial review scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='4 – In the middle ground, PCs should be able to see any change in the reviews once the reviewer sees the author identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Other comments: – Anonymization of author identities is already adopted in most subfields of computer science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' – The real problem is that the heavy emphasis on conference publications which can only be refereed incompletely, and far too few papers end up going through a rigorous refereeing process by a journal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' – We should as a community be more careful about who we pick as reviewers and make sure there is some diversity in that process as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' It would be good to gather statistics on the demographics of reviewers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Perhaps a reminder that statistics will be gathered at the end of the process would be enough to improve things.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' 5 Discussion and limitations Our results suggest support for some form of anonymizing authors in peer review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Doing so can help mitigate biases pertaining to initial impressions about authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' In the literature, there are some stated benefits of not anonymizing authors, such as using author identities to gain confidence in a proof or wacky idea, and also to be circumspect of authors who have a history of submitting low-quality work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' While these stated benefits are debated in the survey responses, if a venue wishes to realize them, the approach followed by ITCS 2023 may allow them to do so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Although this may result in bias appearing in the process after revelation of author identities, in ITCS 2023, we did not find much change in the overall merit scores after author identities were revealed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' (Note that we were unable to check changes in review text or eventual discussions, leaving open the possibility of author identities playing a role there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=') It is also sometimes stated that anonymizing authors is not useful as author identities may be guessed from the contents of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' However, we found that a majority of reviewers were unable to guess the authors’ identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' 4This is certainly an interesting suggestion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' However, it may lead to violation of false alarm requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' To see this, consider a simplified scenario where papers are either from influential or non-influential authors, and where papers by influential authors actually have a higher “true” quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Suppose the true quality of papers by influential authors is 4 out of 5, and that of papers by non-influential reviewers is 2 out of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' The review process is noisy, and suppose the mean score given to any paper is its true score+1 with probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='5 and true score-1 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Finally, in this model, the discussion process simply serves to denoise the scores, thereby accepting papers with a higher true score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' There is thus no bias in the entire process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Now, the data will comprise a set of papers – some from influential authors and others from non-influential authors – each of which has a mean reviewer score of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' However, the acceptance rates for papers with more influential authors will be higher, thereby leading such an analysis to falsely conclude existence of a bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Designing statistical tests which can detect this effect with a guaranteed control on false alarm is an interesting question for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' 10 Another stated benefit of knowing author identities is the ease of checking conflicts of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' This was indeed a challenge at ITCS 2023, where several participants in the survey complained about problems in ensuring that papers are assigned to external reviewers who do not have conflicts of interest with the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Thus an anonymization of authors should be accompanied by an efficient and rigorous process to check conflicts of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' One way to do so is to have an automated system to check for co-authorship and affiliation conflicts, as is done in machine learning conferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' An alternative “manual” option is to have a small set of volunteers who can check conflicts for any external reviewer that a program committee member wishes to invite for reviewing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' A third possible solution, which leads to some reduction in anonymity, is to make author identities anonymous only to external reviewers but not program committee members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Note that importantly, the conference did not impose any restrictions on authors regarding posting their (non anonymous) papers elsewhere such as on preprint servers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Such policy choices are widely de- bated [Ras+22b] in some other research communities, where arguments in favor of such restrictions point to ensuring more thorough anonymization, whereas arguments against such restrictions include allowing free and open dissemination of research and furthermore of unintentionally biasing against the very people that it is supposed to protect [Sha22, Chapter 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' The policies of ITCS 2023, as well as the analysis and discussion in the present paper pertain to the absence of such restrictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' This study also has several limitations that we discuss below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' As briefly mentioned above, we did not have access to any changes in the text of the reviews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='5 Author identities may also have played a role in the discussions between program committee members but we do not have logs of these discussions to draw any inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' One manner in which reviewers may use author identities is to gain confidence in their evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' We attempt to measure this, however, the review questionnaire asked reviewers to self report their expertise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' The confidence and expertise of any reviewer may be related, but not identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' For instance, a self- report of expertise may pertain to the topic of the paper whereas a self-report of confidence may pertain to their evaluation of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' The survey we conducted was anonymous, and hence comes with the usual caveats associated with anonymous surveys such as possible selection biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' We do stratify the responses by the respondents self- reported role in ITCS 2023 (program committee member, external reviewer, or author) but investigation of selection biases or stratification along other attributes is not possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' The experiment was announced to reviewers prior to them starting to review, and they were told that the logs of the review revisions will be analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' It is thus possible that this information may have changed the reviewer’s behavior (Hawthorne effect).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' As for the last point above, we think that future conferences adopting such policies of partial anonymiza- tion should consider imparting transparency with respect to dependence on author identities, in which the program chairs, or the program committee, or even the authors can see the revisions of any review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' All in all, we hope that this investigation will lead to more discussion and evidence-based policy design for an improved peer-review process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Acknowledgments We sincerely thank the ITCS 2023 program chair Yael Kalai for trying out this middle ground approach to author anonymization, for facilitating the analysis, and for valuable inputs throughout the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' We also thank Pravesh Kothari for very helpful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' We are grateful to all the program committee members and reviewers of ITCS 2023 for their efforts in the review process, and all the respondents of our survey for sharing their opinions and suggestions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' This work was reviewed and approved by the CMU Institutional Review Board (IRB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' 5Our preregistration did include analysis of the change of text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' However, we could not access the logs of the review text from the HotCRP conference management platform on which the peer-review process was conducted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' 11 References [Bar18] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Barak and other commentators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' On double blind reviews in theory conferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' en.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content='html.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' [War08] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' Ware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' “Peer review: benefits, perceptions and alternatives”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9AyT4oBgHgl3EQfZPfT/content/2301.00221v1.pdf'} +page_content=' In: Publishing Research Consortium (2008).' metadata={'source': 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b/uNAzT4oBgHgl3EQfc_zP/content/tmp_files/2301.01414v1.pdf.txt @@ -0,0 +1,5709 @@ +arXiv:2301.01414v1 [math.RT] 4 Jan 2023 +DIAGRAMMATICS FOR REAL SUPERGROUPS +SAIMA SAMCHUCK-SCHNARCH AND ALISTAIR SAVAGE +Abstract. We introduce two families of diagrammatic monoidal supercategories. The first family, +depending on an associative superalgebra, generalizes the oriented Brauer category. The second, +depending on an involutive superalgebra, generalizes the unoriented Brauer category. These two +families of supercategories admit natural superfunctors to supercategories of supermodules over +general linear supergroups and supergroups preserving superhermitian forms, respectively. +We +show that these superfunctors are full when the superalgebra is a central real division superalgebra. +As a consequence, we obtain first fundamental theorems of invariant theory for all real forms of +the general linear, orthosymplectic, periplectic, and isomeric supergroups. We also deduce equiva- +lences between monoidal supercategories of tensor supermodules over the real forms of a complex +supergroup. +Contents +1. +Introduction +1 +2. +Monoidal supercategories +5 +3. +Superalgebras and supermodules +7 +4. +Real division superalgebras +13 +5. +The oriented supercategory +16 +6. +The oriented incarnation superfunctor +20 +7. +Superhermitian forms over involutive superalgebras +26 +8. +Superhermitian forms over involutive real division superalgebras +30 +9. +The unoriented supercategory +33 +10. +The unoriented incarnation superfunctor +38 +11. +Unoriented fullness: real case +43 +12. +Unoriented fullness: complex cases +45 +13. +Unoriented fullness: quaternionic case +50 +Appendix A. +Classification of superhermitian forms +55 +Appendix B. +Classification of real forms +59 +References +61 +1. Introduction +Many recent developments in representation theory involve one or more of the following interre- +lated concepts: +(a) Dual pairs. The classic examples are Schur–Weyl duality, which yields a precise relationship +between the symmetric group and the general linear group, and the analogue for the orthog- +onal and symplectic groups, where the symmetric groups are replaced by Brauer algebras. +2020 Mathematics Subject Classification. 18M05, 18M30, 17B10, 18M25. +Key words and phrases. Monoidal category, supercategory, supergroup, string diagram, invariant theory, Deligne +category, interpolating category. + +2 +SAIMA SAMCHUCK-SCHNARCH AND ALISTAIR SAVAGE +(b) Invariant theory. This amounts to giving explicit descriptions of invariants in tensor products +of certain modules, such as the natural modules for classical Lie groups. +(c) Interpolating categories. Here one aims to give uniform descriptions of representations of +families of groups, such as symmetric groups, general linear groups, orthogonal groups, and +symplectic groups. Highly influential in this approach are the interpolating categories intro- +duced by Deligne [Del07]. Such interpolating categories can often be given nice diagrammatic +descriptions, leading to intuitive topological arguments. +In the case of the general linear group, the connection between the above concepts is as follows. +The oriented Brauer category OB(d) is the free rigid symmetric C-linear monoidal category on a +generating object of categorical dimension d. Since the category of modules over the general linear +group GL(m, C), m ∈ N, is rigid symmetric monoidal, there exists a functor +G: OB(m) → GL(m, C)-mod +sending the generating object ↑ of OB(m) to the natural GL(m, C)-module V . The additive Karoubi +envelope of OB(d) is Deligne’s interpolating category for the general linear groups. The endomor- +phism algebra EndOB(d)(↑⊗r) is isomorphic to the group algebra of the symmetric group Sr, and so +the functor G yields an algebra homomorphism +(1.1) +kSr ∼= EndOB(m)(↑⊗r) → EndGL(m,C)(V ⊗r). +One half of Schur–Weyl duality is that the homomorphism (1.1) is surjective. From this, one is able +to deduce that the functor G is full. The connection to invariant theory comes from the fact that G +also induces a surjective homomorphism +(1.2) +HomOB(m)(1, ↑⊗r ⊗ ↓⊗s) → HomGL(m,C)(C, V ⊗r ⊗ (V ∗)⊗s), +where V ∗ is the GL(m, C)-module dual to V , and ↓ is the object of OB(m) dual to ↑. Thus, all +GL(m, C)-invariant elements of V ⊗r ⊗ (V ∗)⊗s lie in the image under G of morphisms in OB(m). +The fullness of G, or of (1.2), is sometimes referred to as the first fundamental theorem of invariant +theory. (Describing the kernel is the second fundamental theorem.) +An analogous picture exists for the orthogonal and symplectic groups. In these cases, the natural +module is self-dual. Thus, the oriented Brauer category is replaced by the unoriented Brauer cate- +gory B(d) of [LZ15], which is the free rigid symmetric k-linear monoidal category on a symmetrically +self-dual object of categorical dimension d. Then, for m ∈ N, one has a full functors +B(m) → O(m, C)-mod +and +B(−2m) → Sp(2m, C)-mod. +Here the endomorphism algebras are Brauer algebras, which surject onto the endomorphism algebras +of tensor powers of the natural module. +In fact, it turns out that the most natural setting for the above picture is that of categories of +supermodules over supergroups. There are full functors +OB(m − n) → GL(m|n, C)-smod +and +B(m − 2n) → OSp(m|2n, C)-smod, +where GL(m|n, C) and OSp(m|2n, C) are the general linear and orthosymplectic supergroups, re- +spectively [CW12, BS12, LSM02, LZ17, LSM02]. The move to the super world also leads to addi- +tional free categories. First, one observes that an isomorphism of a module with its dual can be +even or odd. The even case corresponds to the Brauer category. The odd case leads to the periplec- +tic Brauer supercategory B1, which is the free rigid symmetric k-linear monoidal supercategory on +an odd-self-dual object (which necessarily has categorical dimension zero). Then there is a full +superfunctor +B1 → P(m)-smod, + +DIAGRAMMATICS FOR REAL SUPERGROUPS +3 +where P(m) is the periplectic supergroup [KT17, CE21, DLZ18]. Another free supercategory arises +from the super version of Schur’s lemma. Since we work over the complex numbers, Schur’s lemma +implies that the endomorphism algebra of a simple module is a complex division superalgebra. +In the non-super setting, the only possibility is C. +However, in the super setting, there is one +additional possibility, which is the two-dimensional complex Clifford superalgebra Cl(C). +This +observation leads to the definition of the oriented Brauer-Clifford category OBC of [BCK19], which +is the free rigid symmetric monoidal supercategory on a generating object whose endomorphism +algebra is Cl(C). (As in the periplectic case, the categorical dimension must be zero.) There is a +full superfunctor +OBC → Q(m)-smod, +where Q(m)-smod is the isomeric supergroup (also known as the queer supergroup). +Despite the great success of the above-mentioned approaches to the representation theory of some +of the most important groups and supergroups appearing in mathematics and physics, surprisingly +little is known when we work with real supergroups instead of complex ones. The goal of the current +paper is to initiate this line of research. Let us now describe our main results. +To any associative superalgebra A over a field k, we define a diagrammatic supercategory OBk(A), +which is the free rigid symmetric monoidal supercategory on an object with endomorphism super- +algebra A. Imposing a condition on the categorical dimension yields a quotient category OBk(A; d), +for d ∈ k. This category has essentially appeared in [Sav19, BSW21, MS], although our definition is +slightly more general. When A = k, OBk(k; d) is the oriented Brauer category (over a general field +k) mentioned above. The universal property of OBk(A) implies that, if g is any Lie superalgebra, +and V is a (g, A)-superbimodule, then there is an oriented incarnation superfunctor +OBk(Aop) → g-smod, +sending the generating object of OBk(Aop) to V , where Aop denotes the superalgebra opposite to A. +When we work over the ground field k = R, Schur’s lemma implies that the endomorphism algebra +of a simple supermodule must be one of the ten real division superalgebras. Our first main result +(Theorem 6.10) is that, when A is a central real division superalgebra and V = Am|n, the functor +OBR(Aop; m − n) → gl(m|n, A)-smod +is full. (Note that, since the general linear groups are connected, we can freely replace the general +linear supergroups by the general linear Lie superalgebras.) +The method of proof is to pass to +complexifications and use known results over the complex numbers. +We then turn our attention to the unoriented (i.e. self-dual) cases. Here the situation is a bit +more involved, since we must carefully analyze which types of self-duality can arise. The natural +setting for such self-dualities is over superalgebras equipped with an anti-involution a �→ a⋄. As +mentioned above in the complex setting, the self-duality also has a parity σ ∈ Z2. +To any k- +superalgebra A with anti-involution ⋄, and σ ∈ Z2, we assign a supercategory Bσ +k (A, ⋄) and quotient +supercategories Bσ +k (A, ⋄; d) for d ∈ k. When A = k and the anti-involution is trivial, B0 +k(k, id; d) +is the usual Brauer category, while B1 +k(k, id; 0) is the periplectic Brauer category. We deduce a +basis theorem (Theorem 9.5) for the morphism spaces of Bσ +k (A, ⋄; d) by embedding it into the +superadditive envelope of OBk(A). +Self-duality of a supermodule is realized by a superhermitian or skew-superhermitian form Φ. +To such a form, we can associate the supergroup G(Φ) preserving the form. We then define an +unoriented incarnation superfunctor +FΦ : Bσ +k (A, ⋄) → G(Φ)-smod. + +4 +SAIMA SAMCHUCK-SCHNARCH AND ALISTAIR SAVAGE +It turns out that only four of the ten real division superalgebras admit anti-involutions: the real +numbers, the complex numbers, the quaternions, and the two-dimensional complex Clifford super- +algebra. +Our second main result (Theorem 10.5) is that, in these cases, the functor FΦ is full. +The proof, which occupies Sections 11 to 13, is much more involved than in the oriented case. We +must treat each of the involutive division superalgebras separately, since each one behaves quite +differently. +Taking the oriented and unoriented cases together, our results handle real supergroups cor- +responding to all real forms of the general linear, orthosymplectic, periplectic, and isomeric Lie +superalgebras. (We give a classification of these real forms in Proposition B.3.) Looking at endo- +morphism algebras, as explained above, one immediately obtains analogues of Schur–Weyl duality, +or first fundamental theorems, for these real supergroups. Such results seem to be rare in the litera- +ture. (See [Cal22] for some partial results for certain real groups.) Even in the non-super setting, we +obtain new results, corresponding, for example, to the indefinite orthogonal, unitary, and symplectic +groups; see Theorem 10.7. In fact, in these cases where the module categories are semisimple, we +show that these module categories are isomorphic to quotients of the additive Karoubi envelopes of +our diagrammatic categories by tensor ideals of negligible morphisms; see Theorems 6.11 and 10.7. +These are real analogues of the some of the main results concerning Deligne’s interpolating cate- +gories in the complex case. As another application, we deduce equivalences between supercategories +of tensor supermodules over the different real forms of a complex supergroup; see Corollary 10.9 +and Propositions 11.5, 12.5 and 13.5. +Further directions. We conclude this introduction with a brief discussion of some of the future +research directions that stem from the current work. Many of these are real analogues of promising +work that has been done in the complex case. +While our results show that the oriented and unoriented incarnation functors are full, we leave +a description of the kernels of these functors, also known as the second fundamental theorem, for +future work. When the target module supercategory is semisimple, the kernel is the tensor ideal +of negligible morphisms; see Theorems 6.11 and 10.7. However, this is not the case in general. +For the usual oriented and unoriented Brauer categories, kernels have been described explicitly in +[CW12, LZ21]. +Since the target module supercategories of incarnation functors are idempotent complete, one +has induced functors +Kar(OBk(Aop; m − n)) → gl(m|n; A)-smod +and +Kar(Bσ +k (A, ⋄)) → G(Φ)-smod, +where Kar(C) denotes the additive Karoubi envelope of C. The supermodules that appear in the +image of these functors are the summands of the tensor powers of the natural module (and, in +the oriented case, its dual). It would be interesting to give a more precise description of these +supermodules. For the usual oriented and unoriented Brauer categories, results in this direction +have been obtained in [BS12, CH17, CW12, Hei17]. +The supercategories introduced here have affine analogues [MS, SS22], generalizing the affine +oriented Brauer category of [BCNR17] and the affine Brauer category of [RS19]. +These affine +supercategories act naturally on categories of supermodules over supergroups. We plan to investigate +these actions in future work. +There exist quantum analogues of the oriented and unoriented Brauer categories. These are the +framed HOMFLYPT skein and Kauffman skein categories, respectively. One has analogues of the +results mentioned above, but with supergroups replaced by quantized enveloping superalgebras. We +expect that one can also define quantum analogues of the more general supercategories introduced +in the current paper. When the ground field is R, these should be related to the representation +theory of real quantum groups. + +DIAGRAMMATICS FOR REAL SUPERGROUPS +5 +Acknowledgements. This research of A.S. was supported by NSERC Discovery Grant RGPIN- +2017-03854. We thank Jon Brundan, Inna Entova-Aizenbud, Thorsten Heidersdorf, Allan Merino, +Hadi Salmasian, Nolan Wallach, and Ben Webster for helpful discussions. +2. Monoidal supercategories +In this paper, we will work with strict monoidal supercategories in the sense of [BE17]. +In +this section, we review a few of the more important ideas that are crucial for our exposition and +somewhat less well known. Throughout this section we work over an arbitrary ground field k. +A supercategory is a category enriched in the monoidal category of superspaces and parity pre- +serving linear maps. Thus, its morphism spaces are vector superspaces and composition is parity- +preserving; that is, f ◦ g = ¯f + ¯g, where ¯f denotes the parity of f. +A superfunctor between +supercategories induces a parity-preserving linear map between morphism superspaces. For super- +functors F, G: C → D, a supernatural transformation α: F ⇒ G of parity r ∈ Z2 is a family of +morphisms αX ∈ HomD(FX, GX)r, X ∈ C, such that Gf ◦ αX = (−1)r ¯fαY ◦ Ff for each homo- +geneous f ∈ HomC(X, Y ). Note when r is odd that α is not a natural transformation in the usual +sense due to the sign. A supernatural transformation α: F ⇒ G is a sum α = α0 + α1, where αr is +a supernatural transformation of parity r. +In a strict monoidal supercategory, the super interchange law, which follows from the fact that +⊗ is a superbifunctor, is +(2.1) +(f ′ ⊗ g) ◦ (f ⊗ g′) = (−1) +¯f ¯g(f ′ ◦ f) ⊗ (g ◦ g′). +We denote the unit object by +1 and the identity morphism of an object X by 1X. We will use +the usual calculus of string diagrams, representing the tensor product f ⊗ g of morphisms f and +g diagrammatically by drawing f to the left of g, and the composition f ◦ g by drawing f above +g. Care is needed with horizontal levels in such diagrams due to the signs arising from the super +interchange law: +(2.2) +f +g += +f +g += +(−1) +¯f ¯g +f +g . +Definition 2.1. For a supercategory C, its Π-envelope Cπ is the supercategory with objects given +by formal symbols {ΠrX : X ∈ C, r ∈ Z2} and morphisms defined by +(2.3) +HomCπ(ΠrX, ΠsY ) := Πs−r HomC(X, Y ), +where, on the right-hand side, Π denotes the parity shift operator determined by (ΠV )r := Vr−1 for +a vector superspace V . The composition law in Cπ is induced in the obvious way from the one in C: +writing f s +r for the morphism in HomCπ(ΠrX, ΠsY ) of parity ¯f + r − s defined by f ∈ HomC(X, Y ), +we have that f u +s ◦ gs +r = (f ◦ g)u +r . +A Π-supercategory (D, Π, ζ) is a supercategory D, together with the extra data of a superfunctor +Π: D → D, called the parity shift, and an odd supernatural isomorphism ζ from Π to the identity +superfunctor; see [BE17, Def. 1.7]. The Π-envelope Cπ from Definition 2.1 is a Π-supercategory +with parity shift superfunctor Π: Cπ → Cπ sending object ΠrX to Πr+1X and morphism f s +r to +f s+1 +r+1. Viewing C as a full subcategory of its Π-envelope Cπ via the canonical embedding +(2.4) +C → Cπ, +X �→ Π0X, +f �→ f 0 +0, +the Π-envelope satisfies a universal property: any superfunctor F : C → D to a Π-supercategory D +extends in a canonical way to a superfunctor ˜F : Cπ → D such that ˜F ◦ Π = Π ◦ ˜F. In turn, any +supernatural transformation θ: F ⇒ G between superfunctors F, G: C → D extends in a unique +way to a supernatural transformation ˜θ: ˜F ⇒ ˜G; see [BE17, Lem. 4.2]. + +6 +SAIMA SAMCHUCK-SCHNARCH AND ALISTAIR SAVAGE +Given superalgebras A and B, the supercategory of (A, B)-superbimodules is a Π-supercategory, +as explained in [BE17, Example 1.8]. If V is an (A, B)-supermodule, we will denote its parity shift +by +(2.5) +ΠV := {πv : v ∈ V } +with +πv = v + 1. +Here π is a formal symbol to remind us that πf is an element of ΠV . In order to unify some +expressions where the parity shift may or may not be present, we define πσv, for σ ∈ Z2, by +πσv = +� +πv +if σ = 1, +v +if σ = 0. +For a morphism f : V → W, we have +Πf : ΠV → ΠW, +(Πf)(πv) = (−1) +¯fπf(v). +The isomorphism +(2.6) +Π2V +∼ += +−→ V, +ππv �→ −v +is denoted ξV in the notation of [BE17, Definition 1.7]. +If C is a supercategory, we let Add(C) denote its additive envelope. This is the supercategory +whose objects are formal finite direct sums of objects in C, and whose morphisms are identified with +matrices of morphisms in C in the usual way. The supercategory Add(Cπ), which is the additive +envelope of the Π-envelope of C, is sometimes referred to as the superadditive envelope of C. The +Karoubi envelope Kar(C) of C is the completion of its additive envelope Add(C) at all homogeneous +idempotents. Thus, objects of Kar(C) are pairs (X, e) consisting of a finite direct sum X of objects +of C together with a homogeneous idempotent e ∈ EndAdd(C)(X). Morphisms (X, e) → (Y, f) are +elements of f HomAdd(C)(X, Y )e. If C is a Π-supercategory, the parity shift superfunctors extend by +the usual universal property of Karoubi envelopes to make Kar(C) into a Π-supercategory too. +Now we consider the monoidal situation. We make the category SCat of supercategories and +superfunctors into a symmetric monoidal category following the general construction of [Kel05, +§1.4]. In particular, for supercategories C and D, their k-linear product, denoted C ⊠ D, has as +objects pairs (X, Y ) for X ∈ C and Y ∈ D, and +(2.7) +HomC⊠D((X, Y ), (X′, Y ′)) = HomC(X, X′) ⊗ HomD(Y, Y ′) +with composition defined via (2.1). A strict monoidal supercategory is a supercategory C with an +associative, unital tensor functor −⊗−: C ⊠C → C. See [BE17, Def. 1.4] for the appropriate notions +of (not necessarily strict) monoidal superfunctors between strict monoidal supercategories, and of +monoidal natural transformations between monoidal superfunctors (which are required to be even). +There is also a notion of strict monoidal Π-supercategory; see [BE17, Def. 1.12]. Such a category +is a Π-supercategory in the earlier sense with Π := π ⊗ − for a distinguished object π admitting +an odd isomorphism ζ : π +∼ +−→ +1. The Π-envelope Cπ of a strict monoidal supercategory C is the +Π-supercategory from Definition 2.1, viewed as a strict monoidal Π-supercategory with π := Π1, +tensor product of objects defined by +(2.8) +(ΠrX) ⊗ (ΠsY ) := Πr+s(X ⊗ Y ), +and tensor product (horizontal composition) of morphisms defined by +(2.9) +f s +r ⊗ gv +u := (−1)r(¯g+u+v)+ ¯fv(f ⊗ g)s+v +r+u +for homogeneous morphisms f and g in C. See [BE17, Def. 1.16] for more details and discussion of +its universal property. When working with string diagrams, we will represent the morphism f s +r in Cπ + +DIAGRAMMATICS FOR REAL SUPERGROUPS +7 +by adding horizontal lines labelled by r and s at the bottom and top of the diagram for f : X → Y : +(2.10) +f +r +s +: ΠrX → ΠsY. +Then the rules for horizontal and vertical composition in Cπ become +(2.11) +f +r +s +⊗ +g +u +v += (−1)r(¯g+u+v)+ ¯fv +f +g +r+u +s+v +, +f +s +t +◦ +g +r +s += +f +g +r +t +. +The Karoubi envelope Kar(C) of a strict monoidal Π-supercategory is a strict monoidal Π-supercategory. +If k′ is a field extension of k and C is a supercategory over k, then we define Ck′ to be the +supercategory obtained from C by extension of scalars. Precisely, the objects of Ck′ are the same of +those of C, and we have +HomCk′(X, Y ) := HomC(X, Y ) ⊗k k′, +X, Y ∈ C, +with composition extended in the natural way. Any superfunctor F : C → D naturally extends to a +superfunctor F k′ : Ck′ → Dk′. If C is a (strict) monoidal supercategory or a Π-supercategory, then +so is Ck′. In the special case where k = R, we call CC the complexification of C. +3. Superalgebras and supermodules +In this section, we review some basic properties of superalgebras and supermodules that will be +used in the current paper. We work over a ground field k. +3.1. Associative superalgebras. All vector superspaces and superalgebras are over k unless oth- +erwise indicated. We also assume that all k-supermodules are finite dimensional. We let V0 and V1 +denote the even and odd parts, respectively, of a k-supermodule V . Then its superdimension is +sdimk(V ) = dimk(V0) − dimk(V1). +We let ¯v denote the parity of a homogeneous element v of a k-supermodule V . When we write +equations involving parities of elements, we implicitly assume these elements are homogeneous; we +then extend by linearity. +The term superalgebra refers to a unital associative superalgebra. If A is a superalgebra, its +opposite superalgebra Aop = {aop : a ∈ A} has multiplication given by +aopbop = (−1)¯a¯b(ba)op. +3.2. Supermodules. Throughout this subsection, A denotes a superalgebra and V, W denote right +A-supermodules. We also let g denote a Lie superalgebra. +We let HomA(V, W) denote the k-supermodule of all (that is, not necessarily parity-preserving) +morphisms of A-supermodules. +We also define EndA(V ) := HomA(V, V ). +Thus, for example, +HomA(V, W)0 denotes the k-module of all parity-preserving A-linear maps from V to W. +For a ∈ A, define +(3.1) +ρa : V → V, +v �→ (−1)¯a¯vva. +We also define +flip: V ⊗ W → W ⊗ V, +v ⊗ w �→ (−1)¯v ¯ww ⊗ v. +If V and W are (g, A)-superbimodules, then ρa and flip are homomorphisms of g-supermodules. +Let +V ∗ = Homk(V, k) + +8 +SAIMA SAMCHUCK-SCHNARCH AND ALISTAIR SAVAGE +denote the k-dual of V . This is a left A-module with action given by +(3.2) +(af)(v) := (−1)¯a ¯ff(va). +If V is also a left g-supermodule, then V ∗ is a left g-supermodule, with action given by +(3.3) +(Xf)(v) = −(−1) +¯ +X ¯ff(Xv), +X ∈ g, f ∈ V ∗, v ∈ V. +This action supercommutes with the left A-action given in (3.2). +Let BV be a k-basis for V , which we will sometimes denote by Bk +V when there is some possibility +of confusion about the ground field. Let {v∗ : v ∈ BV } be the dual basis of V ∗ given by +v∗(w) = δvw, +v, w ∈ BV . +We have the evaluation map +ev: V ∗ ⊗ V → k, +f ⊗ v �→ f(v), +and the coevaluation map +coev: k → V ⊗ V ∗, +1 �→ +� +v∈BV +v ⊗ v∗. +The map coev is independent of the choice of basis BV . If V is a left g-supermodule, then ev and +coev are both homomorphisms of g-supermodules. +3.3. Frobenius superalgebras. We now recall some basic definitions and facts about Frobenius +superalgebras. For more details, including proofs in the super case considered here, we refer the +reader to [PS16]. A Frobenius superalgebra is a superalgebra A equipped with a parity-preserving +k-linear map τ = τA : A → k, called the Frobenius form, such that the induced bilinear form +A × A → k, +(a, b) �→ τ(ab), +is nondegenerate. If (A, τ) and (A, τ ′) are two Frobenius superalgebras with the same underlying +superalgebra A, then there exists an even invertible element u ∈ A such that +(3.4) +τ ′(a) = τ(au) +for all a ∈ A. +Every Frobenius superalgebra has a Nakayama automorphism ζ, which is an superalgebra auto- +morphism of A satisfying +(3.5) +τ(ab) = (−1)¯a¯bτ(bζ(a)) = (−1)¯aτ(bζ(a)) = (−1) +¯bτ(bζ(a)) +for all a, b ∈ A, +where the last two equalities follow from the fact that τ(ab) = 0 unless ¯a = ¯b. +A Frobenius +superalgebra is said to be supersymmetric if its Nakayama automorphism is the identity map. We +will not assume that Frobenius superalgebras are supersymmetric in this paper. We will often refer +to A itself as a Frobenius superalgebra, leaving the Frobenius form implied. Our main sources of +examples of Frobenius superalgebras will be the real division superalgebras, to be discussed in detail +in Section 4. See Examples 7.1 for additional examples. If A is a Frobenius superalgebra, then so +is Aop with Frobenius form τAop(aop) = τA(a), a ∈ A. +If BA is a homogeneous k-basis of a Frobenius superalgebra A, we let B∨ +A := {b∨ : b ∈ BA} be +the left dual basis, defined by +τ(b∨c) = δbc, +b, c ∈ BA. +It follows that, for all a ∈ A, we have +(3.6) +a = +� +b∈BA +τ(b∨a)b = +� +b∈BA +τ(ab)b∨. +We also have +(3.7) +b∨ = ¯b +for all b ∈ BA. + +DIAGRAMMATICS FOR REAL SUPERGROUPS +9 +and +(3.8) +� +b∈Bk +A +b ⊗ b∨ is independent of the choice of basis BA. +The basis left dual to {ζ(b) : b ∈ BA} is given by +(3.9) +ζ(b)∨ = ζ(b∨) +for all b ∈ BA. +If V is a right A-supermodule, then the supertrace of the action of a on V is +(3.10) +strV (a) = strk +V (a) := +� +v∈BV +(−1)¯vv∗(va), +where BV is a k-basis of V and {v∗ : v ∈ BV }, is the dual basis of V ∗ +k . We use the superscript k on +strk +V (a) when there is the possibility of confusion about the ground field. We have +strV (ab) = (−1)¯a¯b strV (ba) +for all a, b ∈ A. +It is clear that +(3.11) +strV (a) = (m − n) strA(a) +for V = Am|n. +Lemma 3.1. If (A, τ) is a Frobenius superalgebra, then +(3.12) +strA(a) = strA(ζ(a)) = +� +b∈BA +(−1) +¯bτ(b∨ba) = +� +b∈BA +(−1) +¯bτ(ab∨b), +a ∈ A. +Proof. For all b ∈ BA, we have +τ(b∨a) = b∗(a). +It follows immediately that strA(a) is equal to the first sum in (3.12). +Next, note that τ(c) = τ(ζ(c)) for all c ∈ A. Thus, for all a ∈ A, +strA(a) = +� +b∈BA +(−1) +¯bτ(b∨ba) = +� +b∈BA +(−1) +¯bτ +� +ζ(b∨)ζ(b)ζ(a) +� (3.5) += +� +b∈BA +(−1) +¯bτ +� +aζ(b∨)ζ(b) +� += +� +b∈BA +(−1) +¯bτ +� +ab∨b +� (3.5) += +� +b∈BA +(−1) +¯b(b∨bζ(a)) = strA(ζ(a)). +where, in the fourth equality we changed to a sum over the basis {ζ(b) : b ∈ BA} and used (3.9). +□ +3.4. Supermatrices. We will use the term supermatrix to denote a supermatrix with entries in a +superalgebra A. We let Matp|q,r|s(A) denote the k-supermodule of (p|q) × (r|s) supermatrices, and +set Matp|q(A) := Matp|q,p|q(A). We write a supermatrix X ∈ Matp|q,r|s(A) in block form as +X = +� +X00 +X01 +X10 +X11 +� +, +where X00 is p × r, X01 is p × s, X10 is q × r and X11 is q × s. The even elements of Matp|q,r|s(A) +are those supermatrices X where X00, X11 have even entries and X01, X10 have odd entries. The +odd elements of Matp|q,r|s(A) are those supermatrices X where X00, X11 have odd entries and X01, +X10 have even entries. +We view elements of Am|n as column supermatrices, that is, as (m|n) × (1, 0) supermatrices. +Similarly, we view elements of A as (1|0) × (1|0) supermatrices. Then the right action (v, a) �→ va +of A on Am|n can be viewed as matrix multiplication and we can identify elements of EndA(Am|n) +with Matm|n(A) in the usual way. + +10 +SAIMA SAMCHUCK-SCHNARCH AND ALISTAIR SAVAGE +The supertranspose of a supermatrix is given by +(3.13) +Xst := +� +Xt +00 +(−1) ¯ +XXt +10 +−(−1) ¯ +XXt +01 +Xt +11 +� +, +where Xt denotes the usual transpose of a matrix. Note that +(3.14) +(Xst)st = +� +X00 +−X01 +−X10 +X11 +� +, +so that the supertranspose has order four in general. +The supertrace of a square supermatrix +X ∈ Matp|q(A) is given by +str(X) = tr(X00) − (−1) +¯ +X tr(X11), +where tr denotes the usual matrix trace. +For a supermatrix X ∈ Matp|q,r|s(A), let Xop ∈ Matp|q,r|s(Aop) denote the matrix obtained from +X by replacing each entry a by aop. Then define Xst +op := (Xop)st = (Xst)op. +Lemma 3.2. We have an isomorphism of k-superalgebras +(3.15) +Matm|n(A)op ∼ += +−→ Matm|n(Aop), +Xop �→ Xst +op. +Proof. The map is clearly an isomorphism of k-vector superspaces. +It is also a straightforward +computation to prove that it respects multiplication. +□ +Lemma 3.3. If A is a superalgebra, then +(3.16) +strk +Matm|n(A)(X) = (m − n) strk +A ◦ str(X) +for all X ∈ Matm|n(A). +Proof. Choose the basis {Ersb : 1 ≤ r, s ≤ m + n, b ∈ BA} of Matm|n(A), where Ers denotes the +matrix with a 1 in position (r, s), and a 0 in all other positions. The dual basis of Matm|n(A)∗ is +given by +(Ersb)∗(X) = (−1)p(s)b∗(str(EsrX)). +Then we have +strk +Matm|n(A)(X) = +m+n +� +r,s=1 +� +b∈BA +(−1) +¯b+p(r)b∗(str(EsrErsbX)) = (m − n) +� +b∈BA +(−1) +¯bb∗(str(bX)) += (m − n) +� +b∈BA +(−1) +¯bb∗(b str(X)) = (m − n) strk +A ◦ str(X). +□ +3.5. Lie superalgebras. If g is a Lie superalgebra over k, we let g-smodk denote the supercategory +of finite-dimensional g-supermodules over k with arbitrary (i.e. not necessarily parity-preserving) +homomorphisms. We will be particularly interested in the cases where k is R or C. If k = R, our +notation g-smodR is designed to emphasize that we are speaking of real supermodules, as opposed +to complex supermodules. +The general linear Lie superalgebra gl(VA) associated to a right A-supermodule V is equal to +EndA(V ) as a k-supermodule, with Lie superbracket given by +(3.17) +[X, Y ] := XY − (−1) +¯ +X ¯Y Y X. +In the special case that V = Am|n, we introduce the notation gl(m|n, A) = gl(VA). Identifying +EndA(V ) with Matm|n(A) in the usual way, we have that gl(m|n, A) is equal to Matm|n(A) as a k- +vector superspace, with Lie superbracket given by (3.17). By convention, we define gl(0|0, A) to be +the zero Lie superalgebra, so that gl(0|0, A)-smodk is the supercategory k-smod of k-supermodules. + +DIAGRAMMATICS FOR REAL SUPERGROUPS +11 +Corollary 3.4. We have an isomorphism of Lie k-superalgebras +gl(m|n, A) +∼ += +−→ gl(m|n, Aop), +X �→ −Xst +op. +Proof. The given map is clearly an isomorphism of k-vector spaces. To verify that it also respects +the Lie superbracket, we compute +[−Xst +op, −Y st +op] = Xst +opY st +op − (−1) +¯ +X ¯Y Y st +opXst +op +(3.15) += +(−1) +¯ +X ¯Y (Y X)st +op − (XY )st +op = −[X, Y ]st +op. +□ +3.6. Harish–Chandra superpairs. We will sometimes need to work with supergroups instead +of Lie superalgebras. We review here some basic facts, referring the reader to [DM99] for a more +detailed overview. Instead of directly working with supergroups, it will be simpler to work with +the equivalent category of Harish-Chandra superpairs. +We refer the reader to [Gav20] and the +references cited therein for a proof of this equivalence. A Harish-Chandra superpair over k is a pair +G = (Gred, g), where Gred is an algebraic group over k, g is a Lie superalgebra over k, +• g0 is the Lie algebra of G, +• Gred acts algebraically on g by k-linear transformations, and +• the differential of the action of Gred on g coincides with the action of g0 on g via the +superbracket. +Suppose G = (Gred, g) is a Harish-Chandra superpair. A G-supermodule is a k-supermodule V +that is both a Gred-supermodule and a g-supermodule, and such that the differential of the action +of Gred coincides with the action of g0. The finite-dimensional G-supermodules form a monoidal +supercategory, which we denote by G-smodk. +For G-supermodules V and W, we have +HomG(V, W) = HomGred(V, W) ∩ Homg(V, W). +If Gred is connected, then Homg0(V, W) = HomGred(V, W), and so HomG(V, W) = Homg(V, W). +In this case, the forgetful superfunctor G-smodk → g-smodk is full and faithful. +On the other +hand, if g1 = 0, so that we are working in the purely even setting, then the forgetful functor +G-modk → Gred-modk is full and faithful. (In fact, it is an equivalence of categories.) In this case, +we will often identity G and Gred. +On the other hand, suppose Gred has r+1 connected components, and let X1, . . . , Xr be elements +of Gred, one from each of the r connected components not containing the identity. Then Gred = +H ∪ HX1 ∪ · · · HXr, where H is the identity component of Gred, and we have +(3.18) +HomG(V, W) = HomX1,...,Xr,g(V, W) +:= {f ∈ Homg(V, W) : f(Xtv) = Xtf(v) ∀ v ∈ V, 1 ≤ t ≤ r}. +For example, if k = C, and Gred = O(m, C) is the complex orthogonal group, then we have +HomG(V, W) = HomX,g(V, W), +where X is any element of O(m, C) with det(X) = −1. +3.7. Complexification. If V is a real vector superspace, its complexification is +V C := V ⊗R C. +We view V as an R-vector subspace of V C by identifying v ∈ V with v⊗1. If A is a real superalgebra, +then AC is a complex superalgebra, with product +(a ⊗ y)(b ⊗ z) = ab ⊗ yz, +a, b ∈ A, y, z ∈ C. +Similarly, if g is a Lie superalgebra over R, then its complexification gC is a Lie superalgebra over C. +A real form of a complex vector superspace W is a real vector superspace V such that V C ∼= W as + +12 +SAIMA SAMCHUCK-SCHNARCH AND ALISTAIR SAVAGE +complex vector superspaces. We define real forms of complex associative superalgebras and complex +Lie superalgebras similarly. +Suppose R is either a real associative superalgebra or a real Lie superalgebra. If V is a left +(respectively, right) R-supermodule, then V C is a left (respectively, right) RC-supermodule with the +natural action. Furthermore, every f ∈ HomR(V, W) induces an element f C ∈ HomRC(V C, W C) +given by +f C(v ⊗ z) = f(v) ⊗ z, +v ∈ V, z ∈ C. +We have +ker +� +f C� += ker(f)C, +ker(f) = ker +� +f C� +∩ V, +(3.19) +im +� +f C� += im(f)C, +im(f) = im +� +f C� +∩ V. +(3.20) +In particular +f is injective ⇐⇒ f C is injective +and +f is surjective ⇐⇒ f C is surjective. +The above constructions yield a superfunctor R-smod → RC-smod, which induces a full and faithful +complexification superfunctor +(3.21) +CR : (R-smod)C → RC-smod. +In particular, if g is a real Lie superalgebra and V, W ∈ g-smodR, then we have a canonical isomor- +phism of C-supermodules +(3.22) +Homg(V, W)C ∼ += +−→ HomgC(V C, W C). +If H is a real supergroup acting on a real vector space V , then H also acts on V C = V ⊗R C +by acting on the first factor. If V and W are supermodules over a real Harish-Chandra superpair +G = (Gred, g), then we have an isomorphism of C-supermodules +HomG(V, W)C ∼= HomGred,gC(V C, W C) +:= {X ∈ HomgC(V C, W C) : f(Xv) = Xf(v) ∀ X ∈ Gred, v ∈ V C}. +If Gred has r + 1 connected components, and X1, . . . , Xr are elements of Gred, one from each +connected component not containing the identity, then, using (3.18), we have +(3.23) +HomG(V, W)C ∼= HomX1,...,Xr,gC(V C, W C) +:= {f ∈ HomgC(V C, W C) : f(Xtv) = Xtf(v) ∀ v ∈ V C, 1 ≤ t ≤ r}. +If A is a Frobenius R-superalgebra with Frobenius form τ, then its complexification AC is a +Frobenius C-superalgebra with Frobenius form (which we continue to denote by the same symbol) +(3.24) +τ : AC → C, +a ⊗ z �→ τ(a)z, +a ∈ A, z ∈ C. +It is straightforward to verify that +(3.25) +Matm|n(A)C ∼= Matm|n(AC) +as C-superalgebras and +(3.26) +gl(m|n, A)C ∼= gl(m|n, AC) +as complex Lie superalgebras. + +DIAGRAMMATICS FOR REAL SUPERGROUPS +13 +4. Real division superalgebras +In this section, we discuss real division superalgebras. These will play a key role in our main +applications to the representation theory of real supergroups. +4.1. Real division superalgebras. For our purposes, one of the most important classes of exam- +ples of Frobenius superalgebras are the real division superalgebras, which were classified by Wall +[Wal64]. (See also [Bae20] for a short exposition.) +Proposition 4.1. Every real division superalgebra is isomorphic to exactly one of the following, +where the Z2-grading is given by declaring ε to be odd, and ⋆ denotes complex conjugation. +• Cl0(R) = R; +• Cl1(R) := R ⊕ εR, with ε2 = 1; +• Cl2(R) := C ⊕ εC, with ε2 = 1 and zε = εz⋆ for all z ∈ C; +• Cl3(R) := H ⊕ εH, with ε2 = −1 and zε = εz for all z ∈ H; +• Cl4(R) := H; +• Cl5(R) := H ⊕ εH, with ε2 = 1 and zε = εz for all z ∈ H; +• Cl6(R) := C ⊕ εC, with ε2 = −1 and zε = εz⋆ for all z ∈ C; +• Cl7(R) := R ⊕ εR, with ε2 = −1; +• C; +• Cl(C) := C ⊕ εC, with ε2 = 1 and zε = εz for all z ∈ C. +For 0 ≤ r ≤ 8, we have Clr(R)op ∼= Cl−r(R) as superalgebras, where subscripts are considered +modulo 8. +The notation in Proposition 4.1 is inspired by the fact that Clr(R)⊗RCls(R) is Morita equivalent +to Clr+s(R). Note that C and the complex Clifford algebra Cl(C) are the only complex division +superalgebras. The Clr(R), 0 ≤ r ≤ 7, are real Clifford superalgebras. Recall that a k-superalgebra +is central if its center is k. Thus, the central real division superalgebras are those real division +superalgebras isomorphic to Clr(R) for 0 ≤ r ≤ 7. +Remark 4.2. The complex division superalgebras are isomorphic to their own opposite superalge- +bras. For C, this follows from the fact that C is commutative. For Cl(C), we have Cl(C)op = C⊕εC, +with ε2 = −1, and an isomorphism of C-superalgebras +Cl(C) +∼ += +−→ Cl(C)op, +ε �→ εi. +Convention 4.3 (Frobenius forms on division superalgebras). Note that C and Cl(C) are complex +Frobenius superalgebras, with Frobenius form given by projection projC onto their even part. (They +are also real Frobenius superalgebras with Frobenius form given by projection onto the real part of +their even part.) We will always view the central real division superalgebras as superalgebras over +R. They are real Frobenius superalgebras with Frobenius form Re: D → R given by taking the real +part of the even part of D. In all of these cases, the Nakayama automorphism is given by +(4.1) +ζ(a) = (−1)¯aa, +a ∈ D. +In particular, a real division superalgebra is supersymmetric if and only if it is purely even. +Lemma 4.4. If k ∈ {R, C} and BD is a k-basis for an involutive division k-superalgebra D, then +the basis left dual to B∨ +D = {b∨ : b ∈ BD} is given by +(4.2) +(b∨)∨ = b, +b ∈ BD. +Proof. For all b, c ∈ BD, we have +τ(cb∨) +(3.5) += (−1) +¯b¯cτ(b∨ζ(c)) +(4.1) += (−1) +¯b¯c+¯cτ(b∨c) = δbc. +□ + +14 +SAIMA SAMCHUCK-SCHNARCH AND ALISTAIR SAVAGE +If D is a real division superalgebra, we will use the term D-vector superspace to denote a finite- +dimensional right D-supermodule. Recall the definition of strA given in (3.10). +Lemma 4.5. If D is a real or complex division superalgebra with standard Frobenius form τ, then +strD = (sdimk D)τ. In particular, strD = 0 whenever D is a real or complex division superalgebra +with nonzero odd part. +Proof. If D is a real division superalgebra, choose the basis +BD = + + + + + + + + + + + + + + + + + + + +{1} +if D = R, +{1, i} +if D = C, +{1, i, j, k} +if D = H, +{1, ε} +if D ∈ {Cl1(R), Cl7(R)}, +{1, i, ε, εi} +if D ∈ {Cl2(R), Cl6(R)}, +{1, i, j, k, ε, εi, εj, εk} +if D ∈ {Cl3(R), Cl5(R)}. +If D = C, considered as a complex division superalgebra choose BD = {1}. Finally, if D = Cl(C), +considered as a complex division superalgebra, choose BD = {1, ε}. Then we have b∨ = b−1 for all +b ∈ BD. Hence, using (3.12), we have +strD(a) = +� +b∈BD +(−1) +¯bτ(a) = (sdimk D)τ(a). +□ +4.2. Complexification of real division superalgebras. +Lemma 4.6. We have the following injections of superalgebras, where ⋆ denotes complex conjuga- +tion, +R ֒→ C, +a �→ a, +a ∈ R, +(4.3) +ı: H ֒→ Mat2(C), +i �→ +� +i +0 +0 +−i +� +, +j �→ +� +0 +−1 +1 +0 +� +, +k �→ +� +0 +−i +−i +0 +� +, +(4.4) +Cl1(R) ֒→ Cl(C), +a + εb �→ a + εb, +a, b ∈ R, +(4.5) +Cl2(R) ֒→ Mat1|1(C), +a + εb �→ +� +a +b⋆ +b +a⋆ +� +, +a, b ∈ C, +(4.6) +Cl3(R) ֒→ Mat2(Cl(C)), +a + εb �→ ı(a) + εı(b)i, +a, b ∈ H, +(4.7) +Cl5(R) ֒→ Mat2(Cl(C)), +a + εb �→ ı(a) + εı(b), +a, b ∈ H, +(4.8) +Cl6(R) ֒→ Mat1|1(C), +a + εb �→ +� +a +−b⋆ +b +a⋆ +� +, +a, b ∈ C, +(4.9) +Cl7(R) ֒→ Cl(C), +a + εb �→ a + εib, +a, b ∈ R. +(4.10) +Proof. These are all straightforward verifications. +□ +Remark 4.7. We have an injection of complex superalgebras +Cl(C) ֒→ Mat2(C), +a + εb �→ +� +a +b +b +a +� +, +a, b ∈ C. +Combined with Lemma 4.6, this shows that all of the real division superalgebras can be embedded +in complex supermatrix superalgebras. +The following result gives the complexification of the central real division superalgebras. + +DIAGRAMMATICS FOR REAL SUPERGROUPS +15 +Lemma 4.8. The inclusions of Lemma 4.6 induce isomorphisms of complex superalgebras +RC ∼= C, +HC ∼= Mat2(C), +Cl1(R)C ∼= Cl7(R)C ∼= Cl(C), +Cl2(R)C ∼= Cl6(R)C ∼= Mat1|1(C), +Cl3(R)C ∼= Cl5(R)C ∼= Mat2(Cl(C)). +In particular, for every central real division superalgebra D, its complexification DC is a simple +complex superalgebra. +Proof. It is straightforward to verify that, for each of the injections in Lemma 4.6, every matrix in +the codomain can be written uniquely in the form X + iY , where X and Y are in the image of the +injection. +□ +4.3. General linear Lie superalgebras over division superalgebras. +Lemma 4.9. If D is a real division superalgebra with D1 ̸= 0, then +(a) Dm|n ∼= Dm+n as D-vector superspaces, +(b) Matm|n(D) ∼= Matm+n(D) as superalgebras, +(c) gl(m|n, D) ∼= gl(m + n, D) as Lie superalgebras. +Proof. Suppose D is a real division superalgebra with D1 ̸= 0. Then left multiplication by any +nonzero odd element gives an isomorphism D0|n ∼= Dn|0. Hence Dm|n ∼= Dm|0 ⊕D0|n ∼= Dm|0 ⊕Dn|0 ∼= +Dm+n. This induces isomorphisms of superalgebras +Matm+n(D) ∼= EndD(Dm|n) ∼= EndD(Dm+n) ∼= Matm+n(D). +Passing to the associated Lie superalgebras then gives the isomorphism gl(m|n, D) ∼= gl(m + n, D). +□ +In light of Lemma 4.9, we will often consider only gl(m, D), as opposed to gl(m|n, D), when +D1 ̸= 0. +Remark 4.10. The general linear superalgebras over the central real division superalgebras are +often known by different names and notation. +• gl(m, Cl1(R)) is the split real isomeric Lie superalgebra, often called the split real queer Lie +superalgebra. It is usually denoted q(m, R). +• gl(m, Cl(C)) is the complex isomeric Lie superalgebra, often called the complex queer Lie +superalgebra. It is usually denoted q(m, C). +• gl(m, Cl2(R)) is sometimes denoted q0(m, R). +• gl(m, Cl5(R)) is sometimes denoted q∗(2m). +• gl(m|n, H) is sometimes denoted u∗(2m|2n). +(Recall that gl(m|n, D) ∼= gl(m|n, Dop) by Corollary 3.4.) Many references focus on the realization +of real Lie superalgebras in terms of complex matrices, using the inclusions of Lemma 4.6 and Re- +mark 4.7. However, we feel that the realization in terms of general linear Lie superalgebras over real +division superalgebras is more natural and leads to more uniform and easy-to-understand notation. +We will only use the notation q(m, C) in the complex case: +q(m, C) = gl(m, Cl(C)). +The following proposition shows that the general linear Lie superalgebras over central real division +superalgebras are real forms of general linear and isomeric Lie superalgebras. +Proposition 4.11. We have isomorphisms of complex Lie superalgebras +(a) gl(m|n, R)C ∼= gl(m|n, C), +(b) gl(m|n, H)C ∼= gl(2m|2n, C), + +16 +SAIMA SAMCHUCK-SCHNARCH AND ALISTAIR SAVAGE +(c) gl(m, Cl1(R))C ∼= gl(m, Cl7(R))C ∼= q(m, C), +(d) gl(m, Cl2(R))C ∼= gl(m, Cl6(R))C ∼= gl(m|m, C), +(e) gl(m, Cl3(R))C ∼= gl(m, Cl5(R))C ∼= q(2m, C). +Proof. This follows from (3.26), Lemma 4.8, and the fact that we have canonical isomorphisms of +complex Lie superalgebras +(4.11) +gl(m|n, Matr|s(A)) ∼= gl((mr + ns)|(ms + nr), A) +for any superalgebra A. +□ +5. The oriented supercategory +In this section, we introduce the first of our two main diagrammatic supercategories. +After +defining the supercategory, we prove a basis theorem for morphism spaces. In Sections 5.1 and 5.2, +k denotes an arbitrary field. In Section 5.3, we discuss the special cases k ∈ {R, C}. +5.1. Definition of the supercategory. +Definition 5.1 ([MS, Def. 4.1]). For an associative superalgebra A, we define OBk(A) to be the +strict monoidal supercategory generated by objects ↑ and ↓ and morphisms +: ↑ ⊗ ↑ → ↑ ⊗ ↑ , +a : ↑ → ↑ , a ∈ A, +: ↓ ⊗ ↑ → +1, +: +1 → ↑ ⊗ ↓, +: ↑ ⊗ ↓ → +1, +: +1 → ↓ ⊗ ↑, +subject to the relations +1 = , +λ +a + µ +b = +λa+µb , +b +a += +ab , +(5.1) += +, += +, +a += +a , +(5.2) += +, += +, += += +, +(5.3) += +, += +, +(5.4) +for all a, b ∈ A and λ, µ ∈ k. In the above, the left and right crossings are defined by +(5.5) +:= +, +:= +. +The parity of +a is ¯a, and all the other generating morphisms are even. We refer to the morphisms +a as tokens. +For d ∈ k, we define OBk(A; d) to be the quotient of OBk(A) by the relations +(5.6) +a = d strA(a)1 +1, +a ∈ A, +where strA is given by (3.10). We call d the specialization parameter. +When A is a Frobenius superalgebra, OBk(A) was called the oriented Frobenius Brauer supercat- +egory in [MS, Def. 4.1]. The Frobenius structure on A allows one to enlarge it to the affine oriented +Frobenius Brauer category of [MS, Def. 4.3], which is the central charge zero special case of the +Frobenius Heisenberg supercategory introduced in [Sav19], and further studied in [BSW21, MS]. We +refer the reader to these papers for proofs omitted here, none of which use the Frobenius structure +on A. Our presentation of OBk(A) is slightly different from the one given in [MS, Def. 4.1]. Pre- +cisely, the relations (5.4) are the reflections in the vertical axis of the ones in [MS, (4.4)]. However, +OBk(A) has a symmetry given by reflecting diagrams in the vertical axis. (This is the composition +of the isomorphisms (5.16) and (5.17) in [BSW21].) Hence, the two definitions are equivalent. + +DIAGRAMMATICS FOR REAL SUPERGROUPS +17 +Remark 5.2. +(a) When A = k, we have +a += a +for all a ∈ k. +Thus, we can omit the +generators +a and all the relations involving them. Then we see that OBk(k) is the oriented Brauer +category, which is the free rigid symmetric k-linear monoidal category generated by a single object. +This is the motivation for the notation OBk(A). +(b) The supercategory OBC(Cl(C), 0) is the oriented Brauer–Clifford supercategory introduced +in [BCK19, Def. 3.2]. +Remark 5.3. When A is a real or complex division superalgebra with nonzero odd part, it follows +from Lemma 4.5 that OBk(A; d) = OBk(A; 0) for all d ∈ R. +The relations (5.4) means that ↓ is left dual to ↑. In fact, we also have +(5.7) += +, += +, +and so ↓ is also right dual to ↑. Thus OBk(A) is rigid. Furthermore, we have that +(5.8) +:= += +, +a +:= +a += +a , +a ∈ A. +These relations mean that tokens and crossings slide over all cups and caps in the sense that, for +all orientations of the strands, we have +(5.9) +a += +a , +a += +a , += +, += +. +More precisely, the cups and caps equip OBk(A) with the structure of a strict pivotal supercategory; +see [BSW21, (5.16)]. It follows from the definition of the tokens on downward strands that +b +a += (−1)¯a¯b +ba . +We also have +(5.10) += +, += +, += +, +and += +. +5.2. The basis theorem. We now describe bases for the morphism spaces of OBk(A). Let X = +X1 ⊗ · · · ⊗ Xr and Y = Y1 ⊗ · · · ⊗ Ys be objects of OBk(A) for Xt, Yt ∈ {↑, ↓}. An (X, Y )-matching +is a bijection between the sets +(5.11) +{t : Xt = ↑} ⊔ {t : Yt = ↓} +and +{t : Xt = ↓} ⊔ {t : Yt = ↑}. +A reduced lift of an (X, Y )-matching is a string diagram representing a morphism X → Y such that +• the endpoints of each string are points which correspond under the given matching; +• there are no floating bubbles (i.e. strings with no endpoints) and no tokens on any string; +• there are no self-intersections of strings and no two strings cross each other more than once. +Fix a set ⃗D(X, Y ) consisting of a choice of reduced lift for each (X, Y )-matching. Then let ⃗D•(X, Y ) +denote the set of all morphisms that can be obtained from the elements of ⃗D(X, Y ) by adding one +token to each string according to the following convention. +Convention 5.4. Tokens are placed such that: +• each token is labelled by an element of BA; +• if a string has endpoints at the top and bottom of the diagram, then its token appears near +the bottom of the string (below all crossings); +• if a string has both endpoints at the top of the diagram, then its token appears near the left +endpoint (above all crossings); +• if a string has both endpoints at the bottom of the diagram, then its token appears near the +right endpoint (below all crossings); + +18 +SAIMA SAMCHUCK-SCHNARCH AND ALISTAIR SAVAGE +• all tokens near top endpoints are at the same height, all tokens near bottom endpoints are +at the same height, and the tokens near top endpoints are above the tokens near bottom +endpoints. +For example, for X = ↓ ⊗ ↑ ⊗ ↓ ⊗ ↓ ⊗ ↑ and Y = ↓ ⊗ ↓ ⊗ ↓ ⊗ ↑ ⊗ ↓ ⊗ ↑ ⊗ ↑, +is a possible element of ⃗D(X, Y ) and +b1 +b2 +b3 +b4 +b5 +b6 +, +b1, b2, b3, b4, b5, b6 ∈ BA, +are the corresponding elements of ⃗D•(X, Y ). +While we expect the following theorem to hold for an arbitrary associative superalgebra A, our +proof assumes that A is a Frobenius superalgebra. As explained in Convention 4.3, this assumption +holds whenever A is a real or complex division superalgebra. +Theorem 5.5. Let d ∈ k. For X, Y ∈ OBk(A), the morphism space HomOBk(A;d)(X, Y ) is a free +k-supermodule with basis ⃗D•(X, Y ). +Proof. The supercategory OBk(A) is a sub-supercategory of the affine Frobenius Brauer supercat- +egory AOB(A) defined in [MS, Def. 4.3]. The latter supercategory is the central charge k = 0 case +of the Frobenius Heisenberg category introduced in [Sav19] and further studied in [BSW21]. Thus, +the assertion follows from the basis theorem [MS, Th. 4.7] for AOB(A), which is a special case of +the basis theorem [BSW21, Th. 7.2] for Frobenius Heisenberg categories. +In [BSW21, Th. 7.2], the basis elements carry tokens near the terminus of each strand, which +differs from the placement of tokens in the elements of the ⃗D•(X, Y ). However, it follows from the +relations in OBk(A) that this difference in placement changes the corresponding diagrams by at +most a sign. In addition, [BSW21, Th. 7.2] assumes the Frobenius superalgebra is supersymmetric. +However, the same proof given there works without this assumption, using the defining property of +the Nakayama automorphism wherever supersymmetry is needed, and tracking these applications +throughout the calculations. (See, for example, [Sav19], which works in this generality.) +□ +5.3. Complexifications. Our proof of fullness of the oriented incarnation superfunctor when k = R +and A is a real division superalgebra (Theorem 6.10) will involve the complexification of OBk(A). +In this subsection we state some results about this complexification that we will need. +Proposition 5.6. For any superalgebra A over k = R, and d ∈ R, there are isomorphisms of +monoidal supercategories +R: OBR(A)C ∼ += +−→ OBC(AC) +and +R: OBR(A; d)C ∼ += +−→ OBC(AC; d), +given on objects by ↑ �→ ↑, ↓ �→ ↓ and on morphisms by +�→ +, +�→ +, +�→ +, +�→ +, +�→ +, +a �→ +a⊗1 , +a ∈ A. +Proof. It is clear that the superfunctor R is well-defined. The inverse functor is given on morphisms +by +�→ +, +�→ +, +�→ +, +�→ +, +�→ +, +a⊗z �→ +� +a � +⊗ z, +a ∈ A, z ∈ C. □ + +DIAGRAMMATICS FOR REAL SUPERGROUPS +19 +By Proposition 5.6 and Lemma 4.8, the complexifications OBR(D)C and OBR(D; d)C, where D is a +central real division superalgebra, are related to OBC(R), where R is a supermatrix superalgebra over +a complex division superalgebra A. The following result, which is formulated more generally, relates +these to OBC(A). Recall, from Section 2, the superadditive envelope Add(Cπ) of a supercategory C. +We write morphisms in superadditive envelopes as sums of their components. +Proposition 5.7. For any superalgebra A and r, s ∈ N, r + s ≥ 1, there is a unique monoidal +superfunctor +M: OBk(Matr|s(A)) → Add(OBk(A)π) +given on objects by ↑ �→ ↑⊕r ⊕ Π↑⊕s, ↓ �→ ↓⊕r ⊕ Π↓⊕s, and on morphisms by +�→ +r+s +� +t,u=1 +(−1)p(t)p(u) +t +t +u +u +p(t)+p(u) +p(t)+p(u) , +Etua �→ +u +t p(t) +p(u) +a +, +�→ +r+s +� +t=1 +t t +0 +0 , +�→ +r+s +� +t=1 +t t 0 +0 , +�→ +r+s +� +t=1 +(−1)p(t) t t +0 +0 , +�→ +r+s +� +t=1 +(−1)p(t) +t t 0 +0 , +where +(5.12) +p(t) = +� +0 +if 1 ≤ t ≤ r, +1 +if r < t ≤ r + s. +This superfunctor is full and faithful. For d ∈ k, it induces equivalences of monoidal supercategories +Add +�OBk(Matr|s(A))π +� ∼ += +−→ Add(OBk(A)π), +Add +�OBk(Matr|s(A); d)π +� ∼ += +−→ Add(OBk(A; (r − s)d)π). +Proof. We first consider the non-specialized supercategories. To prove that M is well defined, we +must show that it respects the relations of Definition 5.1. These are all straightforward verifications, +which we leave to the reader. (See the proof of Theorem 9.5 for the details of a similar, but slightly +less straightforward, verification.) +Next we prove that M is full and faithful. +Suppose X1, . . . , Xv, Y1, . . . , Yw ∈ {↑, ↓}, and let +X = X1 ⊗ · · · ⊗ Xv, Y = Y1 ⊗ · · · ⊗ Yw. Then M induces a k-linear map +(5.13) +HomOBk(Matr|s(A))(X, Y ) → +r+s +� +t1,...,tv,u1,...,uw=1 +HomOBk(A) +� +Πp(t1)+···+p(tv)X, Πp(u1)+···+p(uw)Y +� +. +It suffices to assume that v + w is even and +#{a : Xa = ↑} + #{a : Ya = ↓} = v + w +2 += #{a : Xa =↓} + #{a : Ya = ↑}, +otherwise both the domain and image of (5.13) have dimension zero. +(Here, #S denotes the +cardinality of a set S.) By Theorem 5.5, +dimk HomOBk(Matr|s(A))(X, Y ) = +� v+w +2 +� +! +� +(r + s)2 dimk A +�(v+w)/2 . +We use here the fact there the number of (X, Y )-matchings is (v+w +2 )!, and that dimk Matr|s(A) = +(r + s)2 dimk A. On the other hand, Theorem 5.5 implies that the codomain of the map (5.13) has +the same dimension. Thus, it suffices to prove that the map (5.13) is surjective. This follows from +the fact that any string diagram in the summand +HomOBk(A) +� +Πp(t1)+···+p(tv)X, Πp(u1)+···+p(uw)Y +� + +20 +SAIMA SAMCHUCK-SCHNARCH AND ALISTAIR SAVAGE +is the image under (5.13) (up to a sign) of the same diagram with appropriate tokens +Etu placed +near the endpoints of strands. +Finally, we show that M is essentially surjective. This follows from the fact that the generating +objects ↑ and ↓ of OBk(A) are the images of (↑, +E11 ) and (↓, +E11 ), respectively, if m ≥ 1, and the +images of (Π↑, +E11 ) and (Π↓, +E11 ), respectively, if m = 0. +It remains to prove the statement about the specialized supercategories. For 1 ≤ t, u ≤ r + s +and a ∈ A, we have +M +� +Etua +� += M( +) ◦ M( ⊗ +Etua ) ◦ ( +) +(2.11) += +δtu(−1)p(t)+p(t)¯a +a +0 +0 += δtu(−1)p(t)(r − s)d strk +A(a) = (r − s)d strk +A ◦ str(Etua) +(3.16) += +d strk +Matr|s(A)(Etua), +where, in the third equality, we used the fact that strk +A(a) = 0 unless ¯a = 0. +□ +6. The oriented incarnation superfunctor +In this section we introduce the main application of the supercategory OBk(A) to the repre- +sentation theory of Lie superalgebras. We begin by defining a very general oriented incarnation +superfunctor. We then turn our attention to the special cases where k ∈ {R, C} and A is a division +superalgebra over k. When k = C, fullness of the incarnation functor follows from known results. +When k = R, we give a proof of fullness using the complexification of the supercategories involved. +6.1. Definition of the superfunctor. Throughout this subsection, k denotes an arbitrary field. +Recall the maps flip, ev, coev, and ρa from Section 3.2. The following result is the main motivation +for the definition of the supercategory OBk(A). +Theorem 6.1. Suppose that A is an associative superalgebra, g is a Lie superalgebra, and V is +a (g, A)-superbimodule. There exists a unique monoidal superfunctor, which we call the oriented +incarnation superfunctor, +G = GV : OBk(Aop) → g-smodk. +such that G(↑) = V , G(↓) = V ∗, and +(6.1) +G( +) = flip, +G( +) = ev, +G( +aop ) = ρa, +a ∈ A. +This superfunctor also satisfies the following: +G( +) = coev, +G( +) = ev ◦ flip, +G( +) = flip ◦ coev, +(6.2) +G +� +a � += strV (a), +a ∈ A. +(6.3) +If V = Am|n and g = gl(m|n, A) for some m, n ∈ N, then GV induces a monoidal superfunctor +Gm|n : OBk(Aop; m − n) → gl(m|n, A)-smodk. +Proof. We first show that (6.1) and (6.2) indeed yield a superfunctor G. We must show that it +respects the relations (5.1) to (5.4). The first two relations in (5.1) are straightforward. For the +third relation in (5.1), we have +G +� +bop +aop � +(v) = (−1)(¯a+¯b)¯v+¯a¯bvba = (−1)¯a¯bG( +(ba)op )(v) = G( aopbop )(v). +Next, we show that +G( +) = flipV ∗,V , +G( +) = flipV,V ∗, +G( +) = flipV ∗,V ∗ . + +DIAGRAMMATICS FOR REAL SUPERGROUPS +21 +Using the definition (5.5) of the left crossing, the map +G( +) = G +� +� +: V ∗ ⊗ V → V ⊗ V ∗ +is given by +f ⊗ v �→ +� +v∈Bk +V +f ⊗ v ⊗ w ⊗ w∗ �→ +� +v∈Bk +V +(−1)¯v ¯wf ⊗ w ⊗ v ⊗ w∗ +�→ v ⊗ +� +v∈Bk +V +(−1)¯v ¯wf(w)w∗ = (−1) +¯f ¯vv ⊗ f, +where we use the fact that ¯w = ¯f whenever f(w) ̸= 0. The proofs for +and +are analogous. +The relations (5.2) and the first two relations in (5.3) are then straightforward to verify. +For the fourth equality in (5.3), we have +G +� +� +: v �→ +� +w∈Bk +V +v ⊗ w ⊗ w∗ �→ +� +w∈Bk +V +(−1)¯v ¯ww ⊗ v ⊗ w∗ �→ +� +w∈Bk +V +w∗(v)w = v = G +� � +(v). +The verification of the third equality in (5.3) is analogous. +Verification of the relations (5.4) is +straightforward. +To show (6.3), we compute that +G +� +aop +� +: k �→ k +is the map +1 �→ +� +v∈Bk +V +(−1)¯vv∗ ⊗ v �→ +� +v∈Bk +V +(−1)¯vv∗ ⊗ va �→ +� +v∈Bk +V +(−1)¯vv∗(va) +(3.10) += +strV (a). +The fact that G factors through OBk(Aop; m − n) when V = Am|n then follows from (3.11). +It remains to prove that, for any functor as in the first sentence of the theorem, we have (6.2). +Suppose that +G( +): 1 �→ +� +u,v∈BV +auvu ⊗ v∗, +auv ∈ k. +Then, for all v ∈ BV , we have +v = G +� +� +(v) = G +� +� +(v) +G +� +⊗ +� +�−−−−−−−−→ +� +u,w∈BV +auwu ⊗ w∗ ⊗ v +1V ⊗ev +�−−−−→ +� +u∈BV +auvu. +It follows that auv = δuv for all u, v ∈ BV , and so G( +) = coev. The other two equalities in (6.2) +then follow from (5.10). +□ +Remark 6.2. Theorem 6.1 holds in greater generality. +If C is any rigid symmetric monoidal +supercategory (e.g. the category of supermodules over a triangular Hopf superalgebra) with an +object V that has the structure of a right A-supermodule, then (6.1) defines a unique monoidal +superfunctor G: OBk(Aop) → C, and (6.2) and (6.3) hold. The proof of this more general statement +is exactly the same as the proof of Theorem 6.1. We chose to state Theorem 6.1 with the choice +C = g-smodk since that will be our main application. +Remark 6.3. When A is a Frobenius superalgebra, G is essentially the functor of [MS, Th. 5.1]. The +paper [MS] works with right gl(m|n, A)-supermodules and left A-supermodules. In Theorem 6.1, we +have translated to the setting of right A-supermodules by considering OBk(Aop) instead of OBk(A) +and to the setting of left gl(m|n, A)-supermodules using the involution X �→ −X of gl(m|n, A). + +22 +SAIMA SAMCHUCK-SCHNARCH AND ALISTAIR SAVAGE +The natural module V is denoted by V+ in [MS]. Furthermore, in [MS], the dual module V ∗ is +replaced by a supermodule V−, together with a nondegenerate bilinear form V− ⊗ V+ → k. This +form identifies V− with V ∗. +Remark 6.4. By the universal property of Π-envelopes mentioned in Section 2, we have an induced +monoidal superfunctor +G: OBk(Aop)π → gl(VA)-smodk. +The coherence maps of this monoidal superfunctor involve some signs. For example, we have the +coherence map +G(↑) ⊗ G(↑) = ΠV ⊗ ΠV +∼ += +−→ Π2V ⊗ V +∼ += +−−−→ +(2.6) V ⊗ V = G(↑ ⊗ ↑) = V ⊗ V, +πv ⊗ πw �→ (−1)¯vπ2v ⊗ w �→ −(−1)¯vv ⊗ w. +The following result will be useful in later computations. +Lemma 6.5. We have +(6.4) +G( aop ): V ∗ → V ∗, +f �→ af, +f ∈ V ∗, a ∈ A, +where af is defined as in (3.2). +Proof. We have +G( +aop ) = G +� +aop +� +: f �→ +� +v∈Bk +V +f ⊗ v ⊗ v∗ +�→ +� +v∈Bk +V +(−1)¯a( ¯f+¯v)f ⊗ va ⊗ v∗ �→ +� +v∈Bk +V +(−1)¯a( ¯f+¯v)f(va)v∗ = +� +v∈Bk +V +(af)(v)v∗ = af. +□ +6.2. Fullness over the complex numbers. The remainder of this section is dedicated to proving +that the oriented incarnation superfunctor of Theorem 6.1 is full in certain important special cases. +In this subsection we consider the case where k = C and A is a matrix superalgebra over a complex +division superalgebra. We begin with the case where A is complex division superalgebra, which +follows from results in the literature. +Proposition 6.6. If k = C and A is a complex division superalgebra, then the oriented incarnation +functor Gm|n of Theorem 6.1 is full for all m, n ∈ N. +Proof. As explained in Section 4, the only complex division superalgebras are C and Cl(C). When +A = C, the supercategory OBC(C; m − n) is the usual oriented Brauer category, and the result +was proved in [CW12, §8.3]. (Closely related results were obtained in [BS12, Th. 7.8] and [LSM02, +Th. 3.5].) On the other hand, OBC(Cl(C), 0) is the oriented Brauer–Clifford supercategory. (Recall +that, by Remark 5.3, we may assume the specialization parameter is zero.) In this case, fullness +was proved in [BCK19, Th. 4.1]. +□ +In the remainder of this subsection, our goal is to show that the oriented incarnation superfunctor +Gm|n is full when A is the superalgebra of supermatrices over a complex division superalgebra. This +will be key in our proof that it is also full when A is a real division superalgebra (Theorem 6.10). +We begin with a result that holds in a more general setup. +Let A be a superalgebra. Fix m, n, r, s ∈ N with m + n, r + s ≥ 1. In what follows, we will +identify +Matm|n(Matr,s(A)) +and +Mat(mr+ns)|(ms+nr)(A) + +DIAGRAMMATICS FOR REAL SUPERGROUPS +23 +in the natural way. This induces a natural identification of +gl(m|n, Matr,s(A)) +and +gl((mr + ns|ms + nr), A), +and we denote this Lie superalgebra by g. Let +W = Matr|s(A)m|n +and +V = A(mr+ns|ms+nr). +We have an isomorphism of (g, A)-superbimodules +(6.5) +W +∼ += +−→ V ⊕r ⊕ ΠV ⊕s, +v �→ +� +(−1)p(t)vtπp(t)vt +�r+s +t=1 , +where vt ∈ V is the t-th column of v, and p(t) is defined as in (5.12). +Similarly, we have an +isomorphism of (g, A)-superbimodules +(6.6) +W ∗ ∼ += +−→ (V ∗)⊕r ⊕ (ΠV ∗)⊕s, +f �→ (πp(t)ft)r+s +t=1, +where ft ∈ V ∗ denotes the restriction of f to the t-th summand in (6.5). +The next result shows that the diagram of superfunctors +OBk(Matr|s(A)op) +Add(OBk(Aop)π) +g-smodk +M +Gm|n +G(mr+ns|ms+nr) +commutes up to natural isomorphism, where M is the superfunctor of Proposition 5.7. +Proposition 6.7. The isomorphisms (6.5) and (6.6) induce a monoidal supernatural isomorphism +of superfunctors Gm|n +∼ += +−→ G(mr+ns)|(ms+nr)M. +Proof. Let ω denote the supernatural transformation induced by (6.5) and (6.6). To simplify nota- +tion, let G = Gm|n and G′ = G(mr+ns)|(ms+nr). We need to show that +ωY ◦ G(f) = G′M(f) ◦ ωX +for every generating morphism f ∈ { +, +, +, +, +, +a : a ∈ A}, where X and Y denote the +domain and codomain of f, respectively. These are all straightforward verifications, although care +is needed to keep careful track of signs. We give the details for +and +, since the others are +similar. +First consider the case f = +. We have +ω↑⊗↑ : W ⊗ W → +r+s +� +t,u=1 +Πp(t)+p(u)V ⊗ V, +v ⊗ w �→ +� +(−1)p(t)vt+p(u)wu+p(u)vt+p(t)p(u)vt ⊗ wu +�r+s +t,u=1 , +where p(t) is defined as in (5.12). Then we compute that +G′M( +) ◦ ω↑⊗↑ = +r+s +� +t,u=1 +(−1)p(t)p(u) � +Πp(t)+p(u) flip +� +◦ ω↑⊗↑ +and +ω↑⊗↑ ◦ G( +) = ω↑⊗↑ ◦ flip + +24 +SAIMA SAMCHUCK-SCHNARCH AND ALISTAIR SAVAGE +are both the map (see Remark 6.4) +W ⊗ W → +r+s +� +t,u=1 +Πp(t)+p(u)V ⊗ V +v ⊗ w �→ +� +(−1)p(t)vt+p(u)wu+p(u)vt+vtwuvt ⊗ wu +�r+s +t,u=1 . +Now consider f = +. We have +ω↓⊗↑ : W ∗ ⊗ W → +r+s +� +t,u=1 +Πp(t)+p(u)V ∗ ⊗ V, +f ⊗ v �→ +� +(−1)p(u)vu+p(u)ft+p(t)p(u)ft ⊗ vu +�r+s +t,u=1 . +In addition, ω +1 : k → k is the identity map. Then we compute that +G′M( +) ◦ ω↓⊗↑ = +r+s +� +t=1 +G′( +) ◦ ω↓⊗↑ +and +ω +1 ◦ G( +) = G( +) +are both the map +W ∗ ⊗ W → k, +f ⊗ v �→ +r+s +� +t=1 +(−1)p(t)ft(vt). +□ +Proposition 6.8. If D is a complex division superalgebra, then the superfunctor +Gm|n: OBC(Matr|s(D)op) → gl((mr + ns|ms + nr), D)-smodC +is full for all m, n ∈ N. +Proof. This follows from Propositions 5.7, 6.6 and 6.7. +□ +6.3. Fullness over the real numbers. In this subsection, we prove one of our main results: the +oriented incarnation superfunctor of Theorem 6.1 is full when k = R and A is a central real division +superalgebra. +Suppose D is a central real division superalgebra and recall Convention 4.3. For V = Dm|n, we +have canonical isomorphisms +(6.7) +V C = (Dm|n)C ∼ += +−→ (DC)m|n +and +V ∗,C = +� +(Dm|n)∗�C ∼ += +−→ +� +(DC)m|n�∗ +. +The next result shows that the diagram +OBR(Dop)C +OBC(Dop,C) +(gl(m|n, D)-smodR)C +gl(m|n, DC)-smodC +R +∼ += +GC +m|n +Gm|n +Cgl(m|n,D) +∼ += +commutes up to supernatural isomorphism, where Cgl(m|n,D) is defined in (3.21), and R is the su- +perfunctor of Proposition 5.6. +Proposition 6.9. This is a monoidal supernatural isomorphism of superfunctors +Cgl(m|n,D)GC +m|n +∼ += +−→ Gm|nR +determined by (6.7). + +DIAGRAMMATICS FOR REAL SUPERGROUPS +25 +Proof. To simplify notation, we set +G = Gm|n, +GC = GC +m|n, +C = Cgl(m|n,D), +W = (DC)m|n. +Let ω be the monoidal supernatural isomorphism determined by (6.7). For each generating mor- +phism f ∈ { +, +, +, +, +, +aop : a ∈ A}, we must show that +ωY ◦ CGC(f) = GR(f) ◦ ωX, +where X and Y are the domain and codomain of f, respectively. +We have +ω↑⊗↑ ◦ CGC( +) = ω↑⊗↑ ◦ flipV C,V C = flipW,W ◦ω↑⊗↑ = GR( +) ◦ ω↑⊗↑. +For a ∈ D, v ∈ V , and y, z ∈ C, we have +ω↑ ◦ CGC( +aop ⊗ y): v ⊗ z �→ (−1)¯a¯vω↑(va ⊗ yz) +and +GR( +aop ⊗ y) ◦ ω↑ = G( aop⊗y ) ◦ ω↑ : v ⊗ z �→ (−1)¯a¯vω↑(va ⊗ yz). +For f ∈ V ∗, v ∈ V , and y, z ∈ C, we have +ω +1 ◦ CGC( +): (f ⊗ y) ⊗ (v ⊗ z) �→ f(v)yz +and +GR( +) ◦ ω↓⊗↑ = G( +) ◦ ω↓⊗↑ : (f ⊗ y) ⊗ (v ⊗ z) �→ f(v)yz. +Finally, we have +ω↑⊗↓ ◦ CGC( +): 1 �→ +� +v∈BR +V +v ⊗ v∗ +and +GR( +) ◦ ω +1 : 1 �→ +� +v∈BC +V C +v ⊗ v∗. +These are equal since any R-basis of V is also a C basis of V C. The verifications for the remaining +generating morphisms +and +are similar. +□ +Theorem 6.10. When k = R and A is a central real division superalgebra, the oriented incarnation +superfunctor Gm|n of Theorem 6.1 is full for all m, n ∈ N. +Proof. Suppose A = D is a central real division superalgebra and m, n ∈ N. We wish to show that, +for all objects X, Y in OBk(Dop), the R-linear map +G: HomOBk(Dop;m−n)(X, Y ) → Homgl(m|n,D)(G(X), G(Y )) +is surjective. As explained in Section 3.7, this map is surjective if and only if the complexified map +GC : HomOBk(Dop;m−n)(X, Y )C → Homgl(m|n,D)(G(X), G(Y ))C +is surjective. This follows from Propositions 6.8 and 6.9 and Lemma 4.8. +□ +6.4. Consequences for real Lie groups. We assume throughout this subsection that D ∈ {R, H}. +Then OBR(D) is a monoidal category, and there is no need to work in the setting of supercategories. +In fact, OBR(D) is a spherical pivotal category. (We refer the reader to [Sel11, §4.3] for the definition +of spherical pivotal category.) +In any spherical pivotal category C, we have a trace map Tr: � +X∈C EndC(X) → EndC(1). In +terms of string diagrams, this corresponds to closing a diagram off to the right or left: +Tr +� +f +� += +f += +f +, +where the second equality follows from the axioms of a spherical category. We say that a morphism +f ∈ HomC(X, Y ) is negligible if Tr(f ◦ g) = 0 for all g ∈ HomC(Y, X). The negligible morphisms + +26 +SAIMA SAMCHUCK-SCHNARCH AND ALISTAIR SAVAGE +form a two-sided tensor ideal N of C, and the quotient C/N is called the semisimplification of C. +For m ∈ N, let ON R(D; m) denote the tensor ideal of negligible morphisms of Kar(OBR(D; m)). +For an associative k-algebra A and m ∈ N, let gl(m, A)-tmodk denote the monoidal category of +tensor gl(m, A)-modules. By definition, this is the full subcategory of gl(m, A)-modk whose objects +are direct summands of V ⊗r ⊗ (V ∗)⊗s, r, s ∈ N, where V = Am is the natural module. +Theorem 6.11. For D ∈ {R, H} and m ∈ N, the oriented incarnation functor induces an equiva- +lence of monoidal categories +Kar(OBR(Dop; m))/ON R(Dop; m) → gl(m, D)-tmodR. +Proof. Let D ∈ {R, H}. Since the category gl(m, D)-tmodR is idempotent complete, Gm = Gm|0 +induces a monoidal functor +Kar(Gm): Kar(OBR(Dop; m)) = gl(m, D)-tmodR. +By Theorem 6.10, this functor is full. +Every object in gl(m, D)-tmodR is completely reducible, since its complexification is a tensor +module for gl(m, D)C ∼= gl(m, DC), hence completely reducible. Thus, the category gl(m, D)-tmodR +is semisimple. +In addition, by Theorem 5.5, EndOBR(Dop;m)(1) = R1 +1. Thus, by [SW22, Prop. 6.9], +the kernel of Kar(Gm) is equal to ON R(Dop; m). Therefore, Kar(Gm) induces a full and faithful +functor +OBR(Dop; m)/ON R(Dop; m) → gl(m, D)-tmodR. +Finally, since the image of Kar(Gm) contains all summands of tensor powers of the natural +module Dm and its dual (i.e. all tensor modules), it is essentially surjective, hence an equivalence +of categories. +□ +7. Superhermitian forms over involutive superalgebras +In Section 9, we will introduce our second main diagrammatic supercategory. Then, in Section 10, +we will define the corresponding incarnation superfunctor. +These constructions will depend on +superhermitian forms over involutive superalgebras. In the current section, we cover the important +properties of these forms. Then, in Section 8, we further specialize to the case where the superalgebra +is a real division superalgebra. +7.1. Involutive superalgebras. An anti-involution of a superalgebra A is a homomorphism of +associative superalgebras A → Aop squaring to the identity. +Equivalently, it is a k-linear map +⋆: A → A, a �→ a⋆, such that +(7.1) +(ab)⋆ = (−1)¯a¯bb⋆a⋆ +and +(a⋆)⋆ = a +for all a, b ∈ A. +An involutive superalgebra is a pair (A, ⋆), where ⋆ is an anti-involution of an associative superalgebra +A. +We will typically use the notation ⋆ or ⋄ for anti-involutions. If (A, ⋆) is an involutive superalgebra +and V is a right A-supermodule, then we let V ⋆ denote the left A-supermodule that is equal to V +as a k-supermodule, and with A-action given by +(7.2) +a · v := (−1)¯a¯vva⋆. +Recall the definition of the Nakayama automorphism ζ of a Frobenius superalgebra from Sec- +tion 3.3. +An involutive Frobenius superalgebra is a triple (A, ⋆, τ) such that (A, τ) is Frobenius +superalgebra, (A, ⋆) is an involutive superalgebra, and +(7.3) +ζ2(a) = a +and +τ(a⋆) = τ(a) +for all a ∈ A. + +DIAGRAMMATICS FOR REAL SUPERGROUPS +27 +We will discuss our main examples of interest, the involutive real division superalgebras, in +Section 8. However, let us mention here some other important examples. +Examples 7.1. +(a) The identity map is an anti-involution of any supercommutative Frobenius +superalgebra. +(b) As a special case of (a), if A = k[x]/(xn) for some n ≥ 1, we can take ⋆ to be the identity +map and τ(�n−1 +r=0 arxr) = an−1. +(c) If G is a finite group, we can take A = kG, g⋆ = g−1 for all g ∈ G, and τ to be projection +onto the identity element. +Lemma 7.2. If (A, ⋆, τ) is an involutive Frobenius superalgebra, then +(7.4) +ζ(a⋆) = ζ(a)⋆ +for all a ∈ A. +Proof. For all a, b ∈ A, we have +τ(ab) +(7.3) += τ((ab)⋆) = (−1)¯a¯bτ(b⋆a⋆) +(3.5) += (−1)¯a¯bτ(a⋆ζ(b⋆)) +and +τ(ab) +(3.5) += +(7.3) (−1)¯a¯bτ(ζ(b)a) +(7.3) += (−1)¯a¯bτ((ζ(b)a)⋆) = (−1)¯a¯bτ(a⋆ζ(b)⋆). +Then (7.4) follows from the nondegeneracy of the Frobenius form. +□ +The following corollary will play an important role; see Lemma 7.10. +Corollary 7.3. If (A, ⋆, τ) is an involutive Frobenius algebra, then so is (A, ⋄, τ), where a⋄ = ζ(a)⋆. +Lemma 7.4. Suppose (A, ⋆, τ) is an involutive Frobenius superalgebra with Nakayama automor- +phism ζ, and let BA be a homogeneous k-basis of A. Then the left dual basis to B⋆ +A := {b⋆ : b ∈ BA} +is given by +(7.5) +(b⋆)∨ = ζ(b∨)⋆. +Proof. For b, c ∈ BA, we have +τ +� +ζ(c∨)⋆b⋆� (7.1) += +(7.3) (−1) +¯b¯cτ(bζ(c∨)) +(3.5) += τ(c∨b) = δbc. +□ +Lemma 7.5. If (A, ⋆, τ) is an involutive Frobenius superalgebra, then +(7.6) +strA(a) = strA(a⋆) +for all a ∈ A, +where strA is the supertrace map defined in (3.10). +Proof. Using (3.12), we compute +strA(a) = +� +b∈BA +(−1) +¯bτ(b∨ba) +(7.3) += +� +b∈BA +τ +� +a⋆b⋆(b∨)⋆� += +� +b∈BA +(−1) +¯bτ(a⋆b∨b) = strA(a⋆), +where, in the second-to-last equality we changed to a sum over {(b∨)⋆ : b ∈ BA} and used that, for +b, c ∈ BA, +(−1)¯cτ(c⋆(b∨)⋆) = τ(b∨c) = δbc, +and so ((b∨)⋆)∨ = (−1)¯bb⋆. +□ + +28 +SAIMA SAMCHUCK-SCHNARCH AND ALISTAIR SAVAGE +7.2. Supersymmetric forms. Let (A, ⋄) denote an involutive superalgebra. +Definition 7.6. For ν ∈ {±1}, a (ν, ⋄)-supersymmetric form on a right A-supermodule V is a +homogeneous k-bilinear form Φ: V × V → k such that +(7.7) +Φ(v, w) = ν(−1)¯v ¯wΦ(w, v) +and +Φ(va, w) = (−1)¯a ¯wΦ(v, wa⋄) +for all v, w ∈ V and a ∈ A. +If Φ is a nondegenerate (ν, ⋄)-supersymmetric form on V and BV is a k-basis of V , then the left +dual basis B∨ +V = {v∨ : v ∈ BV } of V is defined by +Φ(v∨, w) = δvw. +Note that v∨ = v + ¯Φ. +Recall the left A-actions on V ∗ and V ⋄ given in (3.2) and (7.2). A (ν, ⋄)-supersymmetric form +Φ of parity σ induces a parity-preserving homomorphism of left A-supermodules +(7.8) +V ⋄ → ΠσV ∗, +v �→ πσΦv, +where +Φv(w) = Φ(v, w). +This is an isomorphism if and only if Φ is nondegenerate. +Lemma 7.7. If a right A-supermodule V admits a (ν, ⋄)-supersymmetric form, then +(7.9) +strV (a⋄) = strV (a) +for all a ∈ A. +Proof. Let Φ be a (ν, ⋄)-supersymmetric form on V , let BV be a k-basis of V , and let B∨ +V = {v∨ : +v ∈ BV } denote the left dual basis with respect to Φ. Then +Φ(w, v∨) +(7.7) += ν(−1) ¯w(¯v+¯Φ)Φ(v∨, w) = ν(−1)¯v+¯v ¯Φδvw. +Thus (v∨)∨ = (−1)¯v+¯v ¯Φv. Then we compute +strV (a⋄) = +� +v∈BV +(−1)¯vv∗(va⋄) = +� +v∈BV +(−1)¯vΦ(v∨, va⋄) +(7.7) += +� +v∈BV +(−1)¯v+¯a¯vΦ(v∨a, v) +(7.7) += ν +� +v∈BV +(−1)¯v ¯ΦΦ(v, v∨a) = +� +v∈BV +(−1)¯vΦ(v∨, va) = +� +v∈BV +strV (a), +where, in the second-to-last equality, we changed to a sum over the basis B∨ +V . +□ +7.3. Superhermitian forms. Our main source of examples of (ν, ⋄)-supersymmetric forms will +come from superhermitian forms over involutive Frobenius superalgebras. +Let V be a finitely- +generated right supermodule over an involutive superalgebra (A, ⋆). A homogeneous map ϕ: V × +V → A is a ⋆-sesquilinear form if it is k-bilinear and +(7.10) +ϕ(va, wb) = (−1)¯a( ¯ϕ+¯v)a⋆ϕ(v, w)b, +for all a, b ∈ A, v, w ∈ V. +(In our cases of interest, A will be purely even whenever ¯ϕ = 1.) +If, in addition, there exists +ν ∈ {±1} such that +(7.11) +ϕ(v, w) = ν(−1)¯v ¯wϕ(w, v)⋆ +for all v, w ∈ V, +then we say that ϕ is a (ν, ⋆)-superhermitian form. We say that ϕ is unimodular if the map +V → HomA(V, A), +v �→ ϕ(v, −), +v ∈ V, +is an isomorphism of k-supermodules. If A is a division superalgebra, then ϕ is unimodular if and +only if it is nondegenerate. + +DIAGRAMMATICS FOR REAL SUPERGROUPS +29 +We say that two (ν, ⋆)-superhermitian forms ϕ1 and ϕ2 are equivalent if there exists a homoge- +neous f ∈ AutA(V ) such that +ϕ2(v, w) = ϕ1(f(v), f(w)) +for all v, w ∈ V. +Note that, when A = k, a (ν, id)-superhermitian form is the same as a (ν, id)-supersymmetric form. +Example 7.8. If A = C, ⋆ is complex conjugation, and V is purely even, then a (1, ⋆)-superhermitian +form is the familiar notion of a hermitian form, while a (−1, ⋆)-superhermitian form is a skew- +hermitian form. On the other hand, a (1, id)-superhermitian form is a symmetric C-bilinear form, +while a (−1, id)-superhermitian form is a skew-symmetric C-bilinear form. +Remark 7.9. An even (ν, ⋆)-superhermitian form on V is equivalent to an even (−ν, ⋆)-superhermitian +form on the parity-shifted supermodule ΠV . Thus, for even forms, one can assume ν = 1 without +losing any generality. However, this is not the case for odd forms. An odd (ν, ⋆)-superhermitian +form on V is equivalent to an odd (ν, ⋆)-superhermitian form on the parity shift ΠV . Since odd +forms are important for the periplectic Lie superalgebras we wish to include, we consider general ν +in the current paper. +Lemma 7.10. If (A, ⋆, τ) is an involutive Frobenius superalgebra, and ϕ is a nondegenerate (ν, ⋆)- +superhermitian form on V , then the composite +(7.12) +Φ = τ ◦ ϕ: V × V → k +is a nondegenerate (ν, ⋄)-supersymmetric form on V , with a⋄ = ζ(a)⋆. +Proof. For all v, w ∈ V , we have +Φ(v, w) = τ(ϕ(v, w)) +(7.11) += +ν(−1)¯v ¯wτ(ϕ(w, v)⋆) +(7.3) += ν(−1)¯v ¯wτ(ϕ(w, v)) = ν(−1)¯v ¯wΦ(w, v) +and +Φ(va, w) = τ(ϕ(va, w)) +(7.10) += +(−1)¯a( ¯ϕ+¯v)τ(a⋆ϕ(v, w)) +(3.5) += (−1)¯a ¯wτ(ϕ(v, w)ζ(a⋆)) +(7.10) += +(−1)¯a ¯wτ(ϕ(v, wζ(a⋆))) +(7.4) += (−1)¯a ¯wΦ(v, wζ(a)⋆). +Thus Φ is (ν, ⋄)-supersymmetric, with a⋄ = ζ(a)⋆. +It remains to show that Φ is nondegenerate. +Suppose v ∈ V is nonzero. +Then, since ϕ is +nondegenerate, there exists w ∈ V such that ϕ(v, w) ̸= 0. Since τ is nondegenerate, there exists +a ∈ A such that +0 ̸= τ(ϕ(v, w)a) = τ(ϕ(v, wa)) = Φ(v, wa). +□ +7.4. Adjoint operators. Suppose that (A, ⋆) is an involutive superalgebra, and that ϕ is a uni- +modular (ν, ⋆)-superhermitian form on a right A-supermodule V . +Lemma 7.11. For all X ∈ EndA(V ), there exists a unique X† ∈ EndA(V ) such that +ϕ(v, Xw) = (−1) +¯ +X¯vϕ(X†v, w) +for all v, w ∈ V. +Proof. Fix v ∈ V . The map w �→ (−1) ¯ +X¯vϕ(v, Xw) is an element of HomA(V, A). +Thus, since ϕ is +unimodular, there exists a unique v′ ∈ V such that +ϕ(v′, w) = (−1) +¯ +X¯vϕ(v, Xw) +for all v ∈ V. +We define X†v = v′. It is then straightforward to verify that X† ∈ EndA(V ). +□ +The element X† is called the adjoint to X. + +30 +SAIMA SAMCHUCK-SCHNARCH AND ALISTAIR SAVAGE +Lemma 7.12. The map X �→ X† is an anti-involution on EndA(V ). In particular, (X†)† = X, +and +(7.13) +(XY )† = (−1) +¯ +X ¯Y Y †X† +for all X, Y ∈ EndA(V ). +Proof. This is a straightforward verification. +□ +Now suppose that (A, ⋆, τ) is an involutive Frobenius superalgebra, and consider the nondegen- +erate (ν, ⋄)-supersymmetric form Φ = τ ◦ ϕ as in Lemma 7.10. Then, for all X ∈ EndA(V ), we +have +Φ(v, Xw) = (−1) +¯ +X¯vΦ(X†v, w) +for all v, w ∈ V. +It follows that the definition of X† is the same if we use the (ν, ⋆)-superhermitian form ϕ or the +corresponding (ν, ⋄)-supersymmetric form Φ. +7.5. Harish-Chandra superpairs associated to superhermitian forms. For ϕ either a uni- +modular (ν, ⋆)-superhermitian form or a nondegenerate (hence unimodular) (ν, ⋄)-supersymmetric +form, define +Gred(ϕ) = {X ∈ AutA(V )0 : ϕ(Xv, Xw) = ϕ(v, w) for all v, w ∈ V } +g(ϕ) = {X ∈ EndA(V ) : ϕ(Xv, w) = −(−1) +¯ +X¯vϕ(v, Xw) for all v, w ∈ V } += {X ∈ EndA(V ) : X† = −X}. +We have that g(ϕ) is a Lie superalgebra with the usual bracket: +[X, Y ] = XY − (−1) +¯ +X ¯Y Y X. +The pair G(ϕ) := (Gred(ϕ), g(ϕ)) is a Harish-Chandra superpair; see Section 3.6. +If ϕ1 and ϕ2 are equivalent forms, then the groups Gred(ϕ1) and Gred(ϕ2) are isomorphic, as are +the Lie superalgebras g(ϕ1) and g(ϕ2). +If (A, ⋆, τ) is an involutive Frobenius superalgebra and ϕ is a unimodular (ν, ⋆)-superhermitian +form, then we have the nondegenerate (ν, ⋄)-supersymmetric form Φ = τ ◦ ϕ from Lemma 7.10. It +follows from the nondegeneracy of τ that +g(ϕ) = g(Φ) +and +Gred(ϕ) = Gred(Φ). +8. Superhermitian forms over involutive real division superalgebras +For our purposes, the most important examples of involutive superalgebras come from real divi- +sion superalgebras. In this section, we examine some important properties of the Lie superalgebras +associated to superhermitian forms over involutive real division superalgebras. +For the quaternions, we have the anti-involution +⋆: H → R, +(a + bi + cj + dk)⋆ = a − bi − cj − dk, +a, b, c, d ∈ R, +of quaternionic conjugation. This restricts to complex conjugation on C and the identity map on +R. The complex Clifford superalgebra Cl(C) has anti-involution +(8.1) +⋆: Cl(C) → Cl(C), +(a + εb)⋆ = a⋆ + εb⋆i, +a, b ∈ C, +where, on the right-hand side, ⋆ denotes complex conjugation. Note that this is an anti-involution of +real superalgebras, but not of complex superalgebras. In fact, there are no C-linear anti-involutions +of Cl(C). +The notation ⋆ will always refer to the above involutions when working with the +real division superalgebras R, C, H, and Cl(C). Note that (D, ⋆, Re) is an involutive Frobenius +superalgebra for D ∈ {R, C, H, Cl(C)}, as is (C, id, Re), where Re(a) is the real part of the even +part of a. None of the other real division superalgebras Clr(R), r ∈ {1, 2, 3, 5, 6, 7}, admit anti- +involutions, since they are not isomorphic to their opposite superalgebras. + +DIAGRAMMATICS FOR REAL SUPERGROUPS +31 +If (A, ⋆) is an involutive superalgebra, then the complexification AC is also an involutive super- +algebra, with involution (which we continue to denote by the same symbol) +(8.2) +⋆: AC → AC, +(a ⊗ z)⋆ = a⋆ ⊗ z, +a ∈ A, z ∈ C. +Similarly, if (A, ⋆, τ) is an involutive Frobenius superalgebra, then so is (AC, ⋆, τ); see (3.24). +8.1. Real case. In this subsection we work over the involutive real division superalgebra (R, id). If +ϕ: V × V → R +is a nondegenerate (ν, id)-superhermitian form on an R-supermodule V , then its complexification +ϕC : V C × V C → C +is a nondegenerate (ν, id)-superhermitian form on V C. +Proposition 8.1. We have an isomorphism of complex Lie superalgebras +g(ϕ)C ∼= g(ϕC). +Proof. It is straightforward to verify that the map X ⊗ a �→ Xa, X ∈ g(ϕ), a ∈ C, gives the desired +isomorphism. +□ +8.2. Complex cases. In this subsection we work over the involutive real division superalgebra +(D, ⋆), where D ∈ {C, Cl(C)}. +Proposition 8.2. Suppose ϕ is a nondegenerate (ν, ⋆)-superhermitian form on Dm|n. Then we have +an isomorphism of complex Lie superalgebras +g(ϕ)C ∼= gl(m|n, D). +Proof. It follows from (7.13) with Y = aI that (aX)† = a⋆X† for all a ∈ D and X ∈ g(ϕ). +Multiplication by i gives an isomorphism of C-supermodules +g(ϕ) +∼ += +−→ {X ∈ gl(m|n, D) : X† = X}. +Therefore, for every X ∈ gl(m|n, D), we have +X = 1 +2(X† − X) + 1 +2(X† + X), +with +1 +2(X† − X) ∈ g(ϕ) +and +1 +2(X† + X) ∈ g(ϕ)i. +The result follows. +□ +8.3. Quaternionic case. In this subjection we work over the real involutive division superalgebra +(H, ⋆) and we fix a nondegenerate (ν, ⋆)-superhermitian form ϕ on Hm|n. We will often view the +quaternions as +(8.3) +H = C[j], +j2 = −1, +jzj−1 = z⋆, z ∈ C. +Choosing the C-basis {1, j} for H, we will identify Hm|n with C2m|2n. Similarly, under the inclusion +ı of (4.4), we will identify Matm|n(H) with a subring of Mat2m|2n(C). In particular, we have +(8.4) +Matm|n(H) = {X ∈ Mat2m|2n(C) : X(vj) = (Xv)j for all v ∈ Hm|n = C2m|2n}. +Note that, for v ∈ Hm|n = C2m|2n, we have +(8.5) +vj = Jv⋆, +where +J = Jm+n = + + + + + + +J1 +0 +· · · +0 +0 +J1 +... +... +... +... +... +0 +0 +· · · +0 +J1 + + + + + + +, +J1 = +� +0 +−1 +1 +0 +� +. + +32 +SAIMA SAMCHUCK-SCHNARCH AND ALISTAIR SAVAGE +It follows that, for X ∈ Mat2m|2n(C), +(8.6) +X ∈ Matm|n(H) ⇐⇒ JX⋆J−1 = X. +Consider the C-linear maps +projH +C, projH +jC: H → C, +projH +C(a + jb) = a, +projH +jC(a + jb) = b, +a, b ∈ C. +It is straightforward to verify that +(8.7) +projH +jC(zj) = projH +C(z)⋆ +for all z ∈ C. +Define +ϕ1 := projH +C ◦ϕ, +ϕj := projH +jC ◦ϕ. +These are precisely the components of ϕ with respect to the C-basis {1, j} of H. The following +lemma gives a precise relationship between ϕ1 and ϕj. +Lemma 8.3. For all v, w ∈ Hm|n = C2m|2n, we have +ϕj(vj, w) = −ϕ1(v, w), +ϕ1(vj, w) = ϕj(v, w), +(8.8) +ϕj(v, wj) = ϕ1(v, w)⋆, +ϕ1(v, wj) = −ϕj(v, w)⋆. +(8.9) +Proof. We have +ϕ1(vj, w) + jϕj(vj, w) = ϕ(vj, w) = −jϕ(v, w) = −jϕ1(v, w) + ϕj(v, w). +Comparing components gives (8.8). Similarly +ϕ1(v, wj) + jϕj(v, wj) = ϕ(v, wj) = ϕ(v, w)j +(8.3) += jϕ1(v, w)⋆ − ϕj(v, w)⋆ +implies (8.9). +□ +Lemma 8.4. The form +ϕj : C2m|2n × C2m|2n → C +is nondegenerate and (−ν, id)-supersymmetric. +Proof. It follows from (8.3) that +ϕj(av, wb) = aϕj(v, w)b, +v, w ∈ V, a, b ∈ C. +Also, +ϕ(v, w) = ν(−1)¯v ¯wϕ(w, v)⋆ = ν(−1)¯v ¯w� +ϕ1(w, v) + jϕj(w, v) +�⋆ += ν(−1)¯v ¯w� +ϕ1(w, v)⋆ + ϕj(w, v)⋆j⋆� += ν(−1)¯v ¯wϕ1(w, v)⋆ − jν(−1)¯v ¯wϕj(w, v). +Thus, ϕj : C2m|2n × C2m|2n → C is a (−ν, id)-supersymmetric form. It follows from (8.9) that, for +all v, w ∈ C2m|2n, we have +ϕ(v, w) = ϕj(v, wj)⋆ + jϕj(v, w). +Thus, ϕj is nondegenerate, since ϕ is. +□ +Corollary 8.5. We have +(8.10) +ϕj(v, wj) − ϕj(vj, w) = 2 Re ϕ(v, w). +Proof. Using (8.8) and (8.9), we have +ϕj(v, wj) − ϕj(vj, w) = ϕ1(v, w)⋆ + ϕ1(v, w) = 2 Re ϕ(v, w). +□ +Proposition 8.6. We have an isomorphism of complex Lie superalgebras +g(ϕ)C ∼= g(ϕj). + +DIAGRAMMATICS FOR REAL SUPERGROUPS +33 +Proof. For all X ∈ Matm|n(H) satisfying ϕj(Xv, w) = −(−1) ¯ +X¯vϕj(v, Xw) for all v, w ∈ C2m|2n, we +have +ϕ1(Xv, w) +(8.9) += ϕj(Xv, wj)⋆ = −(−1) +¯ +X¯vϕj(v, X(wj))⋆ +(8.4) += −(−1) +¯ +X¯vϕj(v, (Xw)j)⋆ (8.9) += −(−1) +¯ +X¯vϕ1(v, Xw). +Thus +g(ϕ) = {X ∈ gl(m|n, H) : ϕj(Xv, w) = −(−1) +¯ +X¯vϕj(v, Xw)}. +Furthermore, using Lemma 8.4, for X ∈ g(ϕ), a ∈ C, and v, w ∈ C2m|2n, we have +ϕj((Xa)v, w) = −(−1) +¯ +X¯vaϕj(v, Xw) = −(−1) +¯ +X¯vϕj(v, Xw)a = −(−1) +¯ +X¯vϕj(v, (Xa)w). +It then follows from Proposition 4.11(b) that +g(ϕ)C = {X ∈ gl(2m|2n, C) : ϕj(Xv, w) = −(−1) +¯ +X¯vϕj(v, Xw)} = g(ϕj). +□ +9. The unoriented supercategory +In this section, we introduce the second of our two main diagrammatic supercategories. After +defining the supercategory, we deduce some of its additional properties. We then prove a basis +theorem for morphism spaces. Throughout this section, k is an arbitrary field, and (A, ⋄) is an +involutive superalgebra. +9.1. Definition of the supercategory. +Definition 9.1. For σ ∈ Z2, we define Bσ +k (A, ⋄) to be the strict monoidal supercategory generated +by one object I and morphisms +: I⊗2 → I⊗2, +: I⊗2 → +1, +: +1 → I⊗2, +a : I → I, +a ∈ A, +subject to the relations += +, += +, += = (−1)σ +, += +, += +, +(9.1) +1 = , +λ +a + µ +b = +λa+µb , +b +a = +ab , +a += +a , +a += +a⋄ , +(9.2) +for all a, b ∈ A and λ, µ ∈ k. The parity of +a is ¯a, the morphisms +and +both have parity σ, +and +is even. We refer to the morphisms +a as tokens. +For d ∈ k, we define Bσ +k (A, ⋄; d) to be the quotient of Bσ +k (A, ⋄) by the additional relations +(9.3) +a = d strA(a)1 +1, +a ∈ A, +where strA is given by (3.10). We call d the specialization parameter. +Example 9.2. When A = k and ⋄ = id, we have +a = a for all a ∈ k. Thus, we can omit the +generators +a and all the relations involving them. Then, if σ = 0, the relations (9.1) and (9.3) are +the defining relations of the Brauer category, as given in [LZ15, Th. 2.6]. For general σ, Bσ +k (k, id; d) is +isomorphic to the marked Brauer category of [KT17], although the description there looks somewhat +different since the authors do not use the concept of a monoidal supercategory. +Proposition 9.3. The following relations hold in Bσ +k (A, ⋄) for all a ∈ A: += (−1)σ +, += +, +a += +a , +a = +a⋄ +. +(9.4) + +34 +SAIMA SAMCHUCK-SCHNARCH AND ALISTAIR SAVAGE +Proof. For the second relation in (9.4), we have += +(2.2) += (−1)σ += (−1)σ += +, +where, for the unadorned equalities, we have used the third, sixth, and fourth equalities in (9.1), in +that order. Then, for the first relation in (9.4), we compute += (−1)σ += (−1)σ +(2.2) += += += += (−1)σ +, +where, for the unadorned equalities, we use fourth relation in (9.1), the sixth relation in (9.1), the +second relation in (9.4), the fifth relation in (9.1), and finally the fourth relation in (9.1). +The third relation in (9.4) follows from the fourth relation in (9.2) after composing on the top +and bottom with the crossing +and using the first relation in (9.1). For the last relation in (9.4), +we have +a +(9.1) += (−1)σ +a +(2.2) += (−1)σ+σ¯a +a +(9.2) += (−1)σ+σ¯a +a⋄ +(9.1) += +(2.2) +a⋄ +. +□ +It follows from the defining relations that “bubbles” are central in Bσ +k (A, ⋄): +a = +a +for all a ∈ A. +Note also that +(9.5) +ab +(9.2) += +a +b +(9.2) += +a⋄ +b +(2.2) += (−1)¯a¯b +a⋄ +b +(9.4) += (−1)¯a¯b +b +a +(9.2) += (−1)¯a¯b +ba +and +(9.6) +a +(9.4) += (−1)σ +a +(9.2) += (−1)σ +a +(9.1) += (−1)σ a +(9.2) += (−1)σ +a⋄ . +Remark 9.4. Suppose that (A, ⋄, τ) is an involutive Frobenius superalgebra, σ = 1, and d ̸= 0. In +B1 +k(A, ⋄; d), we have +d strA(a)1 +1 = +a +(9.6) += − +a⋄ = −d strA(a⋄)1 +1 +(7.6) += −d strA(a)1 +1 +for all a ∈ A. +In applications to representation theory, we will assume that the characteristic +of k is not two. In this case, it follows that strA = 0 or 1 +1 = 0. In the former case, we have +B1 +k(A, ⋄; d) = B1 +k(A, ⋄; 0). +In the latter case, B1 +k(A, ⋄; d) is the zero supercategory. Thus, when +σ = 1 and (A, ⋄) can be endowed with the structure of an involutive Frobenius superalgebra, we +will usually assume that d = 0. +For any d ∈ k, we have isomorphisms of monoidal supercategories +(9.7) +Ξ⋄ : Bσ +k (A, ⋄) → Bσ +k (Aop, ⋄) +and +Ξ⋄ : Bσ +k (A, ⋄; d) → Bσ +k (Aop, ⋄; d), +given by applying the involution ⋄ to all tokens. +9.2. The basis theorem. This subsection is dedicated to the proof of a basis theorem giving bases +for the morphisms spaces of the Bσ +k (A, ⋄). Our method involves embedding this supercategory into +the superadditive envelope of the oriented supercategory OBk(A). Even in the case A = k, when +B0 +k(k, id) is the usual Brauer category, this method of proof is new. +Recall, from Section 2, that, for a monoidal supercategory C, we let Add(Cπ) denote its super- +additive envelope. The objects of Add(Cπ) are formal direct sums of objects of the Π-envelope Cπ, + +DIAGRAMMATICS FOR REAL SUPERGROUPS +35 +and morphisms in Add(Cπ) are matrices of morphisms in Cπ, which we will write as sums of their +components. For example, +0 +0 + +σ +σ + +σ +σ + (−1)σ +0 +0 : (↑ ⊕Πσ↓) ⊗ (↑ ⊕Πσ↓) → (↑ ⊕Πσ↓) ⊗ (↑ ⊕Πσ↓) +is a morphism in Add(OBk(A)) with components +0 +0 : ↑ ⊗ ↑ → ↑ ⊗ ↑, +σ +σ : Πσ↑ ⊗ ↓ → Πσ↓ ⊗ ↑, +σ +σ : Πσ↓ ⊗ ↑ → Πσ↑ ⊗ ↓, +(−1)σ +0 +0 : ↓ ⊗ ↓ → ↓ ⊗ ↓, +and all other components equal to zero. +Theorem 9.5. Fix σ ∈ Z2. There exists a unique monoidal superfunctor D: Bσ +k (A, ⋄) → Add(OBk(A)π) +such that D(I) = ↑ ⊕Πσ↓ and +D +� +� += +0 +0 + +σ +σ + +σ +σ + (−1)σ +0 +0 , +D ( +) = +0 +σ + +0 +σ , +D ( +) = +σ +0 + (−1)σ +σ +0 , +D +� +a � += +a +0 +0 + (−1)σ¯a a⋄ +σ +σ , +a ∈ A. +If (A, ⋄) can be endowed with the structure of an involutive Frobenius superalgebra, then, for all +d ∈ k, this induces a monoidal superfunctor +D: Bσ +k (A, ⋄; 2d) → Add(OBk(A; d)π), +where we assume that d = 0 if σ = 1. (See Remark 9.4.) +Proof. We must verify that D respects the defining relations of Bk(A, ⋄) from Definition 9.1. For +the first relation in (9.1), we compute +D +� +� += +0 +0 ++ +σ +σ ++ +σ +σ ++ +0 +0 += +0 +0 ++ +σ +σ ++ +σ +σ ++ +0 +0 += D +� +� +. +The proof of the second relation in (9.1) is similar. +For the third relation in (9.1), we have +D +� +� += D( ⊗ +) ◦ D( +⊗ ) +(2.11) += +� +0 +σ + +0 +σ + (−1)σ +σ +0 + (−1)σ +σ +0 +� +◦ +� +σ +0 + (−1)σ +σ +0 + +0 +σ + (−1)σ +0 +σ +� += +0 +0 ++ +σ +σ += +0 +0 ++ +σ +σ += D +� � +. +Similarly, for the fourth relation in (9.1), we have +D +� +� += D( +⊗ ) ◦ D( ⊗ +) + +36 +SAIMA SAMCHUCK-SCHNARCH AND ALISTAIR SAVAGE +(2.11) += +� +0 +σ + +0 +σ + +σ +0 + +σ +0 +� +◦ +� +σ +0 + (−1)σ +σ +0 + (−1)σ +0 +σ + +0 +σ +� += (−1)σ +� +0 +0 ++ +σ +σ +� += (−1)σ +� +0 +0 ++ +σ +σ +� += (−1)σD +� � +. +The fifth relation in (9.1) is straightforward. For the sixth relation in (9.1), we compute +D +� +� += D( ⊗ +) ◦ D( +⊗ ) +(2.11) += +� +0 +σ + +0 +σ + (−1)σ +σ +0 + (−1)σ +σ +0 +� +◦ +� +0 +0 + +σ +σ + +σ +σ + (−1)σ +0 +0 + +σ +σ + +0 +0 + +0 +0 + (−1)σ +σ +σ +� += +0 +σ + +0 +σ + +σ +0 + (−1)σ +σ +0 +and +D +� +� += D( +⊗ ) ◦ D( ⊗ +) +(2.11) += +� +0 +σ + +0 +σ + +σ +0 + +σ +0 +� +◦ +� +0 +0 + +σ +σ + +σ +σ + (−1)σ +0 +0 + +σ +σ + +0 +0 + +0 +0 + (−1)σ +σ +σ +� += +0 +σ + +0 +σ + +σ +0 ++ (−1)σ +σ +0 +(5.9) += +0 +σ + +0 +σ + +σ +0 ++ (−1)σ +σ +0 +. +The first, second, and fourth relations in (9.2) are straightforward. For the third relation in (9.2), +we compute +D +� +a +b +� += +a +b +0 +0 ++ a⋄ +b⋄ +σ +σ += +ab +0 +0 ++ (−1)¯a¯b b⋄a⋄ +σ +σ +(7.1) += +ab +0 +0 ++ (ab)⋄ +σ +σ += D +� +ab +� +. +Finally, for the last relation in (9.2), we have +D +� +a +� += D( +) ◦ D( a ⊗ +) +(2.11) += +� +0 +σ + +0 +σ +� +◦ +� +a +0 +0 + (−1)σ¯a a +σ +σ + (−1)σ¯a a⋄ +σ +σ + a⋄ +0 +0 +� += (−1)σ¯a +� +a⋄ +0 +σ + a +0 +σ +� += (−1)σ¯a +� +a⋄ +0 +σ + +a +0 +σ +� +and +D +� +a⋄ � += D( +) ◦ D( ⊗ +a⋄ ) +(2.11) += +� +0 +σ + +0 +σ +� +◦ +� +a⋄0 +0 + (−1)σ¯a +a +σ +σ + (−1)σ¯a +a⋄σ +σ + +a0 +0 +� + +DIAGRAMMATICS FOR REAL SUPERGROUPS +37 += (−1)σ¯a +� +a⋄ +0 +σ + +a +0 +σ +� +. +It remains to prove the final statement of the theorem. This statement is clear if d = 0, and so +it suffices to assume σ = 0. In this case, we have +D +� +a � += +a + +a⋄ = +a+a⋄ +where, in the last equality, we used [MS, (4.24)] to convert the clockwise bubble to a counterclockwise +one. Then the assertion follows from the fact that +strA(a + a⋄) +(7.6) += 2 strA(a). +□ +We are now ready to state and prove the basis theorem for Bσ +k (A, ⋄; d). For r, s ∈ N, an (r, s)- +Brauer diagram is a string diagram representing a morphism in HomBσ +k (A,⋄;d)(I⊗r, I⊗s) such that: +• there no tokens on any string and no closed strings (i.e. strings with no endpoints); +• no string has more than one critical point; +• there are no self-intersections of strings and no two strings cross each other more than once. +A perfect matching of a finite set is a partition of that set into subsets of size 2. Numbering the +bottom endpoints 1, . . . , r from left to right and the top endpoints r +1, . . . , r +s from left to right, +each (r, s)-Brauer diagram induces a perfect matching of {1, . . . , r + s}. Let D(r, s) denote a set of +(r, s)-Brauer diagrams, with each perfect matching of {1, . . . , r + s} induced by exactly one element +of D(r, s). +For r, s ∈ N, let D•(r, s) denote the set of all morphisms that can be obtained from elements of +D(r, s) by adding exactly one token to each string according to Convention 5.4. For example, +is a possible element of D(5, 7) and +b1 +b2 +b3 +b4 +b5 +b6 +, +b1, b2, b3, b4, b5, b6 ∈ BA, +are the corresponding elements of D•(5, 7). +We expect the following theorem to hold for an arbitrary involutive superalgebra (A, ⋄). However, +since our proof relies on Theorem 5.5, it assumes that A also admits the structure of a Frobenius +superalgebra. As explained in Convention 4.3, this assumption holds whenever A is a real or complex +division superalgebra. +Theorem 9.6. For all r, s ∈ N and d ∈ k, the set D•(r, s) is k-basis for the morphism space +HomBσ +k (A,⋄;d)(I⊗r, I⊗s) +Proof. We first prove that the elements of D•(r, s) span HomBσ +k (A,⋄;d)(I⊗r, I⊗s) as k-supermodule. +Using the last two relations in (9.2) and the last two relations in (9.4), tokens on strings with +endpoints can be moved near the appropriate endpoints. Then, using the third relation in (9.2), we +can reduce the number of tokens to precisely one on each string. (Recall that a string with no token +is the same as a string with a token labelled by the identity element of A, using the first relation in +(9.2).) Next, using the second relation in (9.2), we can write any diagram as a linear combination +of ones where the tokens are labelled by elements of BA. Finally, using the relations (9.1) and the +first two relations in (9.4), we can move the strings so that they are positioned to agree with some + +38 +SAIMA SAMCHUCK-SCHNARCH AND ALISTAIR SAVAGE +element of D(r, s), together with some bubbles to the right of this element. Then we can evaluate +all bubbles using (9.3). +It remains to prove linear independence of D•(r, s). Under the functor D of Theorem 9.5, each +element of D•(r, s) is sent, up to sign and parity shift, to a sum over all possible orientations of +the strands, with the map a �→ ±a⋄ applied to labels of tokens on downward pointing strands. It +follows from Theorem 5.5 that these images are linearly independent. Therefore, D•(r, s) is linearly +independent. +□ +10. The unoriented incarnation superfunctor +In this section, we introduce the main application of the supercategory Bσ +k (A, ⋄) to the rep- +resentation theory of supergroups. +We begin by defining a very general unoriented incarnation +superfunctor and proving an asymptotic faithfulness result. +We then turn our attention to the +special cases where k ∈ {R, C} and A is a division superalgebra over k. When k = C, fullness of the +incarnation functor follows from known results. When k = R, we state the fullness result, whose +proof is split into three cases, proved in Sections 11 to 13. +10.1. Definition of the superfunctor. In this subsection we work over an arbitrary field k. Let +(A, ⋄) be an involutive superalgebra, let V be a right A-supermodule, and let Φ be a nondegenerate +(ν, ⋄)-supersymmetric form of parity σ on V . Fix a homogeneous k-basis BV of V , and let B∨ +V = +{b∨ : b ∈ BV } be the left dual basis with respect to Φ. +Recall the notation flip and ρa from +Section 3.2. It follows from (7.1) that +(10.1) +ρa⋄ρb⋄ = ρ(ab)⋄, +a, b ∈ A. +Theorem 10.1. There exists a unique monoidal superfunctor, which we call the unoriented incar- +nation superfunctor associated to Φ, +FΦ : Bσ +k (A, ⋄) → G(Φ)-smodk +such that FΦ(I) = V and +(10.2) +FΦ +� +� += ν flip, +FΦ ( +) = Φ, +FΦ( a ) = ρa⋄, +a ∈ A. +This superfunctor also satisfies the following: +FΦ ( +) = Φ′ : k → V ⊗ V, +1 �→ +� +v∈BV +(−1)σ¯vv ⊗ v∨, +and +(10.3) +FΦ +� +a � += strV (a)1 +1 +for all a ∈ A. +(10.4) +If V = Am|n for some m, n ∈ N, then FΦ induces a monoidal superfunctor +FΦ : Bσ +k (A, ⋄; ν(m − n)) → G(Φ)-smodk. +Proof. We first show that (10.2) and (10.3) indeed yield a superfunctor FΦ. We must show that +it respects the relations (9.1) and (9.2). The first two relations in (9.1) are clear. For the third +equality in (9.1), we compute +FΦ +� +� +: v +Φ′⊗1V +�−−−−→ +� +w∈BV +(−1)σ ¯ww ⊗ w∨ ⊗ v +1V ⊗Φ +�−−−−→ +� +w∈BV +Φ(w∨, v)v = v. +For the fourth equality in (9.1), we compute +FΦ +� +� +: v +1V ⊗Φ′ +�−−−−→ +� +w∈BV +(−1)σ(¯v+ ¯w)v ⊗ w ⊗ w∨ Φ⊗1V +�−−−−→ (−1)σ � +w∈BV +Φ(v, w)w∨ = (−1)σv, +where, to simplify the sign, we used the fact that Φ(v, w) = 0 unless ¯v + ¯w = σ. + +DIAGRAMMATICS FOR REAL SUPERGROUPS +39 +The fifth equality in (9.1) follows immediately from the fact that Φ is a (ν, ⋄)-supersymmetric +k-bilinear form. For the sixth equality in (9.1), we compute +FΦ +� +� +: u ⊗ v ⊗ w +ν flip ⊗1V +�−−−−−−→ ν(−1)¯u¯vv ⊗ u ⊗ w +1V ⊗Φ +�−−−−→ ν(−1)(σ+¯u)¯vΦ(u, w)v, +and +FΦ +� +� +u ⊗ v ⊗ w +1V ⊗ν flip +�−−−−−−→ ν(−1)¯v ¯wu ⊗ w ⊗ v +Φ⊗1V +�−−−−→ ν(−1)¯v ¯wΦ(u, w)v = ν(−1)(σ+¯u)¯vΦ(u, w)v, +where, in the final equality, we used the fact that Φ(u, w) = 0 unless ¯w = σ + ¯u. +The first two relations in (9.2) are straightforward, while the third follows from (10.1). For the +fourth relation in (9.2), we compute +FΦ +� +a +� +: u ⊗ v +ρa⊗1V +�−−−−→ (−1)¯a¯uua⋆ ⊗ v +ν flip +�−−−→ ν(−1)¯u¯v+¯a(¯u+¯v)v ⊗ ua⋆, +and +FΦ +� +a � +: u ⊗ v +ν flip +�−−−→ ν(−1)¯u¯vv ⊗ u +1V ⊗ρa +�−−−−→ ν(−1)¯u¯v+¯a(¯u+¯v)v ⊗ ua⋆. +Finally, for the fifth relation in (9.2), we compute +FΦ +� +a +� +: u ⊗ v +ρa⊗1V +�−−−−→ (−1)¯a¯uua⋄ ⊗ v Φ +�−→ (−1)¯a¯uΦ(ua⋄, v) +(7.7) += (−1)¯a(¯u+¯v)Φ(u, va), +and +FΦ +� +a⋄ � +: u ⊗ v +1V ⊗ρa⋄ +�−−−−−→ (−1)¯a(¯u+¯v)u ⊗ va Φ +�−→ (−1)¯a(¯u+¯v)Φ(u, va). +Next we prove (10.4). Using Lemma 7.10, we have +Φ(w, v∨) +(7.7) += ν(−1) ¯w(¯v+σ)Φ(v∨, w) = ν(−1)¯v+σ¯vδvw. +Thus, the k-basis of V left dual to B∨ +V is given by (v∨)∨ = ν(−1)¯v+σ¯vv. Therefore, +Fϕ +� +a � +: 1 Φ′ +�−→ ν +� +v∈BV +(−1)¯vv∨ ⊗ v +1V ⊗ρa +�−−−−→ ν +� +v∈BV +(−1)¯vv∨ ⊗ va⋄ +Φ +�−→ ν +� +v∈BV +(−1)¯vΦ(v∨, va⋄) = ν strV (a), +where, in the final equality, we used Lemma 7.7. The fact that FΦ factors through Bσ +k (A, ⋄; ν(m−n)) +when V = Am|n then follows from (3.11). +It remains to prove that, for any functor as in the first sentence of the theorem, we have Fϕ( +) = +Φ′. Suppose that +Fϕ( +): 1 �→ +� +u,v∈BV +auvu ⊗ v∨, +auv ∈ k. +Then, for all v ∈ BV , we have +v = Fϕ +� � +(v) = Fn +� +� +(v) +Fn +� +⊗ +� +�−−−−−−−→ +� +u,w∈BV +auwu ⊗ w∨ ⊗ v +1V ⊗Φ +�−−−−→ +� +u∈BV +(−1)σ¯uauvu. +It follows that auv = (−1)σ¯vδuv for all u, v ∈ BV , and so Fn( +) = Φ′. +□ + +40 +SAIMA SAMCHUCK-SCHNARCH AND ALISTAIR SAVAGE +10.2. Asymptotic faithfulness. For the remainder of this section, we assume that V = Am|n. +Fix a k-basis BA of A with the property that b⋄ = ±b for all b ∈ BA. +Proposition 10.2. If 2m + 2n ≥ r + s, then the elements FΦ(f), f ∈ D•(r, s), are linearly +independent, over k, in HomG(Φ)(V ⊗r, V ⊗s). +Proof. We have a commutative diagram +HomBσ +k (A,⋄)(I⊗r, I⊗s) +HomBσ +k (A,⋄)(I⊗(r+s), +1) +HomG(Φ) +� +V ⊗r, V ⊗s� +HomG(Φ) +� +V ⊗(r+s), k +� +FΦ +∼ += +FΦ +∼ += +where the horizontal maps are the usual isomorphisms that hold in any rigid monoidal supercategory. +In particular, the top horizontal map is the k-linear isomorphism given on diagrams by +· · · +· · · +�→ +... +· · · +· · · +with inverse +· · · +· · · +�→ ± +· · · +... +· · · +, +where the rectangles denotes some diagram, and the sign (which is needed only when σ = 1) +is determined by the parity of this diagram. Applying the top horizontal map to an element of +D•(r, s), then sliding tokens along strands to the correct position, yields an element of D•(r + s, 0) +up to sign. Therefore, it suffices to prove the theorem in the case where s = 0. +Suppose m, n ∈ N satisfy 2m + 2n ≥ r. If r is odd, then D•(r, 0) = ∅, and the proposition holds +trivially. Therefore, we suppose that r is even. In what follows we number the strand endpoints +1, 2, . . . , r from left to right. Given f ∈ D•(r, 0), we enumerate the strands in f in order of the +numbering of their right endpoint. Let bi, 1 ≤ i ≤ r +2, denote the label of the token on the i-th +strand of f. For 1 ≤ i ≤ r, define +vi = +� +ej +if the j-th strand in f has right endpoint in position i +(ejbj)′ +if the j-th strand in f has left endpoint in position i, +where e1, . . . , em+n denotes the standard A-basis of V = Am|n, and where +{(eib)∨ : 1 ≤ i ≤ m + n, b ∈ BA} +denotes the basis of V left dual to {eib : 1 ≤ i ≤ m + n, b ∈ BA} with respect to Φ. Now define +vf = v1 ⊗ v2 ⊗ · · · ⊗ vr. +For example, +if f = +b1 +b2 b3 +b4 , +then +vf = (e4b4)∨ ⊗ (e1b1)∨ ⊗ (e2b2)∨ ⊗ e1 ⊗ e2 ⊗ (e3b3)∨ ⊗ e3 ⊗ e4. +It is straightforward to verify that +FΦ(f)(vg) = ±δf,g, +for all f, g ∈ D•(r, 0). +It follows that the elements of FΦ(f), f ∈ D•(r, 0), are linearly independent, as desired. +□ + +DIAGRAMMATICS FOR REAL SUPERGROUPS +41 +Proposition 10.3. If 2m + 2n ≥ r + s, then the induced k-supermodule homomorphism +FΦ : HomBσ +k (A,⋄;ν(m−n))(I⊗r, I⊗s) → HomG(Φ)(V ⊗r, V ⊗s) +is injective. +Proof. This follows immediately from Theorem 9.6 and Proposition 10.2. +□ +10.3. Fullness over the real and complex numbers. We are now ready to state the last of our +main results: the fullness of the unoriented incarnation functor in the case of a central real division +superalgebra. We begin by stating the fullness result over the complex numbers. +Proposition 10.4. If Φ is a nondegenerate (ν, id)-supersymmetric form on Cm|n of parity σ, then +the unoriented incarnation superfunctor +FΦ: Bσ +C(C, id; ν(m − n)) → G(Φ)-smodC +of Theorem 10.1 is full. +Proof. First consider the case σ = 0. As explained in Appendix A.1, we may assume that ν = 1. +Then B0 +C(C, id; m − n) is the usual Brauer category and the result was proved in [LZ17, Th. 5.6]; +see also [ES16]. Next consider the case σ = 1, in which case we must have m = n, as explained in +Appendix A.7. By (A.11), we may again assume that ν = 1. Then B0 +C(C, id; 0) is the periplectic +Brauer category, introduced in [KT17] as B(0, −1). In this case, the result was proved in [CE21, +Th. 6.2.1], with the key ingredient being [DLZ18, §4.9]. +□ +In the remainder of the paper, we will be mostly concerned with the case where A is a real division +superalgebra (see Convention 4.3), and where the (ν, ⋄)-supersymmetric form comes from a (ν, ⋆)- +superhermitian form, as in Lemma 7.10. To simplify notation we define, for (D, ⋆) an involutive real +division superalgebra, and d ∈ R, +(10.5) +Bσ +R(D) := Bσ +R(D, ⋄), +Bσ +R(D; d) := Bσ +R(D, ⋄; d), +where a⋄ = (−1)¯aa⋆. +Theorem 10.5. Suppose (D, ⋆) is an involutive central real division superalgebra, and ϕ is a nonde- +generate (ν, ⋆)-superhermitian form on Dm|n of parity σ. Let Φ be the corresponding nondegenerate +(ν, ⋄)-supersymmetric form on Dm|n, as in Lemma 7.10. Then the unoriented incarnation super- +functor +FΦ : Bσ +R(D; ν(m − n)) → G(Φ)-smodR +of Theorem 10.1 is full. +The proof of Theorem 10.5 will be broken into three parts: +• we prove it holds for (D, ⋆) = (R, id) in Proposition 11.4, +• we prove it holds for (D, ⋆) = (C, ⋆) and (D, ⋆) = (Cl(C), ⋆) in Proposition 12.4, and +• we prove it holds for (D, ⋆) = (H, ⋆) in Proposition 13.4. +Remark 10.6. As explained in Section 3.6, the forgetful functor G(Φ)-smodR → g(Φ)-smodR is +full and faithful when Gred(Φ) is connected. +In this case, we can replace the target supercate- +gories G(Φ)-smodC and G(Φ)-smodR in Proposition 10.4 and Theorem 10.5 by g(Φ)-smodC and +g(Φ)-smodR, respectively. It follows from the descriptions of Gred(Φ) in Appendix A that we can +make this replacement whenever σ = 1. In addition, when σ = 0, we can make this replacement in +Theorem 10.5 when (D, ⋆) ∈ {(C, ⋆), (Cl(C), ⋆)}. + +42 +SAIMA SAMCHUCK-SCHNARCH AND ALISTAIR SAVAGE +10.4. Consequences for the semisimple cases. For (D, ⋆) ∈ {(R, id), (C, ⋆), (H, ⋆)}, we see that +Bk(D) is a monoidal category (as opposed to a monoidal supercategory). In fact, the category Bk(D) +is a spherical pivotal category. We also have ⋄ = ⋆. For d ∈ Z, we let NR(D; d) denote the tensor +ideal of negligible morphisms of Kar(BR(D; d)); see Section 6.4. +For a (ν, ⋆)-supersymmetric form Φ on Dm|n, we let G(Φ)-tsmodR denote the monoidal supercat- +egory of tensor G(Φ)-supermodules. By definition, this is the full sub-supercategory of G(Φ)-smodR +whose objects are direct summands of V ⊗r, r ∈ N, where V = Dm|n is the natural supermodule. We +let G(Φ)-tsmod′ +R denote the underlying category of G(Φ)-tsmodR. By definition, this is the category +with the same objects as G(Φ)-tsmodR, but whose morphisms are the even G-supermodule homo- +morphisms. So G(Φ)-tsmod′ +R is a monoidal category (as opposed to a monoidal supercategory). If +n = 0, so that G(Φ) is a real group (as opposed to a supergroup), then we write G(Φ)-tmodR for +the category of tensor modules, defined in the same way. +Theorem 10.7. For (D, ⋆) ∈ {(R, id), (C, ⋆), (H, ⋆)} and p, q, m, n ∈ N, p + q = m, the unoriented +incarnation functor induces equivalences of monoidal categories +Kar(BR(R; m))/NR(R; m) → O(p, q)-tmodR, +Kar(BR(R; −2n))/NR(R; −2n) → OSp(0|2n, R)-tsmod′ +R, +Kar(BR(R; 1 − 2n))/NR(R; 1 − 2n) → OSp(1|2n, R)-tsmod′ +R, +Kar(BR(C; m))/NR(C; m) → U(p, q)-tmodR, +Kar(BR(H; m))/NR(H; m) → Sp(p, q)-tmodR, +Kar(BR(H; −n))/NR(H; −n) → O(m, H)-tsmod′ +R. +Note that, while OSp(0|2n, R) is isomorphic to Sp(2n, R), we write OSp(0|2n, R)-tsmod′ +R in +Theorem 10.7, instead of Sp(2n, R)-tmodR, since we view the natural module R2n as purely odd. +Similarly, in O(n, H)-tsmod′ +R, we view the natural module Hn as purely odd; see Appendix A.5. +Proof. The proof is similar to that of Theorem 6.11. +To simplify notation, we give the proof +for O(p, q), since the other cases are analogous; see Remark 10.8(a). +Let Φ = ϕp,q|2n be the +(ν, ⋆)-supersymmetric form on Rm defined in (A.6), so that G(Φ) = O(p, q). Since the category +O(p, q)-tmodR is idempotent complete, FΦ induces a monoidal functor +Kar(FΦ): Kar(BR(R; m)) → O(p, q)-tmodR. +By Theorem 10.5, this functor is full. +Every object in O(p, q)-tmodR is completely reducible, since its complexification is a tensor +module for O(m, C), hence completely reducible. Thus, the category O(p, q)-tmodR is semisimple. +In addition, by Theorem 9.6, EndBR(R;m)(1) = R1 +1. Thus, by [SW22, Prop. 6.9], the kernel of +Kar(FΦ) is equal to NR(R; m). Therefore, Kar(FΦ) induces a full and faithful functor +BR(R; m)/NR(R; m) → O(p, q)-tmodR. +Finally, since the image of Kar(FΦ) contains all summands of tensor powers of the natural module +Rm (i.e. all tensor modules), it is essentially surjective, hence an equivalence of categories. +□ +Remark 10.8. +(a) Theorem 10.7 involves precisely the supergroups with Lie superalgebras that +are real forms of complex Lie superalgebras whose finite-dimensional modules are all semisimple; +see [DH76, Th. 4.1]. +(b) As explained in Remark 10.6, we can replace OSp(0|2n, R), U(p, q), and Sp(p, q) in The- +orem 10.7 by sp(0|2n, R), u(p, q), and sp(p, q), respectively. However, we cannot replace O(p, q), +OSp(1|2n, R), or O(n, H) by their Lie superalgebras, since the orthogonal groups are not connected. + +DIAGRAMMATICS FOR REAL SUPERGROUPS +43 +Corollary 10.9. If p, p′, q, q′ ∈ N satisfy p + q = p′ + q′, then we have equivalences of monoidal +categories +O(p, q)-tmodR ≃ O(p′, q′)-tmodR, +U(p, q)-tmodR ≃ U(p′, q′)-tmodR, +Sp(p, q)-tmodR ≃ Sp(p′, q′)-tmodR, +sending the natural supermodule to the natural supermodule. +We will extend Corollary 10.9 to equivalences of more general supergroups in Propositions 11.5, +12.5 and 13.5. +Remark 10.10. It is crucial that Corollary 10.9 involves O(p, q) and U(p, q), as opposed to SO(p, q) +and SU(p, q). For example, let V = C2 ∼= R4 denote the natural U(2)-module. By restriction, this +is also the natural module for SU(2). Direct computation shows that +C ∼= EndU(2)(V ) ⊆ EndSU(2)(V ) ∼= H. +On the other hand, all irreducible modules of SU(1, 1) ∼= SL(2, R) have endomorphism algebra +isomorphic to R. Thus, the categories SU(1, 1)-tmodR and SU(2)-tmodR are not equivalent. If +W = C2 ∼= R4 denotes the natural module of U(1, 1), which is also the natural module for SU(1, 1) +by restriction, we have +C ∼= EndU(1,1)(W) ⊆ EndSU(1,1)(W) ∼= Mat2(R). +The equivalence U(2)-tmodR ≃ U(1, 1)-tmodR of Corollary 10.9 sends V to W. Both modules have +endomorphism algebra isomorphic to C only if we use the full unitary groups. +11. Unoriented fullness: real case +In this section, we prove Theorem 10.5 in the case (D, ⋆) = (R, id). Note that, since R is purely +even, we have ⋄ = ⋆. Recall, from Section 7, that, if (A, ⋆) is an involutive superalgebra, then so is +(AC, ⋆). The proof of the following result is analogous to that of Proposition 5.6. +Proposition 11.1. For any real involutive superalgebra (A, ⋆) and σ ∈ Z2, there is an isomorphism +of monoidal supercategories +Bσ +R(A, ⋆)C ∼ += +−→ Bσ +C(AC, ⋆) +given on objects by I �→ I and on morphisms by +�→ +, +�→ +, +�→ +, +a �→ +a⊗1 , +a ∈ A. +For all d ∈ k, this induces an isomorphism of monoidal supercategories +Bσ +R(A, ⋆; d)C +∼ += +−→ Bσ +C(AC, ⋆; d) +Fix m, n ∈ N, set V = Rm|n, and let Φ be a nondegenerate (ν, id)-supersymmetric form on V +of parity σ. (See Appendices A.3 and A.7 for a classification.) We have a natural identification +V C = Cm|n, and we extend Φ to a nondegenerate (ν, id)-supersymmetric form +ΦC : V C × V C → C. +Lemma 11.2. For all G(Φ)-supermodules U and W, we have an isomorphism of C-supermodules +HomG(Φ)(U, W)C ∼ += +−→ HomG(ΦC)(U C, W C), +f ⊗ a �→ f ⊗ a, +f ∈ HomG(Φ)(U, W), a ∈ C. + +44 +SAIMA SAMCHUCK-SCHNARCH AND ALISTAIR SAVAGE +Proof. First suppose that σ = 0. +Then, as explained in Appendix A.1, we may assume that +ν = 1. In this case, the classification of the nondegenerate (1, id)-supersymmetric forms is recalled +in Appendix A.3. We see that n must be even, and Gred(Φ) = O(p, q) × Sp(n, R) for some p, q ∈ N, +p + q = m. The group Sp(n, R) is connected. On the other hand, if m ≥ 1, then O(p, q) has four +connected components if p, q ≥ 1, and two connected components if p = 0 or q = 0. Suppose p, q ≥ 1, +the proof in the other case being analogous. Then choose elements X1, X2, X3 in O(p, q), one from +each of its connected components not containing the identity. By (3.23) and Proposition 8.1, we +have an isomorphism +HomG(Φ)(U, W)C ∼= HomX1,X2,X3,g(ΦC)(U C, W C). +Now, Gred(ΦC) ∼= O(m, C) × Sp(n, C). The group Sp(n, C) is connected, while O(m, C) has two +connected components. Reordering if necessary, we may assume that det(X1) = det(X2) = −1 = +− det(X3). By (3.18), we have +HomX1,X2,X3,g(ΦC)(U C, W C) = HomX1,g(ΦC)(U C, W C) = HomG(ΦC)(U C, W C). +The case σ = 1 is easier. +As explained in Appendix A.7, we have G(Φ) = GL(m, R) and +G(ΦC) = GL(m, C), which are both connected. Then, using (3.22), we have +HomG(Φ)(U, W)C = Homg(Φ)(U, W)C ∼= Homg(ΦC)(U C, W C) = HomG(ΦC)(U C, W C). +□ +It follows from Lemma 11.2 that we have a canonical full and faithful superfunctor +(11.1) +ER : (G(Φ)-smodR)C → G(ΦC)-smodC +sending V to V C. Since R is commutative, we can identify R and Rop. Let SR denote the isomor- +phism of Proposition 11.1 when (A, ⋆) = (R, id). +Proposition 11.3. The diagram +Bσ +R(R; ν(m − n))C +Bσ +C(C; ν(m − n)) +(G(Φ)-smodR)C +G(ΦC)-smodC +SR +FC +Φ +FΦC +ER +commutes. +Proof. To simplify notation, we set S = SR, E = ER, F = FΦC, and FC = FC +Φ. On objects, we have +EFC(I) = V C = F(I) = FS(I). +For morphisms, we need to show that +EFC(f) = FS(f) +for f ∈ +� +, +, +� +, +where X and Y are the domain and codomain of f, respectively. +For f = +, we have +EFC( +) = ν flipV C,V C = FS( +). +For f = +, we have +EFC( +) = FS( +): V C ⊗C V C → C, +u ⊗ v �→ ΦC(u, v). +Finally, we consider f = +. Let BV denote an R-basis of V , which we also view as a C-basis of V C. +Then we have +EFC( +) = FS( +): C → V C ⊗C V C, +1 �→ +� +v∈BV +v ⊗ v∨. +□ +Proposition 11.4. Theorem 10.5 holds when (D, ⋆) = (R, id). + +DIAGRAMMATICS FOR REAL SUPERGROUPS +45 +Proof. We wish to show that the R-linear map +FΦ : HomBσ +R (R;ν(m−n))(I⊗r, I⊗s) → HomG(Φ)(V ⊗r, V ⊗s) +is surjective. This map is surjective if and only if the induced map +(11.2) +FC +Φ : HomBσ +R (R;ν(m−n))(I⊗r, I⊗s)C → HomG(Φ)(V ⊗r, V ⊗s)C +is surjective. To show that (11.2) is surjective, it suffices to show that the diagram +(11.3) +HomBσ +R (R;ν(m−n))(I⊗r, I⊗s)C +HomG(Φ)(V ⊗r, V ⊗s)C +HomBσ +C (C);ν(m−n)(I⊗r, I⊗s) +HomG(ΦC) +� +(V C)⊗r, (V C)⊗s� +SR ∼ += +FC +Φ +∼ += +ER +FΦC +commutes, where surjectivity of the bottom horizontal map follows from Proposition 10.4. Com- +mutativity of this diagram follows from Proposition 11.3. +□ +If ϕp,q|2n is the form defined in (A.6), then G(ϕp,q|2n) = OSp(p, q|2n, R) is the indefinite or- +thosymplectic supergroup. Recall the definition G(Φ)-tsmodR of the monoidal supercategory of +tensor G(Φ)-supermodules from Section 10.4. +Proposition 11.5. If p, p′, q, q′, n ∈ N satisfy p + q = p′ + q′, then we have an equivalence of +monoidal supercategories, +OSp(p, q|2n, R)-tsmodR ≃ OSp(p′, q′|2n, R)-tsmodR +sending the natural supermodule to the natural supermodule. +Proof. Viewing B0 +R(R; m − n) as a subcategory of B0 +R(R; m − n)C, it follows from (3.19) and the +commutativity of (11.3), with σ = 0 and ν = 1, that ker(FΦ) = S−1 +R (ker(FΦC)) ∩ B0 +R(R, m − n). In +particular, ker(FΦ) depends only on ΦC. As noted in Appendix A.2, ΦC depends only on p + q and +n, up to equivalence. Hence, both OSp(p, q|2n)-tsmodR and OSp(p′, q′|2n)-tsmodR are equivalent +to the quotient of B0 +R(R, m − n) by this common kernel. +□ +12. Unoriented fullness: complex cases +In this section, we prove Theorem 10.5 in the case where (D, ⋆) is either (C, ⋆) or (Cl(C), ⋆). +Throughout this section, we assume that (D, ⋆) is one of these two complex involutive superalge- +bras. Recall, from Convention 4.3, that D is also a real Frobenius superalgebra, with Nakayama +automorphism ζ given by (4.1), so that +a⋄ = (−1)¯aa⋆, +a ∈ D. +By Lemma 4.5, we can assume that the specialization parameter d is zero when D = Cl(C). +If V is a complex vector superspace, we let V ⋆ denote the conjugate complex vector superspace, +which is a special case of the construction described in Section 7.1. Precisely, V ⋆ is equal to V as +an R-vector superspace, but the C-action is given by +(12.1) +V ⋆ × C → V ⋆, +(v, a) �→ va⋆. +Above, and elsewhere, the juxtaposition vb, for b ∈ C and v ∈ V or V ⋆, will always denote the +C-action on V (as opposed to the C-action on V ⋆). Recall the notion of complexification from +Section 3.7. We have an isomorphism of C-vector spaces +(12.2) +V C ∼ += +−→ V ⊕ V ⋆, +v �→ +1 +√ +2(v, v), +v ∈ V. + +46 +SAIMA SAMCHUCK-SCHNARCH AND ALISTAIR SAVAGE +Note that C-linearity implies that +v ⊗ a �→ +1 +√ +2(va, va⋆), +v ∈ V, a ∈ C. +Recall, from Section 2, the superadditive envelope Add(Cπ) of a supercategory C. +Proposition 12.1. Fix d ∈ k and σ ∈ Z2. There exists a unique C-linear monoidal superfunctor +SD: Bσ +R(D; d)C → Add(OBC(D; d)π) +such that SD(I) = ↑ ⊕Πσ↓ and +SD +� +� += +0 +0 + +σ +σ + +σ +σ + (−1)σ +0 +0 , +(12.3) +SD ( +) = +0 +σ + +0 +σ , +SD ( +) = +σ +0 + (−1)σ +σ +0 , +(12.4) +SD +� +a � += +a +0 +0 + (−1)σ¯a a⋄ +σ +σ , +a ∈ D. +(12.5) +The functor SD is full and faithful. +Proof. The superfunctor SD is the complexification DC of the superfunctor of Theorem 9.5, with d +replaced by d/2, followed by the superfunctor +Add(OBR(D; d/2))C → Add(OBC(D; d)) +given by imposing the relations +a = a +, +a ∈ C. +Note that the doubling of the specialization parameter comes from the fact that, by Lemma 4.5, +strR +D(1) = sdimR D = 2(sdimC D) = 2 strC +D(1). +It remains to prove that SD is full and faithful. For r, s ∈ N, the functor SD induces a C-linear +map +(12.6) +HomBσ +R (D;d)(I⊗r, I⊗s)C → HomAdd(OBσ +C(D;d)π)((↑ ⊕Πσ ↓)⊗r, (↑ ⊗ ↓)⊗s) +∼= +� +X1,...,Xr,Y1,...,Ys∈{↑,Πσ↓} +HomOBσ +C(D;d)π(X1 ⊗ · · · ⊗ Xr, Y1 ⊗ · · · ⊗ Ys). +By Theorem 9.6, HomBσ +R (D;d)(I⊗r, I⊗s)C has R-basis D•(r, s). Thus, it has dimension +(dimR D)(r+s)/2(r + s − 1)!! +if r + s is even and dimension zero if r + s is odd. +(We use here the fact that the number of +perfect matchings of a set of size 2n is (2n − 1)!! := (2n − 1)(2n − 3) · · · 1.) On the other hand, +by Theorem 9.6, � +X1,...,Xr,Y1,...,Ys∈{↑,Πσ↓} HomOBC(D)(X1 ⊗ · · · ⊗ Xr, Y1 ⊗ · · · ⊗ Ys) has the same +dimension since, when r + s is even, there are (r + s − 1)!! perfect matchings of the r + s endpoints, +2(r+s)/2 choices for the orientations of the strands, and then (dimC D)(r+s)/2 ways to put a token +labelled b ∈ BC +D on each strand. Thus, to prove that (12.6) is an isomorphism, it suffices to prove +that it is surjective. To do this, it is enough to show that each generating morphism of OBσ +C(D; d), +with appropriate parity shifts, is in the image of SD. Noting that +SD +� 1 +2 − i +2 +i +� += +0 +0 +and +SD +� 1 +2 + i +2 +i +� += +σ +σ , +we have +1 +4SD +� +− i i +− i +i − i +i +� += +0 +0 , +1 +2SD +� +a − i +ia +� += +a +0 +0 , +a ∈ D, + +DIAGRAMMATICS FOR REAL SUPERGROUPS +47 +1 +2SD +� +− i +i +� += +0 +σ , +1 +2SD +� ++ i +i +� += +0 +σ , +1 +2SD +� ++ i +i +� += +σ +0 , +(−1)σ 1 +2SD +� +− i +i +� += +σ +0 . +□ +Fix m, n ∈ N, and set V = Dm|n. If D = Cl(C), we assume that n = 0; see Lemma 4.9. Note +that, if ϕ is a (ν, ⋆)-superhermitian form on V , then iϕ is a (−ν, ⋆)-superhermitian form on V . +Therefore, without loss of generality, we let ϕ be a nondegenerate (1, ⋆)-superhermitian form on V +of parity σ. (See Appendices A.4 and A.6 for a classification.) Let Φ = τ ◦ ϕ be the corresponding +nondegenerate (ν, ⋄)-supersymmetric form, with a⋄ = (−1)¯aa⋆; see Lemma 7.10 and (4.1). Recall, +from Section 7.5, that G(Φ) = G(ϕ). +Lemma 12.2. For all G(Φ)-supermodules U and W, we have an isomorphism of C-supermodules +HomG(Φ)(U, W)C ∼ += +−→ Homgl(m|n,D)(U C, W C), +f ⊗ a �→ f ⊗ a, +f ∈ HomG(Φ)(U, W), a ∈ C. +Proof. As explained in Appendices A.4, A.6 and A.7, Gred(Φ) is connected. Thus +HomG(Φ)(U, W)C ∼= Homg(Φ)(U, W)C (3.22) +∼= +Homgl(m|n,D)(U C, W C), +where we used Proposition 8.2 in the final isomorphism. +□ +It follows from Lemma 12.2 that we have a canonical full and faithful superfunctor +(12.7) +ED : (G(Φ)-smodR)C → gl(m|n, D)-smodC +sending V to V C. +Since D is complex division superalgebra, V is naturally a complex vector superspace, and hence +the g(Φ)-supermodule V = Dm|n is naturally a supermodule over g(Φ)C ∼= gl(m|n, D). Recall the +isomorphism Ξ⋄ of (9.7). The next result shows that the diagram +Bσ +R(D; m − n)C +Add(OBC(D; m − n)π) +Add(OBC(Dop; m − n)π) +(G(Φ)-smodR)C +gl(m|n, D)-smodC +SD +FC +Φ +Ξ⋄ +Gm|n +ED +commutes up to supernatural isomorphism. Recall the notation Φv introduced in (7.8), and the +notation (2.5) for elements of a parity shift. +Proposition 12.3. There is an monoidal supernatural isomorphism of superfunctors +η: EDFC +Φ +∼ += +−→ Gm|nΞ⋄SD +determined by +(12.8) +ηI : V C ∼ += +−→ V ⊕ ΠσV ∗, +ηI(v) = +1 +√ +2 (v, πσΦv) . +Proof. To simplify notation, we set G = Gm|n, S = SD, E = Eg, FC = FC +Φ, and Ξ = Ξ⋄. First note +that ηI is the composition of the isomorphisms of C-supermodules +V C +∼ += +−−−→ +(12.2) V ⊕ V ⋆ ∼ += +−→ V ⊕ ΠσV ∗, +where the second isomorphism uses the restriction of (7.8) to C-supermodules, noting that V ⋄ = V ⋆ +as C-supermodules. +Thus ηI is a parity-preserving isomorphism of C-vector superspaces. +It is +straightforward to verify that it is also a homomorphism of g(Φ)C-supermodules. + +48 +SAIMA SAMCHUCK-SCHNARCH AND ALISTAIR SAVAGE +On objects, we have +EFC(I) = V C ηI−→ +∼ += V ⊕ ΠσV ∗ = G(↑ ⊕Πσ↓) = GΞS(I). +For morphisms, we need to show that +ηY ◦ EFC(f) = GΞS(f) ◦ ηX +for f ∈ +� +, +, +, +a : a ∈ D +� +, +where X and Y are the domain and codomain of f, respectively. +For f = +, we have +ηI⊗I ◦ EFC( +) = ηI⊗I ◦ flipV C,V C += (flipV ⊗V +Πσ flipV ⊗V ∗ +Πσ flipV ⊗V ∗ +(−1)σ flipV ∗⊗V ∗) ◦ ηI⊗I = GΞS( +) ◦ ηI⊗I, +where the sign of (−1)σ arises from the isomorphism (2.6). +For f = +, noting that η +1 is the identity map C → C, we have, for v, w ∈ V , +η +1 ◦ EFC( +): V C ⊗C V C → C, +v ⊗ w �→ Φ(v, w), +and +GΞS( +) ◦ ηI⊗I = G +� +0 +σ + +0 +σ +� +◦ ηI⊗I: V C ⊗ V C → C, +v ⊗ w �→ 1 +2(v, πσΦv) ⊗ (w, πσΦw) �→ 1 +2 +� +Φ(v, w) + (−1)¯v ¯wΦ(w, v) +� (7.7) += Φ(v, w), +as desired. (Above, we have used the fact that the maps are uniquely determined by their values +on v ⊗ w, for v, w ∈ V ⊆ V C.) +Next we consider f = +. +Let BC +V be a homogeneous C-basis for V . +Then BC +V ∪ BC +V i is a +homogeneous R-basis for V . It is straightforward to verify that, for all w ∈ BC +V , we have +(wi)∨ = w∨i +and +Φw∨ = w∗, +where ∨ denotes left duals with respect to Φ. +Identifying v ∈ V and πσf ∈ ΠσV ∗ with +(v, 0), (0, πσf) ∈ V ⊕ ΠσV ∗, we can write (v, πσf) as v + πσf. Using this convention, +ηI⊗I ◦ EFC( +): C → (V ⊕ ΠσV ∗) ⊗C (V ⊕ ΠσV ∗) +∼ += +−→ (V ⊗C V ) ⊕ Πσ(V ⊗C V ∗) ⊕ Πσ(V ∗ ⊗C V ) ⊕ (V ∗ ⊗C V ∗) +is the map given by +1 �→ 1 +2 +� +w∈BC +V +(−1)σ ¯w(w + πσΦw) ⊗ (w∨ + πσΦw∨) + 1 +2 +� +w∈BC +V +(−1)σ ¯w(iw + πσΦiw) ⊗ (w∨i + πσΦw∨i) += 1 +2 +� +w∈BC +V +(−1)σ ¯w(w + πσΦw) ⊗ (w∨ + πσΦw∨) + 1 +2 +� +w∈BC +V +(−1)σ ¯w(wi − πσΦwi) ⊗ (w∨i − πσΦw∨i) += +� +w∈BC +V +(−1)σ ¯ww ⊗ πσΦw∨ + +� +w∈BC +V +(−1)σ ¯wπσΦw ⊗ w∨ += +� +w∈BC +V +(−1)σ ¯ww ⊗ πσw∗ + +� +w∈BC +V +(−1) ¯wπσw∗ ⊗ w +�→ +� +w∈BC +V +πσw ⊗ w∗ + +� +w∈BC +V +(−1) ¯wπσw∗ ⊗ w, + +DIAGRAMMATICS FOR REAL SUPERGROUPS +49 +where, in the final equality, we used the fact that the last sum is independent of the choice of basis +to sum over the basis BC,∨ +V +, and the fact that (w∨)∨ = (−1)σ ¯w+ ¯ww. +On the other hand, we have +GΞS( +) ◦ η +1 = G +� +σ +0 + (−1)σ +σ +0 +� +: C → Πσ(V ⊗ V ∗) ⊕ Πσ(V ∗ ⊗C V ), +1 �→ +� +w∈BC +V +πσw ⊗ w∗ + +� +w∈BC +V +(−1) ¯wπσw∗ ⊗ w. +Finally, for f = +a , a ∈ D, we have +ηI ◦ EFC( a ): V C → V ⊕ ΠσV ∗, +v �→ (−1)¯a¯vva⋄ �→ (−1)¯a¯v 1 +√ +2(va⋄, πσΦva⋄) +and +GΞS( a ) ◦ ηI = G +� +(a⋄)op +0 +0 + (−1)σ¯a aop +σ +σ +� +: V C → V ⊕ ΠσV ∗, +v �→ +1 +√ +2(v, πσΦv) +(6.4) +�−−−→ +1 +√ +2 +� +(−1)¯a¯vva⋄, (−1)σ¯aaπσΦv� += (−1)¯a¯v 1 +√ +2(va⋄, πσΦva⋄). +□ +Proposition 12.4. Theorem 10.5 holds when (D, ⋆) is equal to (C, ⋆) or (Cl(C), ⋆). +Proof. To show that the superfunctor FΦ is full, we must show that the R-linear map +FΦ : HomBσ +R (D;m−n)(I⊗r, I⊗s) → HomG(Φ)(V ⊗r, V ⊗s) +is surjective. This map is surjective if and only if the induced map +(12.9) +FC +Φ : HomBσ +R (D;m−n)(I⊗r, I⊗s)C → HomG(Φ)(V ⊗r, V ⊗s)C +is surjective. To show that (12.9) is surjective, it suffices to show that the diagram +HomBσ +R (D;m−n)(I⊗r, I⊗s)C +HomG(Φ)(V ⊗r, V ⊗s)C +HomAdd(OBC(D;m−n)) ((↑ ⊕Πσ↓)⊗r, (↑ ⊕Πσ↓)⊗s) +Homgl(m|n,D) +� +(V C)⊗r, (V C)⊗s� +HomAdd(OBC(Dop;m−n)) ((↑ ⊕Πσ↓)⊗r, (↑ ⊕Πσ↓)⊗s) +Homgl(m|n,D) ((V ⊕ ΠσV ∗)⊗r, (V ⊕ ΠσV ∗)⊗s) +SD ∼ += +FC +Φ +∼ += +ED +∼ += +Ξ⋄ +Gm|n +∼ += +commutes, where the bottom-right vertical isomorphism is induced by the isomorphism (12.8). +Commutativity of this diagram follows from Proposition 12.3. +□ +As a special case of G(Φ), we have the supergroups U(p, q|r, s) and UQ(p, q); see Appen- +dices A.4 and A.6. +Recall the definition G(Φ)-tsmodR of the monoidal supercategory of tensor +G(Φ)-supermodules from Section 10.4. +Proposition 12.5. If p, p′, q, q′, r, r′, s, s′ ∈ N satisfy p + q = p′ + q′ and r + s = r′ + s′, then we +have equivalences of monoidal supercategories +U(p, q|r, s)-tsmodR ≃ U(p′, q′|r′, s′)-tsmodR +and +UQ(p, q)-tsmodR ≃ UQ(p′, q′)-tsmodR, +sending the natural supermodule to the natural supermodule. +Proof. The proof is analogous to that of Proposition 11.5, using the commutative diagram appearing +in the proof of Proposition 12.4. +□ + +50 +SAIMA SAMCHUCK-SCHNARCH AND ALISTAIR SAVAGE +13. Unoriented fullness: quaternionic case +In this section, we prove Theorem 10.5 in the case where (D, ⋆) = (H, ⋆). We will naturally view +C = R + Ri as a subalgebra of H. +Suppose that V is an H-vector superspace. Then, using (8.3), we have an isomorphism of C-vector +spaces +(13.1) +V +∼ += +−→ V ⋆, +v �→ vj, +where V ⋆ denotes the conjugate C-vector superspace, as in (12.1). Combining with (12.2), this +yields an isomorphism of C-vector spaces +(13.2) +V C ∼ += +−→ V ⊕ V, +v �→ +1 +√ +2(v, vj), +v ∈ V. +Note that C-linearity implies that +v ⊗ a �→ +1 +√ +2(va, vja) = +1 +√ +2(va, va⋆j), +v ∈ V, a ∈ C. +Recall that the additive envelope of a linear category C is the category Add(C) whose objects are +formal direct sums of objects in C, and whose morphisms are identified with matrices of morphisms +in C in the usual way. We write morphisms in additive envelopes as sums of their components, as +in Section 9.2. In what follows, we will often be considering the object I ⊕ I ∈ Add(BC). In order to +distinguish the two copies of I, we will color the first blue and the second red. We will then color +diagrams in such a way that the color of their endpoints indicate which copy of I is involved. In +order to make the diagrams also readable without color, blue strands will be made thick and the +blue copy of I will be written in bold. Thus, for example, += +� +1I +0 +0 +0 +� +, += +� +0 +0 +0 +1I +� +, += +� +0 +0 +1I +0 +� +, += +� +0 +1I +0 +0 +� +∈ HomAdd(BC)(I ⊕ I, I ⊕ I), +And += + + + + +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 + + + + ∈ EndAdd(BC)((I ⊕ I)⊗2) = EndAdd(BC)((I ⊗ I) ⊕ (I ⊗ I) ⊕ (I ⊗ I) ⊕ (I ⊗ I)). +Proposition 13.1. There exists a unique C-linear monoidal functor +SH : Bσ +R(H; d)C → Add(Bσ +C(C; −2d)) +such that SH(I) = I ⊕ I and +SH +� +� += − +− +− +− +, +SH ( +) = +− +, +SH ( +) = +− +, +(13.3) +SH +� +i +� += i − i , +SH +� +j +� += − , +SH +� +k +� += i + i . +(13.4) +The functor SH is full and faithful. +Proof. To show that SH is well defined, we must show that it respects the relations (9.1) and (9.2). +For the first relation in (9.1), we verify that +SH +� +� += ++ ++ ++ += ++ ++ ++ += SH +� +� +. +The proof of the second relation in (9.1) is similar, since both sides are mapped by SH to the negative +of the sum over all possible colorings of the strands. + +DIAGRAMMATICS FOR REAL SUPERGROUPS +51 +For the third relation in (9.1), we have +SH +� +� += ++ += ++ += SH +� � +. +The proof of the fourth relation in (9.1) is analogous. +To verify the fifth relation in (9.1), we compute +SH +� +� += +− += +− += ( +− +) = SH ( +) . +For the sixth relation in (9.1), we have +SH +� +� += +− ++ +− += +− ++ +− += SH +� +� +. +The first two relations in (9.2) are straightforward. For the third relation in (9.2), it suffices to +consider the cases where a, b ∈ {1, i, j, k}. These are all straightforward computations. For example, +we have +SH +� +i +i � += − − = SH +� +−1 +� +, +SH +� +j +j +� += − − = − − = SH +� +−1 +� +, +SH +� +j +i +� += i + i += SH +� +k +� +, +SH +� +i +j � += −i − i += SH +� +−k +� +. +The other cases are analogous. It is also straightforward to verify the fourth relation in (9.2) by +considering the cases a ∈ {1, i, j, k}. For the fifth relation in (9.2), we again consider the cases +a ∈ {1, i, j, k}. When a = i, we have +SH +� +i +� += −i +− i += SH +� +−i � += SH +� +i⋆ � +. +The other cases are analogous. +Finally, we show that SH respects (9.3). First note that, by (9.6), any bubble with a purely +imaginary token is zero. By (5.1), any bubble with a token labelled by a ∈ R is equal to a times a +bubble with no token. Then the fact that SH respects (9.3) follows from the computation +SH +� +� += − +− += −2 +, +and the fact that, by Lemma 4.5, +strR +H(a) = 4a +and +strC +C(a) = a +for all a ∈ R. +It remains to prove that SH is full and faithful. For r, s ∈ N, the functor SH induces a C-linear +map +(13.5) +HomBσ +R (H;d)(I⊗r, I⊗s)C → HomAdd(Bσ +C (C;−2d))((I ⊕ I)⊗r, (I ⊕ I)⊗s). +By Theorem 9.6, HomBσ +R (H;d)(I⊗r, I⊗s)C has C-basis D•(r, s). Thus, it has dimension +4(r+s)/2(r + s − 1)!! = 2r+s(r + s − 1)!! +if r + s is even and dimension zero if r + s is odd. (We use here the fact that the number of perfect +matchings of a set of size 2n is (2n)!! := (2n − 1)(2n − 3) · · · 1, and that dimR H = 4.) On the other +hand, HomAdd(Bσ +C (C;−2d))((I⊕I)⊗r, (I⊕I)⊗s) has the same dimension, since, when r +s is even, there +are (r + s − 1)!! perfect matchings of the r + s endpoints, and then 2r+s ways to choose one of the +colors blue or red for the endpoints of the strings. Thus, to prove that (13.5) is an isomorphism, it +suffices to prove that it is surjective. To do this, it is enough to show that all possible colorings of +the generating morphisms of Bσ +C(C; −2d) lie in the image of SH. +First note that +SH +� 1 +2 + i +2 +i +� += , +SH +�1 +2 − i +2 +i +� += , +(13.6) + +52 +SAIMA SAMCHUCK-SCHNARCH AND ALISTAIR SAVAGE +SH +� +− 1 +2 +j − i +2 +k +� += , +SH +� +1 +2 +j − i +2 +k +� += . +(13.7) +Next, we compute +− 1 +4SH +� ++ i i ++ i +i − i +i +� += +, +SH +�1 +2 +a + i +2 +ia +� += +a , a ∈ C, +1 +2SH +� +j − i +k +� += += +, +1 +2SH +� +j + i +k +� += += +. +Then, composing with the morphisms in (13.7) to change colors of strands, we see that all possible +colorings of the generating morphisms lie in the image of SH, as desired. +□ +Fix m, n ∈ N, set V = Hm|n, and let ϕ be a nondegenerate (ν, ⋆)-superhermitian form on V +of parity σ. (See Appendices A.5 and A.7 for a classification.) Recall, from Lemma 8.4, that the +induced form ϕj : C2m|2n × C2m|2n → C is nondegenerate, (−ν, id)-supersymmetric, and of parity +σ. +Let Φ denote the (ν, ⋆)-supersymmetric form defined as in (7.12), where we recall that the +Frobenius form τ on H is projection onto the real part. Recall, from Section 7.5, that G(Φ) = G(ϕ) +and g(Φ) = g(ϕ). +Lemma 13.2. For all G(Φ)-supermodules U and W, we have an isomorphism of C-supermodules +HomG(Φ)(U, W)C ∼ += +−→ HomG(ϕj)(U C, W C), +f ⊗ a �→ f ⊗ a, +f ∈ HomG(Φ)(U, W), a ∈ C. +Proof. First suppose that σ = 0. Then, as explained in Appendix A.1, we may assume that ν = 1. +By the results of Appendix A.5, we see that Gred(Φ) has two connected components when n ≥ 1. +(The case n = 0 is easier, and similar to the σ = 1 case discussed below.) Fix X ∈ Gred(Φ) with +det(X) = −1, so that X is in the connected component of Gred(Φ) not containing the identity. +Then, using Proposition 8.6 we have +HomG(Φ)(U, W)C (3.23) += +HomX,g(ϕ)C(U C, W C) ∼= HomX,g(ϕj)(U C, W C) +(3.18) += +HomG(ϕj)(U C, W C), +where, in the final equality, we used the fact that Gred(ϕj) has two connected components, and X +lies in the connected component not containing the identity, as explained in Appendix A.2. +The case σ = 1 is easier. As explained in Appendix A.7, G(Φ) and G(ϕj) are both connected. +Then, using (3.22), we have +HomG(Φ)(U, W)C = Homg(Φ)(U, W)C ∼= Homg(ϕj)(U C, W C) = HomG(ϕj)(U C, W C). +□ +It follows from Lemma 13.2 that we have a canonical full and faithful superfunctor +(13.8) +EH : (G(Φ)-smodR)C → G(ϕj)-smodC +sending V to V C. The next result shows that the diagram +Bσ +R(H, ν(m − n))C +Add(Bσ +C(C, ν(2n − 2m))) +(G(Φ)-smodR)C +G(ϕj)-smodC +SH +FC +Φ +Fϕj +EH +commutes up to supernatural isomorphism. +Proposition 13.3. There is a monoidal supernatural isomorphism of functors +θ: EHFC +Φ +∼ += +−→ FϕjSH +determined by +(13.9) +θI : V C ∼ += +−→ V ⊕ V, +θI(v) = +1 +√ +2(v, vj), +v ∈ V. + +DIAGRAMMATICS FOR REAL SUPERGROUPS +53 +Proof. To simplify notation, we set S = SH and E = EH. First note that θI is the isomorphism +(13.2). On objects, we have +EFC +Φ(I) = V C θI−→ +∼ += V ⊕ V = Fϕj(I ⊕ I) = FϕjSH(I). +Here, and it what follows, in the isomorphism V C ∼= V ⊕ V , the first copy of V and its elements +are denoted by bold blue characters and the second copy of V and its elements are denoted by red +non-bold characters. This matches our diagrammatic conventions introduced earlier. +For morphisms, we need to show that +θY ◦ EFC +Φ(f) = FϕjS(f) ◦ θX +for f ∈ +� +, +, +, +i , +j +� +, +where X and Y are the domain and codomain of f, respectively. (Since +k = +i ◦ +j , we do not +need to check f = +k .) +For f = +, we have +θI⊗I ◦ EFC +Φ( +) = νθI⊗I ◦ flipV C,V C = ν flipV ⊕V ,V ⊕V ◦θI⊗I = FϕjS( +) ◦ θI⊗I. +Note that the negative sign appearing in the definition of S( +) cancels with the negative sign arising +from the fact that ϕj is (−ν, id)-supersymmetric. +For f = +, noting that θ +1 is the identity map C → C, we have, for v, w ∈ V , +θ +1 ◦ EFC +Φ( +): V C ⊗C V C → C, +v ⊗ w �→ Re ϕ(v, w), +and +FϕjS( +) ◦ θI⊗I = Fϕj( +− +) ◦ θI⊗I: V C ⊗C V C → C, +v ⊗ w �→ 1 +2(v, vj) ⊗ (w, wj) �→ 1 +2(ϕj(v, wj) − ϕj(vj, w)) +(8.10) += +Re ϕ(v, w). +(Above, we have used the fact that the maps are uniquely determined by their values on v ⊗ w, for +v, w ∈ V ⊆ V C.) +Next we consider f = +. Choose a C-basis BC +V of V . Then {v, vi : v ∈ BC +V } is an R-basis of +V , and it is straightforward to verify that (vi)∨ = v∨i. +Identifying w ∈ V and v ∈ V and with +(w, 0), (0, v) ∈ V ⊕ V , we can write (w, v) as w + v. Using this convention, we have that +θI⊗I ◦ EFC +Φ( +): C → (V ⊕ V ) ⊗C (V ⊕ V ) +is the map given by +1 �→ +� +v∈BR +V +(−1)σ¯vv ⊗ v∨ = +� +v∈BC +V +(−1)σ¯v(v ⊗ v∨ + vi ⊗ v∨i) +�→ 1 +2 +� +v∈BC +V +(−1)σ¯v� +(v + vj) ⊗ (v∨ + v∨j) + (vi + vij) ⊗ (v∨i + v∨ij) +� += +� +v∈BC +V +(−1)σ¯vv ⊗ v∨j + +� +v∈BC +V +(−1)σ¯vvj ⊗ v∨. +On the other hand, for all v, w ∈ BC +V , we have +ϕj(w∨j, v) +(8.8) += −ϕ1(w∨, v) = −δvw +and +ϕj(w∨, vj) +(8.9) += ϕ1(w∨, v)⋆ = δvw. +Thus, the C-bases left dual to BC +V and {vj : v ∈ BC +V } with respect to ϕj are {−v∨j : v ∈ BC +V } and +{v∨ : v ∈ BC +V }, respectively. Therefore, +FϕjS( +) ◦ η +1 = Fϕj( +− +): C → (V ⊗C V ) ⊕ (V ⊗C V ), + +54 +SAIMA SAMCHUCK-SCHNARCH AND ALISTAIR SAVAGE +1 �→ +� +v∈BC +V +(−1)σ¯vvj ⊗ v∨ + +� +v∈BC +V +(−1)σ¯vv ⊗ v∨j. +For f = +i , we have +θI ◦ EFC +Φ +� +i +� +: V C → V ⊕ V , +v �→ −vi �→ +1 +√ +2(−vi, −vij) = +1 +√ +2(−vi, vji) +and +FϕjS +� +i +� +◦ θI = Fϕj +� +i − i +� +◦ ηI : V C → V ⊕ V , +v �→ +1 +√ +2(v, vj) �→ +1 +√ +2(−vi, vji). +For f = +j , we have +θI ◦ EFC +Φ +� +j +� +: V C → V ⊕ V , +v �→ −vj �→ +1 +√ +2(−vj, v) +and +FϕjS +� +j +� +◦ θI = Fϕj +� +− +� +◦ ηI : V C → V ⊕ V , +v �→ +1 +√ +2(v, vj) �→ +1 +√ +2(−vj, v). +□ +Proposition 13.4. Theorem 10.5 holds when (D, ⋆) = (H, ⋆). +Proof. We wish to show that, for all r, s ∈ N, the R-linear map +FΦ : HomBσ +R (H;ν(m−n))(I⊗r, I⊗s) → HomG(Φ)(V ⊗r, V ⊗s) +is surjective. This map is surjective if and only if the map +(13.10) +FC +Φ : HomBσ +R (H,⋆;ν(m−n))(I⊗r, I⊗s)C → HomG(Φ)(V ⊗r, V ⊗s)C +is surjective. To show that (13.10) is surjective, it suffices to show that the diagram +HomBσ +R (H;ν(m−n))(I⊗r, I⊗s)C +HomG(Φ)(V ⊗r, V ⊗s) ⊗R C +HomAdd(Bσ +C (C;ν(2n−2m))) ((I ⊕ I)⊗r, (I ⊕ I)⊗s) +HomG(ϕj) +� +(V C)⊗r, (V C)⊗s� +HomG(ϕj)((V ⊕ V )⊗r, (V ⊕ V )⊗s) +SH ∼ += +FC +Φ +∼ += +EH +Fϕj +∼ += +commutes, where the diagonal isomorphism is induced by the isomorphism (13.9), and surjectivity +of the bottom-left vertical arrow follows from Proposition 10.4. Commutativity of this diagram +follows from Proposition 13.3. +□ +As a special case of G(Φ), we have the quaternionic orthosymplectic supergroups OSp∗(n|p, q); +see Appendix A.5. +Recall the definition G(Φ)-tsmodR of the monoidal supercategory of tensor +G(Φ)-supermodules from Section 10.4. +Proposition 13.5. If p, p′, q, q′, n ∈ N satisfy p + q = p′ + q′, then we have an equivalence of +monoidal supercategories +OSp∗(n|p, q)-tsmodR ≃ OSp∗(n|p′, q′)-tsmodR +sending the natural supermodule to the natural supermodule. +Proof. The proof is analogous to that of Proposition 11.5, using the commutative diagram appearing +in the proof of Proposition 13.4. +□ + +DIAGRAMMATICS FOR REAL SUPERGROUPS +55 +Appendix A. Classification of superhermitian forms +In this appendix, we give the classification of (ν, ⋆)-superhermitian forms for the various choices +of involutive division superalgebra. In each case, we also give the corresponding Harish-Chandra +superpair. The explicit descriptions given in this appendix show that the Lie superalgebras of the +form g(ϕ), together with the Lie superalgebras gl(r|s, D) for a real division superalgebra D, give all +the real forms of gl(m|n, C), osp(m|2n, C), p(m, C), and q(m, C); see Proposition B.3. +A.1. Preliminaries. For any supermatrix X, let +X♯ := (X⋆)st = (Xst)⋆. +The isomorphism (3.15), together with the isomorphism of superalgebras Matm|n(Aop) +∼ += +−→ Matm|n(A), +Xop �→ X⋆, shows that +(A.1) +(XY )♯ = (−1) +¯ +X ¯Y Y ♯X♯ +whenever the product XY is defined. It follows from (3.14) that, for X ∈ Mat(m|n)×(r|s)(A), we +have +(X♯)♯ = Sm|nXSr|s +where +Sp|q = +� +Ip +0 +0 +−Iq +� +. +We will often omit the subscripts on Sp,q when its size is clear from the context. Since we identify +Am|n = Mat(m|n)×(1|0)(A), and S1|0 = I1, we have +(A.2) +� +v0 +v1 +�♯ += +� +v⋆,tr +0 +−v⋆,tr +1 +� +and +(v♯)♯ = Sv +for +v = +� +v0 +v1 +� +∈ Am|n. +Every sesquilinear form on Am|n is of the form +(A.3) +ϕ(v, w) = v♯Mw +where +M ∈ Matm|n(A) is homogeneous. +Lemma A.1. The form ϕ given by (A.3) is (ν, ⋆)-superhermitian if and only if M♯ = ν(−1) ¯ +MMS. +Proof. For v, w ∈ Am|n, we have +(A.4) +ϕ(w, v)⋆ = ϕ(w, v)♯ (A.1) += (−1)¯v ¯w+ ¯ +M(¯v+ ¯w)v♯M♯(w♯)♯ (A.2) += (−1)¯v ¯w+ ¯ +M(¯v+ ¯w)v♯M♯Sw. +Now, if ¯ϕ = +¯ +M = 0, then ϕ(w, v) = 0 = ϕ(v, w) unless ¯v + ¯w = 0. Then (A.4) implies that +ϕ is (⋆, ν)-superhermitian if and only if M = νM♯S. On the other hand, if ¯ϕ = +¯ +M = 1, then +ϕ(w, v) = 0 = ϕ(v, w) unless ¯v + ¯w = 1. Then (A.4) implies that ϕ is (⋆, ν)-superhermitian if and +only if M = −νM♯S. Combining both cases, and using the fact that S2 = I, the result follows. +□ +Lemma A.2. For ϕ as in (A.3), we have +X† = (M♯S)−1X♯M♯S. +Proof. For v, w ∈ Am|n and X ∈ gl(m|n, A), we have +(−1) +¯ +X¯vϕ(X†v, w) +(A.1) += v♯(X†)♯Mw +and +ϕ(v, Xw) = v♯MXw. +Thus +(X†)♯M = MX =⇒ M♯SX†S = X♯M♯ =⇒ X† = (M♯S)−1X♯M♯S. +□ +For the remainder of this section (D, ⋆), will denote an involutive division superalgebra over +k ∈ {R, C}. Our goal is to classify the (ν, ⋆)-superhermitian forms over such superalgebras. An +even (ν, ⋆)-superhermitian form V is equivalent to an even (−ν, ⋆)-superhermitian form on ΠV . +Thus, for even forms, we will only treat the (1, ⋆)-superhermitian case. For odd forms, we will need + +56 +SAIMA SAMCHUCK-SCHNARCH AND ALISTAIR SAVAGE +to treat both the ν = 1 and ν = −1 cases. (However, these lead to isomorphic Lie algebras; see +(A.11).) +An even (1, ⋆)-superhermitian form on a D-supermodule V = V0 ⊕V1 corresponds, when viewing +V as a (non-super) k-module, to a superhermitian form on V0 and an skew-superhermitian form +on V1. This allows us to use well-known results classifying such forms up to equivalence. (See, for +example, [Lew82].) They are typically classified by dimension or signature. +A.2. Case (C, id), k = C, even form. There are no even nondegenerate (1, id)-superhermitian +forms on Cm|n when n ∈ 2Z + 1. Every even nondegenerate (1, id)-superhermitian form on Cm|2n +is equivalent to +(A.5) +ϕm|2n : (v, w) �→ vt + + +Im +0 +0 +0 +0 +In +0 +−In +0 + + w. +Then G(ϕm|2n) is the complex orthosymplectic supergroup OSp(m|2n, C). We have +Gred(ϕm|2n) = O(m, C) × Sp(2n, C) +and +g(ϕm|2n) = osp(m|2n, C). +The group Sp(2n, C) is connected, whereas, when m ≥ 1, O(m, C) has two connected components: +the identity component SO(n, C) and the elements of O(m, C) with determinant −1. It follows +that, for any nondegenerate even (ν, id)-superhermitian form ϕ on Cm|n, the group Gred(ϕ) has two +connected components. Any X ∈ Gred(ϕ) with det(X) = −1 is in the connected component not +containing the identity. +A.3. Case (R, id), k = R, even form. There are no even nondegenerate (1, id)-superhermitian +forms on Rm|n when n ∈ 2Z + 1. Every even nondegenerate (1, id)-superhermitian form on Rm|2n +is equivalent to +(A.6) +ϕp,q|2n: (v, w) �→ vt + + + + +Ip +0 +0 +0 +0 +−Iq +0 +0 +0 +0 +0 +In +0 +0 +−In +0 + + + + w, +v, w ∈ Rm|2n, +for some p, q ∈ N, p+q = m. Furthermore, ϕp,q|2n ∼ ϕp′,q′|2n if and only if (p′, q′) = (p, q) or (p′, q′) = +(q, p). The supergroup G(ϕp,q|2n) is the indefinite orthosymplectic supergroup OSp(p, q|2n, R). We +have +Gred(ϕp,q|2n) = O(p, q) × Sp(2n, R) +and +g(ϕp,q|2n) = osp(p, q|2n, R). +The real symplectic group Sp(2n, R) is connected. When p, q ≥ 1, the indefinite orthogonal group +O(p, q) has four connected components, corresponding to the two choices ±1 of determinant for the +restriction to the two subspaces Rp × 0q and 0p × Rq of Rm. When p = 0 or q = 0, but p + q ≥ 1, +O(p, q) has two connected components. +Since ϕC +p,q|2n is equivalent to the form ϕm|2n of (A.5), it follows from Proposition 8.1 that +osp(p, q|2n, R)C = g(ϕp,q|2n)C ∼= g(ϕC +p,q|2n) ∼= osp(m|2n, C). +A.4. Case (C, ⋆), k = R, even form. Every even nondegenerate (1, ⋆)-superhermitian form on +Cm|n is equivalent to +(A.7) +ϕp,q|r,s: (v, w) �→ v♯ + + + + +Ip +0 +0 +0 +0 +−Iq +0 +0 +0 +0 +iIr +0 +0 +0 +0 +−iIs + + + + w + +DIAGRAMMATICS FOR REAL SUPERGROUPS +57 +for some p, q, r, s ∈ N, p + q = m, r + s = n. Furthermore, ϕp,q|r,s ∼ ϕp′,q′|r′,s′ if and only if +(p′, q′, r′, s′) ∈ {(p, q, r, s), (q, p, s, r), (r, s, p, q), (s, r, q, p)}. +We call U(p, q|r, s) := G(ϕp,q|r,s) the +indefinite unitary supergroup. We have +Gred(ϕp,q|r,s) = U(p, q) × U(r, s) +and +g(ϕp,q|r,s) = u(p, q|r, s). +Since the indefinite unitary group U(p, q) is connected for all p, q ∈ N, it follows that Gred(ϕ) is +connected for any nondegenerate even (ν, ⋆)-superhermitian form ϕ on Cm|n. +It follows from Proposition 8.2 that we have an isomorphism of complex Lie superalgebras +u(p, q|r, s)C = g(ϕp,q|r,s)C ∼= gl(m|n, C). +A.5. Case (H, ⋆), k = R, even form. Every even nondegenerate (1, ⋆)-superhermitian form on +Hm|n is equivalent to +(A.8) +ϕp,q|n: (v, w) �→ v♯ + + +Ip +0 +0 +0 +−Iq +0 +0 +0 +jIn + + w +for some p, q ∈ N, p + q = m. (See, for example, [Lew82, §5, §6].) We have +ϕp,q|n ∼ ϕp′,q′|n′ ⇐⇒ (p′, q′, n) ∈ {(p, q, n), (q, p, n)}. +We call OSp∗(n|p, q) := G(ϕp,q|n) the quaternionic orthosymplectic supergroup. We have +Gred(ϕp,q|n) = O(n, H) × U(p, q, H) +and +g(ϕp,q|n) = osp∗(n|p, q), +where O(n, H) is the quaternionic orthogonal group, sometimes denoted O∗(2n) in the literature, and +U(p, q, H) is the indefinite quaternionic unitary group, which is equal to the indefinite symplectic +group Sp(p, q). +The notation for g(ϕp,q|n) is not consistent in the literature. +For example, it +is denoted osp∗(n|2m, 2p) in [Ser83]. The indefinite symplectic group Sp(p, q) is connected; see +[Kna02, Prop. 1.145]. The quaternionic orthogonal group O(n, H) has two connected components +when n ≥ 1. Viewing elements of O(n, H) as 2n × 2n complex matrices, the identity component, +SO(n, H), of O(n, H) consists of those elements with determinant 1, while the other component +consists of those elements with determinant −1. +It follows that, for any nondegenerate even (ν, ⋆)- +superhermitian form ϕ on Hm|n, n ≥ 1, the group Gred(ϕ) has two connected components. Any +X ∈ Gred(ϕ) with det(X) = −1 is in the connected component not containing the identity. +Lemma A.3. We have an isomorphism of complex Lie superalgebras +g(ϕp,q|n)C ∼= osp(2n|2m, C). +(Note the reversal in the order of m and n on the right-hand side.) +Proof. Consider the form (ϕp,q|n)j in the notation of Section 8.3. Since a skew-supersymmetric form +on C2m|2n is equivalent to a supersymmetric form on C2n|2m (via the parity shift map C2m|2n → +C2n|2m), we see that (ϕp,q|n)j is equivalent to the form ϕ2n|2m given by replacing m and n in (A.5) +by 2n and 2m, respectively. Thus, the result follows from Proposition 8.6. +□ +A.6. Case (Cl(C), ⋆), k = R, even form. In light of Lemma 4.9, we assume in this case that +n = 0. Every even nondegenerate (1, ⋆)-superhermitian form on Cl(C)m is equivalent to +(A.9) +ϕp,q : (v, w) �→ v♯ +� +Ip +0 +0 +−Iq +� +w +for unique p, q ∈ N, p + q = m. +We call UQ(p, q) := G(ϕp,q) the indefinite isomeric unitary +supergroup. We have +Gred(ϕp,q) = U(p, q). + +58 +SAIMA SAMCHUCK-SCHNARCH AND ALISTAIR SAVAGE +The notation for g(ϕp,q) is not consistent in the literature. It is sometimes denoted by uq(p, q). +Its simple quotient is denoted upsq(n, p) in [Ser83]. Since the indefinite unitary group U(p, q) is +connected, it follows that, for any nondegenerate (ν, ⋆)-superhermitian form on Cl(C)m, the group +Gred(ϕ) is connected. +It follows from Proposition 8.2 that we have an isomorphism of complex Lie superalgebras +g(ϕp,q)C ∼= gl(m, Cl(C)) = q(m, C). +A.7. Odd forms. Let k ∈ {R, C} and let (D, ⋆) be an arbitrary involutive division k-superalgebra. +If k = R and ϕ is a nondegenerate (ν, ⋆)-superhermitian form on Cl(C)m|n, then ε(1 + i)ϕ is a +nondegenerate (ν, ⋆)-superhermitian form on Cl(C) of parity ¯ϕ + 1. +Since we already treated the +even forms above, we assume in this subsection that D ∈ {R, C, H}. +An odd form on Dm|n can only be nondegenerate when m = n. Any odd nondegenerate (ν, ⋆)- +superhermitian form on Dm|m is equivalent to +(A.10) +ϕν +m : (v, w) �→ v♯ +� +0 +Im +−νIm +0 +� +w. +(Recall, from (A.2), the sign appearing in v♯.) With this form, it follows from Lemma A.2 that +� +X00 +X01 +X10 +X11 +�† += +� +X♯ +11 +−νX♯ +01 +νX♯ +10 +X♯ +00. +� +. +Thus, +g(ϕν +m) = +��X +Y +Z +−X♯ +� +: X, Y, Z ∈ Matm(D), Y = νY ♯, Z = −νZ♯ +� +. +In particular, we have an isomorphism of Lie superalgebras +(A.11) +g(ϕν +m) +∼ += +−→ g(ϕ−ν +m ), +X �→ X#. +We also have +Gred(ϕν +m) = +��X +0 +0 +(X♯)−1 +� +: A = GL(m, D) +� +∼= GL(m, D). +It follows that Gred(ϕ) is connected for any nondegenerate odd (ν, ⋆)-superhermitian form ϕ on +Dm|n. When (D, ⋆) is equal to (R, id) or (C, id), the Lie superalgebras +g(ϕν +m) = p(m, R) +and +g(ϕν +m) = p(m, C) +are the real and complex periplectic Lie superalgebras, respectively. The notation for the other cases +is less standard. When (D, ⋆) = (C, ⋆), the Lie superalgebra g(ϕm) is sometimes denoted up(m). +When (D, ⋆) = (H, ⋆), g(ϕν +m) is sometimes denoted p∗(m). +Their simple quotients are denoted +usπ(m) and sπ∗(m), respectively, in [Ser83]. +It follows from Propositions 8.1, 8.2 and 8.6 and (A.11) that we have isomorphisms of complex +Lie superalgebras +g(ϕν +m)C ∼= + + + + + +p(m, C) +if (D, ⋆) = (R, id), +gl(m|m, C) +if (D, ⋆) = (C, ⋆), +p(2m, C) +if (D, ⋆) = (H, ⋆). + +DIAGRAMMATICS FOR REAL SUPERGROUPS +59 +Appendix B. Classification of real forms +A classification of the real forms of the classical Lie algebras can be found in [FH91, §26.1]. The +classification of the real simple Lie superalgebras was first given in [Ser83]. In particular, [Ser83, +Table 3] lists all the real forms of the simple subquotients of gl(m|n, C), osp(m|2n, C), p(m, C), and +q(m, C). However, because gl(m|n, C), p(m, C) and q(m, C) are not simple, they are not covered +by this classification. Since the real forms of these Lie superalgebras do not seem to have appeared +in the literature, we give a classification here. +Throughout this subsection g denotes one of the superalgebras gl(m|n, C), q(m, C), or p(m, C), +m, n ∈ N. Let +g′ = [g, g] +and +g′′ = g′/Z(g′), +where [g, g] denotes the ideal of g generated by [X, Y ], X, Y ∈ g, and Z(g′) = {X ∈ g′ : [X, g′] = 0} +denotes the center of g′. For m, n ∈ N, +gl(m|n, C)′ = sl(m|n, C) = {X ∈ gl(m|n, C) : str(X) = 0}, +q(m, C)′ = {X ∈ q(m, C) = gl(m, Cl(C)) : tr(X)1 = 0}, +p(m, C)′ = +�� +X00 +X01 +X10 +−Xt +00 +� +∈ gl(m|m, C) : tr(X00) = 0, Xt +01 = X01, Xt +10 = −X10 +� +. +Since, for m ̸= n, +Z(sl(m|n, C)) = 0, +Z(sl(m|m, C)) = CI2m, +Z(q(m, C)′) = CIm, +Z(p(m, C)′) = 0, +we have +gl(m|n, C)′′ = sl(m|n, C), +gl(m|m, C)′′ = sl(m|m, C)/CI2m = psl(m|m, C), +q(m, C)′′ = q(m, C)′/CIm, +p(m, C)′′ = p(m, C)′. +The adjoint action of g on itself is given by +Ad: g → Endg(g), +Ad(X)(Y ) = [X, Y ], +X, Y ∈ g. +This restricts to give an action Ad′ : g → Endg(g′) of g on g′. +Lemma B.1. We have ker(Ad′) = Z(g). +Proof. We show this for the case g = gl(m|n, C), since the other cases are analogous. Suppose +X = �m+n +r,s=1 arsErs, ars ∈ C, is a homogeneous element of ker(Ad′), where Ers denotes the matrix +with a 1 in position (r, s), and a 0 in all other positions. Then, for all 1 ≤ t, u ≤ m + n, we have +0 = [X, Etu] = +m+n +� +r=1 +artEru ± +m+n +� +s=1 +ausEts. +Thus, ars = 0 whenever r ̸= s. So X is diagonal, hence even. Then, for all 1 ≤ t, u ≤ m + n, we +have +0 = [X, Etu] = (att − auu)Etu. +Thus att = auu, and so X ∈ CIm+n = Z(gl(m|n, C)). +□ +A conjugate-linear involution of g (also called a real structure on g) is an automorphism κ of g, +considered as a real Lie superalgebra, satisfying κ2 = id and κ(aX) = a⋆κ(X) for all a ∈ C. Every +real form of g is isomorphic to +gκ = {X ∈ g : κ(X) = X} + +60 +SAIMA SAMCHUCK-SCHNARCH AND ALISTAIR SAVAGE +for some conjugate-linear involution κ of g. +If κ is a conjugate-linear involution of g, then κ restricts to a conjugate-linear involution κ′ of +g′, which, in turn, induces a conjugate-linear involution κ′′ of g′′. +Lemma B.2. Suppose κ and χ are conjugate-linear involutions of g such that κ′′ = χ′′. Then +κ′ = χ′ and κ(X) − χ(Z) ∈ Z(g) for all X ∈ g. +Proof. Since the odd part of g′′ is equal to the odd part of g, we have κ|g1 = χ|g1. We see by +inspection that [g1, g1] = g′. Thus, κ′ = χ′. Then, for all X ∈ g, Y ∈ g′, we have +[κ(X), χ(Y )] = [κ(X), κ(Y )] = κ([X, Y ]) = χ([X, Y ]) = [χ(X), χ(Y )]. +It follows that Ad′(κ(X)) = Ad′(χ(X)). Hence, by Lemma B.1, we have κ(X) − χ(X) ∈ Z(g). +□ +Proposition B.3. Every real form of gl(m|n, C), osp(m|2n, C), p(m, C), q(m, C), m, n ∈ N, is +isomorphic to either +• gl(r|s; D) for a real division superalgebra D, or +• g(ϕ) for a (ν, ⋆)-superhermitian form ϕ on Dr|s, where (D, ⋆) is an involutive real division +superalgebra, +for some r, s ∈ N. +Proof. In the case of osp(m|2n, C), which is simple, we see from [Ser83, Table 3] that the real forms +are osp(p, q|2n, R), p + q = m and, when m is even, osp∗(m +2 |p, q), p + q = n. Then the result follows +from Appendices A.3 and A.5. +Now assume that g is one of the Lie superalgebras gl(m|n, C), q(m, C), or p(m, C), m, n ∈ N. +All of the real forms described in Appendices A.3 to A.7 induce real forms of g′′. Comparing to +the real forms given [Ser83, Table 3] shows that we obtain all real forms of g′′ in this way. Thus, it +remains to show that, up to isomorphism, a real form of g is determined by the corresponding real +form of g′′. Equivalently, it suffices to show that, if κ and χ are conjugate-linear involutions of g +such that κ′′ = χ′′, then κ = χ. +Suppose κ and χ are conjugate-linear involutions of g such that κ′′ = χ′′. Then, by Lemma B.2, +κ′ = χ′. If g = p(m, C), then Z(g) = 0, and it follows from Lemma B.2 that κ = χ, and we are +done. +Now suppose that g = g(m|n, C), m ̸= n, or g = q(m, C). Then g = g′ ⊕ CI, where I denotes +the identity matrix. Both κ and χ must leave Z(g) = CI invariant, hence the corresponding real +forms must be isomorphic to (g′)κ ⊕ R = (g′)χ ⊕ R, where R denotes the one-dimensional abelian +real Lie algebra. +Finally, suppose that g = gl(m|m, C). We have a short exact sequence of Lie superalgebras +0 → sl(m|m, C) → gl(m|m, C) str +−→ Z(g) = C → 0. +Since κ and χ agree on sl(m|m, C), it follows from Lemma B.2 that there exists a ∈ C such that +κ(X) = χ(X) + a str(X)I, +X ∈ g. +Then, for all X ∈ g, we have +X = κ2(X) = κ (χ(X) + a str(X)I) = X + a⋆ str(X)⋆χ(I) + a str(X)I. +Thus, +a⋆ str(X)⋆χ(I) + a str(X) = 0 +for all X ∈ g. +Choosing X = E11 and X = iE11, implies that a⋆χ(I) + a = 0 and a⋆χ(I) − a = 0. Hence a = 0, +and so κ = χ, as desired. +□ + +DIAGRAMMATICS FOR REAL SUPERGROUPS +61 +References +[Bae20] +J. Baez. The tenfold way. Notices Amer. Math. 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Department of Mathematics and Statistics, University of Ottawa, Ottawa, ON K1N 6N5, +Canada +Email address: ssamc090@uottawa.ca +(A.S.) Department of Mathematics and Statistics, University of Ottawa, Ottawa, ON K1N 6N5, +Canada +URL: alistairsavage.ca, ORCiD: orcid.org/0000-0002-2859-0239 +Email address: alistair.savage@uottawa.ca + diff --git a/uNAzT4oBgHgl3EQfc_zP/content/tmp_files/load_file.txt b/uNAzT4oBgHgl3EQfc_zP/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6233d4e05d0055e9751b75529abad0d315376a27 --- /dev/null +++ b/uNAzT4oBgHgl3EQfc_zP/content/tmp_files/load_file.txt @@ -0,0 +1,2944 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf,len=2943 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='01414v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='RT] 4 Jan 2023 DIAGRAMMATICS FOR REAL SUPERGROUPS SAIMA SAMCHUCK-SCHNARCH AND ALISTAIR SAVAGE Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We introduce two families of diagrammatic monoidal supercategories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The first family, depending on an associative superalgebra, generalizes the oriented Brauer category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The second, depending on an involutive superalgebra, generalizes the unoriented Brauer category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' These two families of supercategories admit natural superfunctors to supercategories of supermodules over general linear supergroups and supergroups preserving superhermitian forms, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We show that these superfunctors are full when the superalgebra is a central real division superalgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' As a consequence, we obtain first fundamental theorems of invariant theory for all real forms of the general linear, orthosymplectic, periplectic, and isomeric supergroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We also deduce equiva- lences between monoidal supercategories of tensor supermodules over the real forms of a complex supergroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Contents 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Introduction 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Monoidal supercategories 5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Superalgebras and supermodules 7 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Real division superalgebras 13 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The oriented supercategory 16 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The oriented incarnation superfunctor 20 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Superhermitian forms over involutive superalgebras 26 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Superhermitian forms over involutive real division superalgebras 30 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The unoriented supercategory 33 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The unoriented incarnation superfunctor 38 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Unoriented fullness: real case 43 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Unoriented fullness: complex cases 45 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Unoriented fullness: quaternionic case 50 Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Classification of superhermitian forms 55 Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Classification of real forms 59 References 61 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Introduction Many recent developments in representation theory involve one or more of the following interre- lated concepts: (a) Dual pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The classic examples are Schur–Weyl duality, which yields a precise relationship between the symmetric group and the general linear group, and the analogue for the orthog- onal and symplectic groups, where the symmetric groups are replaced by Brauer algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' 18M05, 18M30, 17B10, 18M25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Monoidal category, supercategory, supergroup, string diagram, invariant theory, Deligne category, interpolating category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' 2 SAIMA SAMCHUCK-SCHNARCH AND ALISTAIR SAVAGE (b) Invariant theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' This amounts to giving explicit descriptions of invariants in tensor products of certain modules, such as the natural modules for classical Lie groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' (c) Interpolating categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Here one aims to give uniform descriptions of representations of families of groups, such as symmetric groups, general linear groups, orthogonal groups, and symplectic groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Highly influential in this approach are the interpolating categories intro- duced by Deligne [Del07].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Such interpolating categories can often be given nice diagrammatic descriptions, leading to intuitive topological arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' In the case of the general linear group, the connection between the above concepts is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The oriented Brauer category OB(d) is the free rigid symmetric C-linear monoidal category on a generating object of categorical dimension d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Since the category of modules over the general linear group GL(m, C), m ∈ N, is rigid symmetric monoidal, there exists a functor G: OB(m) → GL(m, C)-mod sending the generating object ↑ of OB(m) to the natural GL(m, C)-module V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The additive Karoubi envelope of OB(d) is Deligne’s interpolating category for the general linear groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The endomor- phism algebra EndOB(d)(↑⊗r) is isomorphic to the group algebra of the symmetric group Sr, and so the functor G yields an algebra homomorphism (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1) kSr ∼= EndOB(m)(↑⊗r) → EndGL(m,C)(V ⊗r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' One half of Schur–Weyl duality is that the homomorphism (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1) is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' From this, one is able to deduce that the functor G is full.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The connection to invariant theory comes from the fact that G also induces a surjective homomorphism (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2) HomOB(m)(1, ↑⊗r ⊗ ↓⊗s) → HomGL(m,C)(C, V ⊗r ⊗ (V ∗)⊗s), where V ∗ is the GL(m, C)-module dual to V , and ↓ is the object of OB(m) dual to ↑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Thus, all GL(m, C)-invariant elements of V ⊗r ⊗ (V ∗)⊗s lie in the image under G of morphisms in OB(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The fullness of G, or of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2), is sometimes referred to as the first fundamental theorem of invariant theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' (Describing the kernel is the second fundamental theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=') An analogous picture exists for the orthogonal and symplectic groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' In these cases, the natural module is self-dual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Thus, the oriented Brauer category is replaced by the unoriented Brauer cate- gory B(d) of [LZ15], which is the free rigid symmetric k-linear monoidal category on a symmetrically self-dual object of categorical dimension d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Then, for m ∈ N, one has a full functors B(m) → O(m, C)-mod and B(−2m) → Sp(2m, C)-mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Here the endomorphism algebras are Brauer algebras, which surject onto the endomorphism algebras of tensor powers of the natural module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' In fact, it turns out that the most natural setting for the above picture is that of categories of supermodules over supergroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' There are full functors OB(m − n) → GL(m|n, C)-smod and B(m − 2n) → OSp(m|2n, C)-smod, where GL(m|n, C) and OSp(m|2n, C) are the general linear and orthosymplectic supergroups, re- spectively [CW12, BS12, LSM02, LZ17, LSM02].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The move to the super world also leads to addi- tional free categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' First, one observes that an isomorphism of a module with its dual can be even or odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The even case corresponds to the Brauer category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The odd case leads to the periplec- tic Brauer supercategory B1, which is the free rigid symmetric k-linear monoidal supercategory on an odd-self-dual object (which necessarily has categorical dimension zero).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Then there is a full superfunctor B1 → P(m)-smod, DIAGRAMMATICS FOR REAL SUPERGROUPS 3 where P(m) is the periplectic supergroup [KT17, CE21, DLZ18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Another free supercategory arises from the super version of Schur’s lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Since we work over the complex numbers, Schur’s lemma implies that the endomorphism algebra of a simple module is a complex division superalgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' In the non-super setting, the only possibility is C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' However, in the super setting, there is one additional possibility, which is the two-dimensional complex Clifford superalgebra Cl(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' This observation leads to the definition of the oriented Brauer-Clifford category OBC of [BCK19], which is the free rigid symmetric monoidal supercategory on a generating object whose endomorphism algebra is Cl(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' (As in the periplectic case, the categorical dimension must be zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=') There is a full superfunctor OBC → Q(m)-smod, where Q(m)-smod is the isomeric supergroup (also known as the queer supergroup).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Despite the great success of the above-mentioned approaches to the representation theory of some of the most important groups and supergroups appearing in mathematics and physics, surprisingly little is known when we work with real supergroups instead of complex ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The goal of the current paper is to initiate this line of research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Let us now describe our main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' To any associative superalgebra A over a field k, we define a diagrammatic supercategory OBk(A), which is the free rigid symmetric monoidal supercategory on an object with endomorphism super- algebra A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Imposing a condition on the categorical dimension yields a quotient category OBk(A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' d), for d ∈ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' This category has essentially appeared in [Sav19, BSW21, MS], although our definition is slightly more general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' When A = k, OBk(k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' d) is the oriented Brauer category (over a general field k) mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The universal property of OBk(A) implies that, if g is any Lie superalgebra, and V is a (g, A)-superbimodule, then there is an oriented incarnation superfunctor OBk(Aop) → g-smod, sending the generating object of OBk(Aop) to V , where Aop denotes the superalgebra opposite to A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' When we work over the ground field k = R, Schur’s lemma implies that the endomorphism algebra of a simple supermodule must be one of the ten real division superalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Our first main result (Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='10) is that, when A is a central real division superalgebra and V = Am|n, the functor OBR(Aop;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' m − n) → gl(m|n, A)-smod is full.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' (Note that, since the general linear groups are connected, we can freely replace the general linear supergroups by the general linear Lie superalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=') The method of proof is to pass to complexifications and use known results over the complex numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We then turn our attention to the unoriented (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' self-dual) cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Here the situation is a bit more involved, since we must carefully analyze which types of self-duality can arise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The natural setting for such self-dualities is over superalgebras equipped with an anti-involution a �→ a⋄.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' As mentioned above in the complex setting, the self-duality also has a parity σ ∈ Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' To any k- superalgebra A with anti-involution ⋄, and σ ∈ Z2, we assign a supercategory Bσ k (A, ⋄) and quotient supercategories Bσ k (A, ⋄;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' d) for d ∈ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' When A = k and the anti-involution is trivial, B0 k(k, id;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' d) is the usual Brauer category, while B1 k(k, id;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' 0) is the periplectic Brauer category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We deduce a basis theorem (Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='5) for the morphism spaces of Bσ k (A, ⋄;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' d) by embedding it into the superadditive envelope of OBk(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Self-duality of a supermodule is realized by a superhermitian or skew-superhermitian form Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' To such a form, we can associate the supergroup G(Φ) preserving the form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We then define an unoriented incarnation superfunctor FΦ : Bσ k (A, ⋄) → G(Φ)-smod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' 4 SAIMA SAMCHUCK-SCHNARCH AND ALISTAIR SAVAGE It turns out that only four of the ten real division superalgebras admit anti-involutions: the real numbers, the complex numbers, the quaternions, and the two-dimensional complex Clifford super- algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Our second main result (Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='5) is that, in these cases, the functor FΦ is full.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The proof, which occupies Sections 11 to 13, is much more involved than in the oriented case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We must treat each of the involutive division superalgebras separately, since each one behaves quite differently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Taking the oriented and unoriented cases together, our results handle real supergroups cor- responding to all real forms of the general linear, orthosymplectic, periplectic, and isomeric Lie superalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' (We give a classification of these real forms in Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=') Looking at endo- morphism algebras, as explained above, one immediately obtains analogues of Schur–Weyl duality, or first fundamental theorems, for these real supergroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Such results seem to be rare in the litera- ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' (See [Cal22] for some partial results for certain real groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=') Even in the non-super setting, we obtain new results, corresponding, for example, to the indefinite orthogonal, unitary, and symplectic groups;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' see Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' In fact, in these cases where the module categories are semisimple, we show that these module categories are isomorphic to quotients of the additive Karoubi envelopes of our diagrammatic categories by tensor ideals of negligible morphisms;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' see Theorems 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='11 and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' These are real analogues of the some of the main results concerning Deligne’s interpolating cate- gories in the complex case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' As another application, we deduce equivalences between supercategories of tensor supermodules over the different real forms of a complex supergroup;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' see Corollary 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='9 and Propositions 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='5, 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='5 and 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Further directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We conclude this introduction with a brief discussion of some of the future research directions that stem from the current work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Many of these are real analogues of promising work that has been done in the complex case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' While our results show that the oriented and unoriented incarnation functors are full, we leave a description of the kernels of these functors, also known as the second fundamental theorem, for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' When the target module supercategory is semisimple, the kernel is the tensor ideal of negligible morphisms;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' see Theorems 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='11 and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' However, this is not the case in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For the usual oriented and unoriented Brauer categories, kernels have been described explicitly in [CW12, LZ21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Since the target module supercategories of incarnation functors are idempotent complete, one has induced functors Kar(OBk(Aop;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' m − n)) → gl(m|n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' A)-smod and Kar(Bσ k (A, ⋄)) → G(Φ)-smod, where Kar(C) denotes the additive Karoubi envelope of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The supermodules that appear in the image of these functors are the summands of the tensor powers of the natural module (and, in the oriented case, its dual).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' It would be interesting to give a more precise description of these supermodules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For the usual oriented and unoriented Brauer categories, results in this direction have been obtained in [BS12, CH17, CW12, Hei17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The supercategories introduced here have affine analogues [MS, SS22], generalizing the affine oriented Brauer category of [BCNR17] and the affine Brauer category of [RS19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' These affine supercategories act naturally on categories of supermodules over supergroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We plan to investigate these actions in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' There exist quantum analogues of the oriented and unoriented Brauer categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' These are the framed HOMFLYPT skein and Kauffman skein categories, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' One has analogues of the results mentioned above, but with supergroups replaced by quantized enveloping superalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We expect that one can also define quantum analogues of the more general supercategories introduced in the current paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' When the ground field is R, these should be related to the representation theory of real quantum groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' DIAGRAMMATICS FOR REAL SUPERGROUPS 5 Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' This research of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' was supported by NSERC Discovery Grant RGPIN- 2017-03854.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We thank Jon Brundan, Inna Entova-Aizenbud, Thorsten Heidersdorf, Allan Merino, Hadi Salmasian, Nolan Wallach, and Ben Webster for helpful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Monoidal supercategories In this paper, we will work with strict monoidal supercategories in the sense of [BE17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' In this section, we review a few of the more important ideas that are crucial for our exposition and somewhat less well known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Throughout this section we work over an arbitrary ground field k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' A supercategory is a category enriched in the monoidal category of superspaces and parity pre- serving linear maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Thus, its morphism spaces are vector superspaces and composition is parity- preserving;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' that is, f ◦ g = ¯f + ¯g, where ¯f denotes the parity of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' A superfunctor between supercategories induces a parity-preserving linear map between morphism superspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For super- functors F, G: C → D, a supernatural transformation α: F ⇒ G of parity r ∈ Z2 is a family of morphisms αX ∈ HomD(FX, GX)r, X ∈ C, such that Gf ◦ αX = (−1)r ¯fαY ◦ Ff for each homo- geneous f ∈ HomC(X, Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Note when r is odd that α is not a natural transformation in the usual sense due to the sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' A supernatural transformation α: F ⇒ G is a sum α = α0 + α1, where αr is a supernatural transformation of parity r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' In a strict monoidal supercategory, the super interchange law, which follows from the fact that ⊗ is a superbifunctor, is (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1) (f ′ ⊗ g) ◦ (f ⊗ g′) = (−1) ¯f ¯g(f ′ ◦ f) ⊗ (g ◦ g′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We denote the unit object by 1 and the identity morphism of an object X by 1X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We will use the usual calculus of string diagrams, representing the tensor product f ⊗ g of morphisms f and g diagrammatically by drawing f to the left of g, and the composition f ◦ g by drawing f above g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Care is needed with horizontal levels in such diagrams due to the signs arising from the super interchange law: (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2) f g = f g = (−1) ¯f ¯g f g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For a supercategory C, its Π-envelope Cπ is the supercategory with objects given by formal symbols {ΠrX : X ∈ C, r ∈ Z2} and morphisms defined by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='3) HomCπ(ΠrX, ΠsY ) := Πs−r HomC(X, Y ), where, on the right-hand side, Π denotes the parity shift operator determined by (ΠV )r := Vr−1 for a vector superspace V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The composition law in Cπ is induced in the obvious way from the one in C: writing f s r for the morphism in HomCπ(ΠrX, ΠsY ) of parity ¯f + r − s defined by f ∈ HomC(X, Y ), we have that f u s ◦ gs r = (f ◦ g)u r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' A Π-supercategory (D, Π, ζ) is a supercategory D, together with the extra data of a superfunctor Π: D → D, called the parity shift, and an odd supernatural isomorphism ζ from Π to the identity superfunctor;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' see [BE17, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The Π-envelope Cπ from Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1 is a Π-supercategory with parity shift superfunctor Π: Cπ → Cπ sending object ΠrX to Πr+1X and morphism f s r to f s+1 r+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Viewing C as a full subcategory of its Π-envelope Cπ via the canonical embedding (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='4) C → Cπ, X �→ Π0X, f �→ f 0 0, the Π-envelope satisfies a universal property: any superfunctor F : C → D to a Π-supercategory D extends in a canonical way to a superfunctor ˜F : Cπ → D such that ˜F ◦ Π = Π ◦ ˜F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' In turn, any supernatural transformation θ: F ⇒ G between superfunctors F, G: C → D extends in a unique way to a supernatural transformation ˜θ: ˜F ⇒ ˜G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' see [BE17, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' 6 SAIMA SAMCHUCK-SCHNARCH AND ALISTAIR SAVAGE Given superalgebras A and B, the supercategory of (A, B)-superbimodules is a Π-supercategory, as explained in [BE17, Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' If V is an (A, B)-supermodule, we will denote its parity shift by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='5) ΠV := {πv : v ∈ V } with πv = v + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Here π is a formal symbol to remind us that πf is an element of ΠV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' In order to unify some expressions where the parity shift may or may not be present, we define πσv, for σ ∈ Z2, by πσv = � πv if σ = 1, v if σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For a morphism f : V → W, we have Πf : ΠV → ΠW, (Πf)(πv) = (−1) ¯fπf(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The isomorphism (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='6) Π2V ∼ = −→ V, ππv �→ −v is denoted ξV in the notation of [BE17, Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' If C is a supercategory, we let Add(C) denote its additive envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' This is the supercategory whose objects are formal finite direct sums of objects in C, and whose morphisms are identified with matrices of morphisms in C in the usual way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The supercategory Add(Cπ), which is the additive envelope of the Π-envelope of C, is sometimes referred to as the superadditive envelope of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The Karoubi envelope Kar(C) of C is the completion of its additive envelope Add(C) at all homogeneous idempotents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Thus, objects of Kar(C) are pairs (X, e) consisting of a finite direct sum X of objects of C together with a homogeneous idempotent e ∈ EndAdd(C)(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Morphisms (X, e) → (Y, f) are elements of f HomAdd(C)(X, Y )e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' If C is a Π-supercategory, the parity shift superfunctors extend by the usual universal property of Karoubi envelopes to make Kar(C) into a Π-supercategory too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Now we consider the monoidal situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We make the category SCat of supercategories and superfunctors into a symmetric monoidal category following the general construction of [Kel05, §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' In particular, for supercategories C and D, their k-linear product, denoted C ⊠ D, has as objects pairs (X, Y ) for X ∈ C and Y ∈ D, and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='7) HomC⊠D((X, Y ), (X′, Y ′)) = HomC(X, X′) ⊗ HomD(Y, Y ′) with composition defined via (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' A strict monoidal supercategory is a supercategory C with an associative, unital tensor functor −⊗−: C ⊠C → C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' See [BE17, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='4] for the appropriate notions of (not necessarily strict) monoidal superfunctors between strict monoidal supercategories, and of monoidal natural transformations between monoidal superfunctors (which are required to be even).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' There is also a notion of strict monoidal Π-supercategory;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' see [BE17, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Such a category is a Π-supercategory in the earlier sense with Π := π ⊗ − for a distinguished object π admitting an odd isomorphism ζ : π ∼ −→ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The Π-envelope Cπ of a strict monoidal supercategory C is the Π-supercategory from Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1, viewed as a strict monoidal Π-supercategory with π := Π1, tensor product of objects defined by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='8) (ΠrX) ⊗ (ΠsY ) := Πr+s(X ⊗ Y ), and tensor product (horizontal composition) of morphisms defined by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='9) f s r ⊗ gv u := (−1)r(¯g+u+v)+ ¯fv(f ⊗ g)s+v r+u for homogeneous morphisms f and g in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' See [BE17, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='16] for more details and discussion of its universal property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' When working with string diagrams, we will represent the morphism f s r in Cπ DIAGRAMMATICS FOR REAL SUPERGROUPS 7 by adding horizontal lines labelled by r and s at the bottom and top of the diagram for f : X → Y : (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='10) f r s : ΠrX → ΠsY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Then the rules for horizontal and vertical composition in Cπ become (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='11) f r s ⊗ g u v = (−1)r(¯g+u+v)+ ¯fv f g r+u s+v , f s t g r s = f g r t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The Karoubi envelope Kar(C) of a strict monoidal Π-supercategory is a strict monoidal Π-supercategory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' If k′ is a field extension of k and C is a supercategory over k, then we define Ck′ to be the supercategory obtained from C by extension of scalars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Precisely, the objects of Ck′ are the same of those of C, and we have HomCk′(X, Y ) := HomC(X, Y ) ⊗k k′, X, Y ∈ C, with composition extended in the natural way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Any superfunctor F : C → D naturally extends to a superfunctor F k′ : Ck′ → Dk′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' If C is a (strict) monoidal supercategory or a Π-supercategory, then so is Ck′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' In the special case where k = R, we call CC the complexification of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Superalgebras and supermodules In this section, we review some basic properties of superalgebras and supermodules that will be used in the current paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We work over a ground field k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Associative superalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' All vector superspaces and superalgebras are over k unless oth- erwise indicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We also assume that all k-supermodules are finite dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We let V0 and V1 denote the even and odd parts, respectively, of a k-supermodule V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Then its superdimension is sdimk(V ) = dimk(V0) − dimk(V1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We let ¯v denote the parity of a homogeneous element v of a k-supermodule V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' When we write equations involving parities of elements, we implicitly assume these elements are homogeneous;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' we then extend by linearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The term superalgebra refers to a unital associative superalgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' If A is a superalgebra, its opposite superalgebra Aop = {aop : a ∈ A} has multiplication given by aopbop = (−1)¯a¯b(ba)op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Supermodules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Throughout this subsection, A denotes a superalgebra and V, W denote right A-supermodules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We also let g denote a Lie superalgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We let HomA(V, W) denote the k-supermodule of all (that is, not necessarily parity-preserving) morphisms of A-supermodules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We also define EndA(V ) := HomA(V, V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Thus, for example, HomA(V, W)0 denotes the k-module of all parity-preserving A-linear maps from V to W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For a ∈ A, define (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1) ρa : V → V, v �→ (−1)¯a¯vva.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We also define flip: V ⊗ W → W ⊗ V, v ⊗ w �→ (−1)¯v ¯ww ⊗ v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' If V and W are (g, A)-superbimodules, then ρa and flip are homomorphisms of g-supermodules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Let V ∗ = Homk(V, k) 8 SAIMA SAMCHUCK-SCHNARCH AND ALISTAIR SAVAGE denote the k-dual of V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' This is a left A-module with action given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2) (af)(v) := (−1)¯a ¯ff(va).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' If V is also a left g-supermodule, then V ∗ is a left g-supermodule, with action given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='3) (Xf)(v) = −(−1) ¯ X ¯ff(Xv), X ∈ g, f ∈ V ∗, v ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' This action supercommutes with the left A-action given in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Let BV be a k-basis for V , which we will sometimes denote by Bk V when there is some possibility of confusion about the ground field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Let {v∗ : v ∈ BV } be the dual basis of V ∗ given by v∗(w) = δvw, v, w ∈ BV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We have the evaluation map ev: V ∗ ⊗ V → k, f ⊗ v �→ f(v), and the coevaluation map coev: k → V ⊗ V ∗, 1 �→ � v∈BV v ⊗ v∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The map coev is independent of the choice of basis BV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' If V is a left g-supermodule, then ev and coev are both homomorphisms of g-supermodules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Frobenius superalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We now recall some basic definitions and facts about Frobenius superalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For more details, including proofs in the super case considered here, we refer the reader to [PS16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' A Frobenius superalgebra is a superalgebra A equipped with a parity-preserving k-linear map τ = τA : A → k, called the Frobenius form, such that the induced bilinear form A × A → k, (a, b) �→ τ(ab), is nondegenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' If (A, τ) and (A, τ ′) are two Frobenius superalgebras with the same underlying superalgebra A, then there exists an even invertible element u ∈ A such that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='4) τ ′(a) = τ(au) for all a ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Every Frobenius superalgebra has a Nakayama automorphism ζ, which is an superalgebra auto- morphism of A satisfying (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='5) τ(ab) = (−1)¯a¯bτ(bζ(a)) = (−1)¯aτ(bζ(a)) = (−1) ¯bτ(bζ(a)) for all a, b ∈ A, where the last two equalities follow from the fact that τ(ab) = 0 unless ¯a = ¯b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' A Frobenius superalgebra is said to be supersymmetric if its Nakayama automorphism is the identity map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We will not assume that Frobenius superalgebras are supersymmetric in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We will often refer to A itself as a Frobenius superalgebra, leaving the Frobenius form implied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Our main sources of examples of Frobenius superalgebras will be the real division superalgebras, to be discussed in detail in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' See Examples 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1 for additional examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' If A is a Frobenius superalgebra, then so is Aop with Frobenius form τAop(aop) = τA(a), a ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' If BA is a homogeneous k-basis of a Frobenius superalgebra A, we let B∨ A := {b∨ : b ∈ BA} be the left dual basis, defined by τ(b∨c) = δbc, b, c ∈ BA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' It follows that, for all a ∈ A, we have (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='6) a = � b∈BA τ(b∨a)b = � b∈BA τ(ab)b∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We also have (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='7) b∨ = ¯b for all b ∈ BA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' DIAGRAMMATICS FOR REAL SUPERGROUPS 9 and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='8) � b∈Bk A b ⊗ b∨ is independent of the choice of basis BA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The basis left dual to {ζ(b) : b ∈ BA} is given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='9) ζ(b)∨ = ζ(b∨) for all b ∈ BA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' If V is a right A-supermodule, then the supertrace of the action of a on V is (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='10) strV (a) = strk V (a) := � v∈BV (−1)¯vv∗(va), where BV is a k-basis of V and {v∗ : v ∈ BV }, is the dual basis of V ∗ k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We use the superscript k on strk V (a) when there is the possibility of confusion about the ground field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We have strV (ab) = (−1)¯a¯b strV (ba) for all a, b ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' It is clear that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='11) strV (a) = (m − n) strA(a) for V = Am|n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' If (A, τ) is a Frobenius superalgebra, then (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='12) strA(a) = strA(ζ(a)) = � b∈BA (−1) ¯bτ(b∨ba) = � b∈BA (−1) ¯bτ(ab∨b), a ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For all b ∈ BA, we have τ(b∨a) = b∗(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' It follows immediately that strA(a) is equal to the first sum in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Next, note that τ(c) = τ(ζ(c)) for all c ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Thus, for all a ∈ A, strA(a) = � b∈BA (−1) ¯bτ(b∨ba) = � b∈BA (−1) ¯bτ � ζ(b∨)ζ(b)ζ(a) � (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='5) = � b∈BA (−1) ¯bτ � aζ(b∨)ζ(b) � = � b∈BA (−1) ¯bτ � ab∨b � (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='5) = � b∈BA (−1) ¯b(b∨bζ(a)) = strA(ζ(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' where, in the fourth equality we changed to a sum over the basis {ζ(b) : b ∈ BA} and used (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Supermatrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We will use the term supermatrix to denote a supermatrix with entries in a superalgebra A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We let Matp|q,r|s(A) denote the k-supermodule of (p|q) × (r|s) supermatrices, and set Matp|q(A) := Matp|q,p|q(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We write a supermatrix X ∈ Matp|q,r|s(A) in block form as X = � X00 X01 X10 X11 � , where X00 is p × r, X01 is p × s, X10 is q × r and X11 is q × s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The even elements of Matp|q,r|s(A) are those supermatrices X where X00, X11 have even entries and X01, X10 have odd entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The odd elements of Matp|q,r|s(A) are those supermatrices X where X00, X11 have odd entries and X01, X10 have even entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We view elements of Am|n as column supermatrices, that is, as (m|n) × (1, 0) supermatrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Similarly, we view elements of A as (1|0) × (1|0) supermatrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Then the right action (v, a) �→ va of A on Am|n can be viewed as matrix multiplication and we can identify elements of EndA(Am|n) with Matm|n(A) in the usual way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' 10 SAIMA SAMCHUCK-SCHNARCH AND ALISTAIR SAVAGE The supertranspose of a supermatrix is given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='13) Xst := � Xt 00 (−1) ¯ XXt 10 −(−1) ¯ XXt 01 Xt 11 � , where Xt denotes the usual transpose of a matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Note that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='14) (Xst)st = � X00 −X01 −X10 X11 � , so that the supertranspose has order four in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The supertrace of a square supermatrix X ∈ Matp|q(A) is given by str(X) = tr(X00) − (−1) ¯ X tr(X11), where tr denotes the usual matrix trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For a supermatrix X ∈ Matp|q,r|s(A), let Xop ∈ Matp|q,r|s(Aop) denote the matrix obtained from X by replacing each entry a by aop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Then define Xst op := (Xop)st = (Xst)op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We have an isomorphism of k-superalgebras (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='15) Matm|n(A)op ∼ = −→ Matm|n(Aop), Xop �→ Xst op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The map is clearly an isomorphism of k-vector superspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' It is also a straightforward computation to prove that it respects multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' If A is a superalgebra, then (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='16) strk Matm|n(A)(X) = (m − n) strk A ◦ str(X) for all X ∈ Matm|n(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Choose the basis {Ersb : 1 ≤ r, s ≤ m + n, b ∈ BA} of Matm|n(A), where Ers denotes the matrix with a 1 in position (r, s), and a 0 in all other positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The dual basis of Matm|n(A)∗ is given by (Ersb)∗(X) = (−1)p(s)b∗(str(EsrX)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Then we have strk Matm|n(A)(X) = m+n � r,s=1 � b∈BA (−1) ¯b+p(r)b∗(str(EsrErsbX)) = (m − n) � b∈BA (−1) ¯bb∗(str(bX)) = (m − n) � b∈BA (−1) ¯bb∗(b str(X)) = (m − n) strk A ◦ str(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Lie superalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' If g is a Lie superalgebra over k, we let g-smodk denote the supercategory of finite-dimensional g-supermodules over k with arbitrary (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' not necessarily parity-preserving) homomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We will be particularly interested in the cases where k is R or C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' If k = R, our notation g-smodR is designed to emphasize that we are speaking of real supermodules, as opposed to complex supermodules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The general linear Lie superalgebra gl(VA) associated to a right A-supermodule V is equal to EndA(V ) as a k-supermodule, with Lie superbracket given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='17) [X, Y ] := XY − (−1) ¯ X ¯Y Y X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' In the special case that V = Am|n, we introduce the notation gl(m|n, A) = gl(VA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Identifying EndA(V ) with Matm|n(A) in the usual way, we have that gl(m|n, A) is equal to Matm|n(A) as a k- vector superspace, with Lie superbracket given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' By convention, we define gl(0|0, A) to be the zero Lie superalgebra, so that gl(0|0, A)-smodk is the supercategory k-smod of k-supermodules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' DIAGRAMMATICS FOR REAL SUPERGROUPS 11 Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We have an isomorphism of Lie k-superalgebras gl(m|n, A) ∼ = −→ gl(m|n, Aop), X �→ −Xst op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The given map is clearly an isomorphism of k-vector spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' To verify that it also respects the Lie superbracket, we compute [−Xst op, −Y st op] = Xst opY st op − (−1) ¯ X ¯Y Y st opXst op (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='15) = (−1) ¯ X ¯Y (Y X)st op − (XY )st op = −[X, Y ]st op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Harish–Chandra superpairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We will sometimes need to work with supergroups instead of Lie superalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We review here some basic facts, referring the reader to [DM99] for a more detailed overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Instead of directly working with supergroups, it will be simpler to work with the equivalent category of Harish-Chandra superpairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We refer the reader to [Gav20] and the references cited therein for a proof of this equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' A Harish-Chandra superpair over k is a pair G = (Gred, g), where Gred is an algebraic group over k, g is a Lie superalgebra over k, g0 is the Lie algebra of G, Gred acts algebraically on g by k-linear transformations, and the differential of the action of Gred on g coincides with the action of g0 on g via the superbracket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Suppose G = (Gred, g) is a Harish-Chandra superpair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' A G-supermodule is a k-supermodule V that is both a Gred-supermodule and a g-supermodule, and such that the differential of the action of Gred coincides with the action of g0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The finite-dimensional G-supermodules form a monoidal supercategory, which we denote by G-smodk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For G-supermodules V and W, we have HomG(V, W) = HomGred(V, W) ∩ Homg(V, W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' If Gred is connected, then Homg0(V, W) = HomGred(V, W), and so HomG(V, W) = Homg(V, W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' In this case, the forgetful superfunctor G-smodk → g-smodk is full and faithful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' On the other hand, if g1 = 0, so that we are working in the purely even setting, then the forgetful functor G-modk → Gred-modk is full and faithful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' (In fact, it is an equivalence of categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=') In this case, we will often identity G and Gred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' On the other hand, suppose Gred has r+1 connected components, and let X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' , Xr be elements of Gred, one from each of the r connected components not containing the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Then Gred = H ∪ HX1 ∪ · · · HXr, where H is the identity component of Gred, and we have (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='18) HomG(V, W) = HomX1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=',Xr,g(V, W) := {f ∈ Homg(V, W) : f(Xtv) = Xtf(v) ∀ v ∈ V, 1 ≤ t ≤ r}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For example, if k = C, and Gred = O(m, C) is the complex orthogonal group, then we have HomG(V, W) = HomX,g(V, W), where X is any element of O(m, C) with det(X) = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Complexification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' If V is a real vector superspace, its complexification is V C := V ⊗R C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We view V as an R-vector subspace of V C by identifying v ∈ V with v⊗1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' If A is a real superalgebra, then AC is a complex superalgebra, with product (a ⊗ y)(b ⊗ z) = ab ⊗ yz, a, b ∈ A, y, z ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Similarly, if g is a Lie superalgebra over R, then its complexification gC is a Lie superalgebra over C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' A real form of a complex vector superspace W is a real vector superspace V such that V C ∼= W as 12 SAIMA SAMCHUCK-SCHNARCH AND ALISTAIR SAVAGE complex vector superspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We define real forms of complex associative superalgebras and complex Lie superalgebras similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Suppose R is either a real associative superalgebra or a real Lie superalgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' If V is a left (respectively, right) R-supermodule, then V C is a left (respectively, right) RC-supermodule with the natural action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Furthermore, every f ∈ HomR(V, W) induces an element f C ∈ HomRC(V C, W C) given by f C(v ⊗ z) = f(v) ⊗ z, v ∈ V, z ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We have ker � f C� = ker(f)C, ker(f) = ker � f C� ∩ V, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='19) im � f C� = im(f)C, im(f) = im � f C� ∩ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='20) In particular f is injective ⇐⇒ f C is injective and f is surjective ⇐⇒ f C is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The above constructions yield a superfunctor R-smod → RC-smod, which induces a full and faithful complexification superfunctor (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='21) CR : (R-smod)C → RC-smod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' In particular, if g is a real Lie superalgebra and V, W ∈ g-smodR, then we have a canonical isomor- phism of C-supermodules (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='22) Homg(V, W)C ∼ = −→ HomgC(V C, W C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' If H is a real supergroup acting on a real vector space V , then H also acts on V C = V ⊗R C by acting on the first factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' If V and W are supermodules over a real Harish-Chandra superpair G = (Gred, g), then we have an isomorphism of C-supermodules HomG(V, W)C ∼= HomGred,gC(V C, W C) := {X ∈ HomgC(V C, W C) : f(Xv) = Xf(v) ∀ X ∈ Gred, v ∈ V C}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' If Gred has r + 1 connected components, and X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' , Xr are elements of Gred, one from each connected component not containing the identity, then, using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='18), we have (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='23) HomG(V, W)C ∼= HomX1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=',Xr,gC(V C, W C) := {f ∈ HomgC(V C, W C) : f(Xtv) = Xtf(v) ∀ v ∈ V C, 1 ≤ t ≤ r}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' If A is a Frobenius R-superalgebra with Frobenius form τ, then its complexification AC is a Frobenius C-superalgebra with Frobenius form (which we continue to denote by the same symbol) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='24) τ : AC → C, a ⊗ z �→ τ(a)z, a ∈ A, z ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' It is straightforward to verify that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='25) Matm|n(A)C ∼= Matm|n(AC) as C-superalgebras and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='26) gl(m|n, A)C ∼= gl(m|n, AC) as complex Lie superalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' DIAGRAMMATICS FOR REAL SUPERGROUPS 13 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Real division superalgebras In this section, we discuss real division superalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' These will play a key role in our main applications to the representation theory of real supergroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Real division superalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For our purposes, one of the most important classes of exam- ples of Frobenius superalgebras are the real division superalgebras, which were classified by Wall [Wal64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' (See also [Bae20] for a short exposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=') Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Every real division superalgebra is isomorphic to exactly one of the following, where the Z2-grading is given by declaring ε to be odd, and ⋆ denotes complex conjugation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Cl0(R) = R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Cl1(R) := R ⊕ εR, with ε2 = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Cl2(R) := C ⊕ εC, with ε2 = 1 and zε = εz⋆ for all z ∈ C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Cl3(R) := H ⊕ εH, with ε2 = −1 and zε = εz for all z ∈ H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Cl4(R) := H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Cl5(R) := H ⊕ εH, with ε2 = 1 and zε = εz for all z ∈ H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Cl6(R) := C ⊕ εC, with ε2 = −1 and zε = εz⋆ for all z ∈ C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Cl7(R) := R ⊕ εR, with ε2 = −1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Cl(C) := C ⊕ εC, with ε2 = 1 and zε = εz for all z ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For 0 ≤ r ≤ 8, we have Clr(R)op ∼= Cl−r(R) as superalgebras, where subscripts are considered modulo 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The notation in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1 is inspired by the fact that Clr(R)⊗RCls(R) is Morita equivalent to Clr+s(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Note that C and the complex Clifford algebra Cl(C) are the only complex division superalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The Clr(R), 0 ≤ r ≤ 7, are real Clifford superalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Recall that a k-superalgebra is central if its center is k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Thus, the central real division superalgebras are those real division superalgebras isomorphic to Clr(R) for 0 ≤ r ≤ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The complex division superalgebras are isomorphic to their own opposite superalge- bras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For C, this follows from the fact that C is commutative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For Cl(C), we have Cl(C)op = C⊕εC, with ε2 = −1, and an isomorphism of C-superalgebras Cl(C) ∼ = −→ Cl(C)op, ε �→ εi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Convention 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='3 (Frobenius forms on division superalgebras).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Note that C and Cl(C) are complex Frobenius superalgebras, with Frobenius form given by projection projC onto their even part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' (They are also real Frobenius superalgebras with Frobenius form given by projection onto the real part of their even part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=') We will always view the central real division superalgebras as superalgebras over R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' They are real Frobenius superalgebras with Frobenius form Re: D → R given by taking the real part of the even part of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' In all of these cases, the Nakayama automorphism is given by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1) ζ(a) = (−1)¯aa, a ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' In particular, a real division superalgebra is supersymmetric if and only if it is purely even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' If k ∈ {R, C} and BD is a k-basis for an involutive division k-superalgebra D, then the basis left dual to B∨ D = {b∨ : b ∈ BD} is given by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2) (b∨)∨ = b, b ∈ BD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For all b, c ∈ BD, we have τ(cb∨) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='5) = (−1) ¯b¯cτ(b∨ζ(c)) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1) = (−1) ¯b¯c+¯cτ(b∨c) = δbc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' □ 14 SAIMA SAMCHUCK-SCHNARCH AND ALISTAIR SAVAGE If D is a real division superalgebra, we will use the term D-vector superspace to denote a finite- dimensional right D-supermodule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Recall the definition of strA given in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' If D is a real or complex division superalgebra with standard Frobenius form τ, then strD = (sdimk D)τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' In particular, strD = 0 whenever D is a real or complex division superalgebra with nonzero odd part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' If D is a real division superalgebra, choose the basis BD = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 {1} if D = R, {1, i} if D = C, {1, i, j, k} if D = H, {1, ε} if D ∈ {Cl1(R), Cl7(R)}, {1, i, ε, εi} if D ∈ {Cl2(R), Cl6(R)}, {1, i, j, k, ε, εi, εj, εk} if D ∈ {Cl3(R), Cl5(R)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' If D = C, considered as a complex division superalgebra choose BD = {1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Finally, if D = Cl(C), considered as a complex division superalgebra, choose BD = {1, ε}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Then we have b∨ = b−1 for all b ∈ BD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Hence, using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='12), we have strD(a) = � b∈BD (−1) ¯bτ(a) = (sdimk D)τ(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Complexification of real division superalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We have the following injections of superalgebras, where ⋆ denotes complex conjuga- tion, R ֒→ C, a �→ a, a ∈ R, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='3) ı: H ֒→ Mat2(C), i �→ � i 0 0 −i � , j �→ � 0 −1 1 0 � , k �→ � 0 −i −i 0 � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='4) Cl1(R) ֒→ Cl(C), a + εb �→ a + εb, a, b ∈ R, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='5) Cl2(R) ֒→ Mat1|1(C), a + εb �→ � a b⋆ b a⋆ � , a, b ∈ C, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='6) Cl3(R) ֒→ Mat2(Cl(C)), a + εb �→ ı(a) + εı(b)i, a, b ∈ H, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='7) Cl5(R) ֒→ Mat2(Cl(C)), a + εb �→ ı(a) + εı(b), a, b ∈ H, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='8) Cl6(R) ֒→ Mat1|1(C), a + εb �→ � a −b⋆ b a⋆ � , a, b ∈ C, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='9) Cl7(R) ֒→ Cl(C), a + εb �→ a + εib, a, b ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='10) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' These are all straightforward verifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' □ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We have an injection of complex superalgebras Cl(C) ֒→ Mat2(C), a + εb �→ � a b b a � , a, b ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Combined with Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='6, this shows that all of the real division superalgebras can be embedded in complex supermatrix superalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The following result gives the complexification of the central real division superalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' DIAGRAMMATICS FOR REAL SUPERGROUPS 15 Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The inclusions of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='6 induce isomorphisms of complex superalgebras RC ∼= C, HC ∼= Mat2(C), Cl1(R)C ∼= Cl7(R)C ∼= Cl(C), Cl2(R)C ∼= Cl6(R)C ∼= Mat1|1(C), Cl3(R)C ∼= Cl5(R)C ∼= Mat2(Cl(C)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' In particular, for every central real division superalgebra D, its complexification DC is a simple complex superalgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' It is straightforward to verify that, for each of the injections in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='6, every matrix in the codomain can be written uniquely in the form X + iY , where X and Y are in the image of the injection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' General linear Lie superalgebras over division superalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' If D is a real division superalgebra with D1 ̸= 0, then (a) Dm|n ∼= Dm+n as D-vector superspaces, (b) Matm|n(D) ∼= Matm+n(D) as superalgebras, (c) gl(m|n, D) ∼= gl(m + n, D) as Lie superalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Suppose D is a real division superalgebra with D1 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Then left multiplication by any nonzero odd element gives an isomorphism D0|n ∼= Dn|0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Hence Dm|n ∼= Dm|0 ⊕D0|n ∼= Dm|0 ⊕Dn|0 ∼= Dm+n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' This induces isomorphisms of superalgebras Matm+n(D) ∼= EndD(Dm|n) ∼= EndD(Dm+n) ∼= Matm+n(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Passing to the associated Lie superalgebras then gives the isomorphism gl(m|n, D) ∼= gl(m + n, D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' □ In light of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='9, we will often consider only gl(m, D), as opposed to gl(m|n, D), when D1 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The general linear superalgebras over the central real division superalgebras are often known by different names and notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' gl(m, Cl1(R)) is the split real isomeric Lie superalgebra, often called the split real queer Lie superalgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' It is usually denoted q(m, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' gl(m, Cl(C)) is the complex isomeric Lie superalgebra, often called the complex queer Lie superalgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' It is usually denoted q(m, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' gl(m, Cl2(R)) is sometimes denoted q0(m, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' gl(m, Cl5(R)) is sometimes denoted q∗(2m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' gl(m|n, H) is sometimes denoted u∗(2m|2n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' (Recall that gl(m|n, D) ∼= gl(m|n, Dop) by Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=') Many references focus on the realization of real Lie superalgebras in terms of complex matrices, using the inclusions of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='6 and Re- mark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' However, we feel that the realization in terms of general linear Lie superalgebras over real division superalgebras is more natural and leads to more uniform and easy-to-understand notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We will only use the notation q(m, C) in the complex case: q(m, C) = gl(m, Cl(C)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The following proposition shows that the general linear Lie superalgebras over central real division superalgebras are real forms of general linear and isomeric Lie superalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We have isomorphisms of complex Lie superalgebras (a) gl(m|n, R)C ∼= gl(m|n, C), (b) gl(m|n, H)C ∼= gl(2m|2n, C), 16 SAIMA SAMCHUCK-SCHNARCH AND ALISTAIR SAVAGE (c) gl(m, Cl1(R))C ∼= gl(m, Cl7(R))C ∼= q(m, C), (d) gl(m, Cl2(R))C ∼= gl(m, Cl6(R))C ∼= gl(m|m, C), (e) gl(m, Cl3(R))C ∼= gl(m, Cl5(R))C ∼= q(2m, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' This follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='26), Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='8, and the fact that we have canonical isomorphisms of complex Lie superalgebras (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='11) gl(m|n, Matr|s(A)) ∼= gl((mr + ns)|(ms + nr), A) for any superalgebra A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The oriented supercategory In this section, we introduce the first of our two main diagrammatic supercategories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' After defining the supercategory, we prove a basis theorem for morphism spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' In Sections 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2, k denotes an arbitrary field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' In Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='3, we discuss the special cases k ∈ {R, C}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Definition of the supercategory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1 ([MS, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For an associative superalgebra A, we define OBk(A) to be the strict monoidal supercategory generated by objects ↑ and ↓ and morphisms : ↑ ⊗ ↑ → ↑ ⊗ ↑ , a : ↑ → ↑ , a ∈ A, : ↓ ⊗ ↑ → 1, : 1 → ↑ ⊗ ↓, : ↑ ⊗ ↓ → 1, : 1 → ↓ ⊗ ↑, subject to the relations 1 = , λ a + µ b = λa+µb , b a = ab , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1) = , = , a = a , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2) = , = , = = , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='3) = , = , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='4) for all a, b ∈ A and λ, µ ∈ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' In the above, the left and right crossings are defined by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='5) := , := .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The parity of a is ¯a, and all the other generating morphisms are even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We refer to the morphisms a as tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For d ∈ k, we define OBk(A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' d) to be the quotient of OBk(A) by the relations (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='6) a = d strA(a)1 1, a ∈ A, where strA is given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We call d the specialization parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' When A is a Frobenius superalgebra, OBk(A) was called the oriented Frobenius Brauer supercat- egory in [MS, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The Frobenius structure on A allows one to enlarge it to the affine oriented Frobenius Brauer category of [MS, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='3], which is the central charge zero special case of the Frobenius Heisenberg supercategory introduced in [Sav19], and further studied in [BSW21, MS].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We refer the reader to these papers for proofs omitted here, none of which use the Frobenius structure on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Our presentation of OBk(A) is slightly different from the one given in [MS, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Pre- cisely, the relations (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='4) are the reflections in the vertical axis of the ones in [MS, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='4)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' However, OBk(A) has a symmetry given by reflecting diagrams in the vertical axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' (This is the composition of the isomorphisms (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='16) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='17) in [BSW21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=') Hence, the two definitions are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' DIAGRAMMATICS FOR REAL SUPERGROUPS 17 Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' (a) When A = k, we have a = a for all a ∈ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Thus, we can omit the generators a and all the relations involving them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Then we see that OBk(k) is the oriented Brauer category, which is the free rigid symmetric k-linear monoidal category generated by a single object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' This is the motivation for the notation OBk(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' (b) The supercategory OBC(Cl(C), 0) is the oriented Brauer–Clifford supercategory introduced in [BCK19, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' When A is a real or complex division superalgebra with nonzero odd part, it follows from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='5 that OBk(A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' d) = OBk(A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' 0) for all d ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The relations (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='4) means that ↓ is left dual to ↑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' In fact, we also have (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='7) = , = , and so ↓ is also right dual to ↑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Thus OBk(A) is rigid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Furthermore, we have that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='8) := = , a := a = a , a ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' These relations mean that tokens and crossings slide over all cups and caps in the sense that, for all orientations of the strands, we have (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='9) a = a , a = a , = , = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' More precisely, the cups and caps equip OBk(A) with the structure of a strict pivotal supercategory;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' see [BSW21, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='16)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' It follows from the definition of the tokens on downward strands that b a = (−1)¯a¯b ba .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We also have (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='10) = , = , = , and = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The basis theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We now describe bases for the morphism spaces of OBk(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Let X = X1 ⊗ · · · ⊗ Xr and Y = Y1 ⊗ · · · ⊗ Ys be objects of OBk(A) for Xt, Yt ∈ {↑, ↓}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' An (X, Y )-matching is a bijection between the sets (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='11) {t : Xt = ↑} ⊔ {t : Yt = ↓} and {t : Xt = ↓} ⊔ {t : Yt = ↑}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' A reduced lift of an (X, Y )-matching is a string diagram representing a morphism X → Y such that the endpoints of each string are points which correspond under the given matching;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' there are no floating bubbles (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' strings with no endpoints) and no tokens on any string;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' there are no self-intersections of strings and no two strings cross each other more than once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Fix a set ⃗D(X, Y ) consisting of a choice of reduced lift for each (X, Y )-matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Then let ⃗D•(X, Y ) denote the set of all morphisms that can be obtained from the elements of ⃗D(X, Y ) by adding one token to each string according to the following convention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Convention 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Tokens are placed such that: each token is labelled by an element of BA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' if a string has endpoints at the top and bottom of the diagram, then its token appears near the bottom of the string (below all crossings);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' if a string has both endpoints at the top of the diagram, then its token appears near the left endpoint (above all crossings);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' if a string has both endpoints at the bottom of the diagram, then its token appears near the right endpoint (below all crossings);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' 18 SAIMA SAMCHUCK-SCHNARCH AND ALISTAIR SAVAGE all tokens near top endpoints are at the same height, all tokens near bottom endpoints are at the same height, and the tokens near top endpoints are above the tokens near bottom endpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For example, for X = ↓ ⊗ ↑ ⊗ ↓ ⊗ ↓ ⊗ ↑ and Y = ↓ ⊗ ↓ ⊗ ↓ ⊗ ↑ ⊗ ↓ ⊗ ↑ ⊗ ↑, is a possible element of ⃗D(X, Y ) and b1 b2 b3 b4 b5 b6 , b1, b2, b3, b4, b5, b6 ∈ BA, are the corresponding elements of ⃗D•(X, Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' While we expect the following theorem to hold for an arbitrary associative superalgebra A, our proof assumes that A is a Frobenius superalgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' As explained in Convention 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='3, this assumption holds whenever A is a real or complex division superalgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Let d ∈ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For X, Y ∈ OBk(A), the morphism space HomOBk(A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='d)(X, Y ) is a free k-supermodule with basis ⃗D•(X, Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The supercategory OBk(A) is a sub-supercategory of the affine Frobenius Brauer supercat- egory AOB(A) defined in [MS, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The latter supercategory is the central charge k = 0 case of the Frobenius Heisenberg category introduced in [Sav19] and further studied in [BSW21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Thus, the assertion follows from the basis theorem [MS, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='7] for AOB(A), which is a special case of the basis theorem [BSW21, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2] for Frobenius Heisenberg categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' In [BSW21, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2], the basis elements carry tokens near the terminus of each strand, which differs from the placement of tokens in the elements of the ⃗D•(X, Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' However, it follows from the relations in OBk(A) that this difference in placement changes the corresponding diagrams by at most a sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' In addition, [BSW21, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2] assumes the Frobenius superalgebra is supersymmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' However, the same proof given there works without this assumption, using the defining property of the Nakayama automorphism wherever supersymmetry is needed, and tracking these applications throughout the calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' (See, for example, [Sav19], which works in this generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=') □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Complexifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Our proof of fullness of the oriented incarnation superfunctor when k = R and A is a real division superalgebra (Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='10) will involve the complexification of OBk(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' In this subsection we state some results about this complexification that we will need.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For any superalgebra A over k = R, and d ∈ R, there are isomorphisms of monoidal supercategories R: OBR(A)C ∼ = −→ OBC(AC) and R: OBR(A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' d)C ∼ = −→ OBC(AC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' d), given on objects by ↑ �→ ↑, ↓ �→ ↓ and on morphisms by �→ , �→ , �→ , �→ , �→ , a �→ a⊗1 , a ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' It is clear that the superfunctor R is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The inverse functor is given on morphisms by �→ , �→ , �→ , �→ , �→ , a⊗z �→ � a � ⊗ z, a ∈ A, z ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' □ DIAGRAMMATICS FOR REAL SUPERGROUPS 19 By Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='6 and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='8, the complexifications OBR(D)C and OBR(D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' d)C, where D is a central real division superalgebra, are related to OBC(R), where R is a supermatrix superalgebra over a complex division superalgebra A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The following result, which is formulated more generally, relates these to OBC(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Recall, from Section 2, the superadditive envelope Add(Cπ) of a supercategory C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We write morphisms in superadditive envelopes as sums of their components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For any superalgebra A and r, s ∈ N, r + s ≥ 1, there is a unique monoidal superfunctor M: OBk(Matr|s(A)) → Add(OBk(A)π) given on objects by ↑ �→ ↑⊕r ⊕ Π↑⊕s, ↓ �→ ↓⊕r ⊕ Π↓⊕s, and on morphisms by �→ r+s � t,u=1 (−1)p(t)p(u) t t u u p(t)+p(u) p(t)+p(u) , Etua �→ u t p(t) p(u) a , �→ r+s � t=1 t t 0 0 , �→ r+s � t=1 t t 0 0 , �→ r+s � t=1 (−1)p(t) t t 0 0 , �→ r+s � t=1 (−1)p(t) t t 0 0 , where (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='12) p(t) = � 0 if 1 ≤ t ≤ r, 1 if r < t ≤ r + s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' This superfunctor is full and faithful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For d ∈ k, it induces equivalences of monoidal supercategories Add �OBk(Matr|s(A))π � ∼ = −→ Add(OBk(A)π), Add �OBk(Matr|s(A);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' d)π � ∼ = −→ Add(OBk(A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' (r − s)d)π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We first consider the non-specialized supercategories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' To prove that M is well defined, we must show that it respects the relations of Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' These are all straightforward verifications, which we leave to the reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' (See the proof of Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='5 for the details of a similar, but slightly less straightforward, verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=') Next we prove that M is full and faithful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Suppose X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' , Xv, Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' , Yw ∈ {↑, ↓}, and let X = X1 ⊗ · · · ⊗ Xv, Y = Y1 ⊗ · · · ⊗ Yw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Then M induces a k-linear map (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='13) HomOBk(Matr|s(A))(X, Y ) → r+s � t1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=',tv,u1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=',uw=1 HomOBk(A) � Πp(t1)+···+p(tv)X, Πp(u1)+···+p(uw)Y � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' It suffices to assume that v + w is even and #{a : Xa = ↑} + #{a : Ya = ↓} = v + w 2 = #{a : Xa =↓} + #{a : Ya = ↑}, otherwise both the domain and image of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='13) have dimension zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' (Here, #S denotes the cardinality of a set S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=') By Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='5, dimk HomOBk(Matr|s(A))(X, Y ) = � v+w 2 � !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' � (r + s)2 dimk A �(v+w)/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We use here the fact there the number of (X, Y )-matchings is (v+w 2 )!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=', and that dimk Matr|s(A) = (r + s)2 dimk A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' On the other hand, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='5 implies that the codomain of the map (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='13) has the same dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Thus, it suffices to prove that the map (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='13) is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' This follows from the fact that any string diagram in the summand HomOBk(A) � Πp(t1)+···+p(tv)X, Πp(u1)+···+p(uw)Y � 20 SAIMA SAMCHUCK-SCHNARCH AND ALISTAIR SAVAGE is the image under (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='13) (up to a sign) of the same diagram with appropriate tokens Etu placed near the endpoints of strands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Finally, we show that M is essentially surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' This follows from the fact that the generating objects ↑ and ↓ of OBk(A) are the images of (↑, E11 ) and (↓, E11 ), respectively, if m ≥ 1, and the images of (Π↑, E11 ) and (Π↓, E11 ), respectively, if m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' It remains to prove the statement about the specialized supercategories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For 1 ≤ t, u ≤ r + s and a ∈ A, we have M � Etua � = M( ) ◦ M( ⊗ Etua ) ◦ ( ) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='11) = δtu(−1)p(t)+p(t)¯a a 0 0 = δtu(−1)p(t)(r − s)d strk A(a) = (r − s)d strk A ◦ str(Etua) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='16) = d strk Matr|s(A)(Etua), where, in the third equality, we used the fact that strk A(a) = 0 unless ¯a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The oriented incarnation superfunctor In this section we introduce the main application of the supercategory OBk(A) to the repre- sentation theory of Lie superalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We begin by defining a very general oriented incarnation superfunctor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We then turn our attention to the special cases where k ∈ {R, C} and A is a division superalgebra over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' When k = C, fullness of the incarnation functor follows from known results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' When k = R, we give a proof of fullness using the complexification of the supercategories involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Definition of the superfunctor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Throughout this subsection, k denotes an arbitrary field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Recall the maps flip, ev, coev, and ρa from Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The following result is the main motivation for the definition of the supercategory OBk(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Suppose that A is an associative superalgebra, g is a Lie superalgebra, and V is a (g, A)-superbimodule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' There exists a unique monoidal superfunctor, which we call the oriented incarnation superfunctor, G = GV : OBk(Aop) → g-smodk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' such that G(↑) = V , G(↓) = V ∗, and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1) G( ) = flip, G( ) = ev, G( aop ) = ρa, a ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' This superfunctor also satisfies the following: G( ) = coev, G( ) = ev ◦ flip, G( ) = flip ◦ coev, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2) G � a � = strV (a), a ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='3) If V = Am|n and g = gl(m|n, A) for some m, n ∈ N, then GV induces a monoidal superfunctor Gm|n : OBk(Aop;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' m − n) → gl(m|n, A)-smodk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We first show that (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2) indeed yield a superfunctor G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We must show that it respects the relations (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1) to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The first two relations in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1) are straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For the third relation in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1), we have G � bop aop � (v) = (−1)(¯a+¯b)¯v+¯a¯bvba = (−1)¯a¯bG( (ba)op )(v) = G( aopbop )(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Next, we show that G( ) = flipV ∗,V , G( ) = flipV,V ∗, G( ) = flipV ∗,V ∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' DIAGRAMMATICS FOR REAL SUPERGROUPS 21 Using the definition (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='5) of the left crossing, the map G( ) = G � � : V ∗ ⊗ V → V ⊗ V ∗ is given by f ⊗ v �→ � v∈Bk V f ⊗ v ⊗ w ⊗ w∗ �→ � v∈Bk V (−1)¯v ¯wf ⊗ w ⊗ v ⊗ w∗ �→ v ⊗ � v∈Bk V (−1)¯v ¯wf(w)w∗ = (−1) ¯f ¯vv ⊗ f, where we use the fact that ¯w = ¯f whenever f(w) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The proofs for and are analogous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The relations (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2) and the first two relations in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='3) are then straightforward to verify.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For the fourth equality in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='3), we have G � � : v �→ � w∈Bk V v ⊗ w ⊗ w∗ �→ � w∈Bk V (−1)¯v ¯ww ⊗ v ⊗ w∗ �→ � w∈Bk V w∗(v)w = v = G � � (v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The verification of the third equality in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='3) is analogous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Verification of the relations (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='4) is straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' To show (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='3), we compute that G � aop � : k �→ k is the map 1 �→ � v∈Bk V (−1)¯vv∗ ⊗ v �→ � v∈Bk V (−1)¯vv∗ ⊗ va �→ � v∈Bk V (−1)¯vv∗(va) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='10) = strV (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The fact that G factors through OBk(Aop;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' m − n) when V = Am|n then follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' It remains to prove that, for any functor as in the first sentence of the theorem, we have (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Suppose that G( ): 1 �→ � u,v∈BV auvu ⊗ v∗, auv ∈ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Then, for all v ∈ BV , we have v = G � � (v) = G � � (v) G � ⊗ � �−−−−−−−−→ � u,w∈BV auwu ⊗ w∗ ⊗ v 1V ⊗ev �−−−−→ � u∈BV auvu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' It follows that auv = δuv for all u, v ∈ BV , and so G( ) = coev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The other two equalities in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2) then follow from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' □ Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1 holds in greater generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' If C is any rigid symmetric monoidal supercategory (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' the category of supermodules over a triangular Hopf superalgebra) with an object V that has the structure of a right A-supermodule, then (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1) defines a unique monoidal superfunctor G: OBk(Aop) → C, and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='3) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The proof of this more general statement is exactly the same as the proof of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We chose to state Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1 with the choice C = g-smodk since that will be our main application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' When A is a Frobenius superalgebra, G is essentially the functor of [MS, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The paper [MS] works with right gl(m|n, A)-supermodules and left A-supermodules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' In Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1, we have translated to the setting of right A-supermodules by considering OBk(Aop) instead of OBk(A) and to the setting of left gl(m|n, A)-supermodules using the involution X �→ −X of gl(m|n, A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' 22 SAIMA SAMCHUCK-SCHNARCH AND ALISTAIR SAVAGE The natural module V is denoted by V+ in [MS].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Furthermore, in [MS], the dual module V ∗ is replaced by a supermodule V−, together with a nondegenerate bilinear form V− ⊗ V+ → k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' This form identifies V− with V ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' By the universal property of Π-envelopes mentioned in Section 2, we have an induced monoidal superfunctor G: OBk(Aop)π → gl(VA)-smodk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The coherence maps of this monoidal superfunctor involve some signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For example, we have the coherence map G(↑) ⊗ G(↑) = ΠV ⊗ ΠV ∼ = −→ Π2V ⊗ V ∼ = −−−→ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='6) V ⊗ V = G(↑ ⊗ ↑) = V ⊗ V, πv ⊗ πw �→ (−1)¯vπ2v ⊗ w �→ −(−1)¯vv ⊗ w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The following result will be useful in later computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We have (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='4) G( aop ): V ∗ → V ∗, f �→ af, f ∈ V ∗, a ∈ A, where af is defined as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We have G( aop ) = G � aop � : f �→ � v∈Bk V f ⊗ v ⊗ v∗ �→ � v∈Bk V (−1)¯a( ¯f+¯v)f ⊗ va ⊗ v∗ �→ � v∈Bk V (−1)¯a( ¯f+¯v)f(va)v∗ = � v∈Bk V (af)(v)v∗ = af.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Fullness over the complex numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The remainder of this section is dedicated to proving that the oriented incarnation superfunctor of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1 is full in certain important special cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' In this subsection we consider the case where k = C and A is a matrix superalgebra over a complex division superalgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We begin with the case where A is complex division superalgebra, which follows from results in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' If k = C and A is a complex division superalgebra, then the oriented incarnation functor Gm|n of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1 is full for all m, n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' As explained in Section 4, the only complex division superalgebras are C and Cl(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' When A = C, the supercategory OBC(C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' m − n) is the usual oriented Brauer category, and the result was proved in [CW12, §8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' (Closely related results were obtained in [BS12, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='8] and [LSM02, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=') On the other hand, OBC(Cl(C), 0) is the oriented Brauer–Clifford supercategory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' (Recall that, by Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='3, we may assume the specialization parameter is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=') In this case, fullness was proved in [BCK19, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' □ In the remainder of this subsection, our goal is to show that the oriented incarnation superfunctor Gm|n is full when A is the superalgebra of supermatrices over a complex division superalgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' This will be key in our proof that it is also full when A is a real division superalgebra (Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We begin with a result that holds in a more general setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Let A be a superalgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Fix m, n, r, s ∈ N with m + n, r + s ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' In what follows, we will identify Matm|n(Matr,s(A)) and Mat(mr+ns)|(ms+nr)(A) DIAGRAMMATICS FOR REAL SUPERGROUPS 23 in the natural way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' This induces a natural identification of gl(m|n, Matr,s(A)) and gl((mr + ns|ms + nr), A), and we denote this Lie superalgebra by g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Let W = Matr|s(A)m|n and V = A(mr+ns|ms+nr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We have an isomorphism of (g, A)-superbimodules (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='5) W ∼ = −→ V ⊕r ⊕ ΠV ⊕s, v �→ � (−1)p(t)vtπp(t)vt �r+s t=1 , where vt ∈ V is the t-th column of v, and p(t) is defined as in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Similarly, we have an isomorphism of (g, A)-superbimodules (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='6) W ∗ ∼ = −→ (V ∗)⊕r ⊕ (ΠV ∗)⊕s, f �→ (πp(t)ft)r+s t=1, where ft ∈ V ∗ denotes the restriction of f to the t-th summand in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The next result shows that the diagram of superfunctors OBk(Matr|s(A)op) Add(OBk(Aop)π) g-smodk M Gm|n G(mr+ns|ms+nr) commutes up to natural isomorphism, where M is the superfunctor of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The isomorphisms (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='5) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='6) induce a monoidal supernatural isomorphism of superfunctors Gm|n ∼ = −→ G(mr+ns)|(ms+nr)M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Let ω denote the supernatural transformation induced by (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='5) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' To simplify nota- tion, let G = Gm|n and G′ = G(mr+ns)|(ms+nr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We need to show that ωY ◦ G(f) = G′M(f) ◦ ωX for every generating morphism f ∈ { , , , , , a : a ∈ A}, where X and Y denote the domain and codomain of f, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' These are all straightforward verifications, although care is needed to keep careful track of signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We give the details for and , since the others are similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' First consider the case f = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We have ω↑⊗↑ : W ⊗ W → r+s � t,u=1 Πp(t)+p(u)V ⊗ V, v ⊗ w �→ � (−1)p(t)vt+p(u)wu+p(u)vt+p(t)p(u)vt ⊗ wu �r+s t,u=1 , where p(t) is defined as in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Then we compute that G′M( ) ◦ ω↑⊗↑ = r+s � t,u=1 (−1)p(t)p(u) � Πp(t)+p(u) flip � ω↑⊗↑ and ω↑⊗↑ ◦ G( ) = ω↑⊗↑ ◦ flip 24 SAIMA SAMCHUCK-SCHNARCH AND ALISTAIR SAVAGE are both the map (see Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='4) W ⊗ W → r+s � t,u=1 Πp(t)+p(u)V ⊗ V v ⊗ w �→ � (−1)p(t)vt+p(u)wu+p(u)vt+vtwuvt ⊗ wu �r+s t,u=1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Now consider f = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We have ω↓⊗↑ : W ∗ ⊗ W → r+s � t,u=1 Πp(t)+p(u)V ∗ ⊗ V, f ⊗ v �→ � (−1)p(u)vu+p(u)ft+p(t)p(u)ft ⊗ vu �r+s t,u=1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' In addition, ω 1 : k → k is the identity map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Then we compute that G′M( ) ◦ ω↓⊗↑ = r+s � t=1 G′( ) ◦ ω↓⊗↑ and ω 1 ◦ G( ) = G( ) are both the map W ∗ ⊗ W → k, f ⊗ v �→ r+s � t=1 (−1)p(t)ft(vt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' □ Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' If D is a complex division superalgebra, then the superfunctor Gm|n: OBC(Matr|s(D)op) → gl((mr + ns|ms + nr), D)-smodC is full for all m, n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' This follows from Propositions 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='7, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='6 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Fullness over the real numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' In this subsection, we prove one of our main results: the oriented incarnation superfunctor of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1 is full when k = R and A is a central real division superalgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Suppose D is a central real division superalgebra and recall Convention 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For V = Dm|n, we have canonical isomorphisms (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='7) V C = (Dm|n)C ∼ = −→ (DC)m|n and V ∗,C = � (Dm|n)∗�C ∼ = −→ � (DC)m|n�∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The next result shows that the diagram OBR(Dop)C OBC(Dop,C) (gl(m|n, D)-smodR)C gl(m|n, DC)-smodC R ∼ = GC m|n Gm|n Cgl(m|n,D) ∼ = commutes up to supernatural isomorphism, where Cgl(m|n,D) is defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='21), and R is the su- perfunctor of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' This is a monoidal supernatural isomorphism of superfunctors Cgl(m|n,D)GC m|n ∼ = −→ Gm|nR determined by (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' DIAGRAMMATICS FOR REAL SUPERGROUPS 25 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' To simplify notation, we set G = Gm|n, GC = GC m|n, C = Cgl(m|n,D), W = (DC)m|n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Let ω be the monoidal supernatural isomorphism determined by (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For each generating mor- phism f ∈ { , , , , , aop : a ∈ A}, we must show that ωY ◦ CGC(f) = GR(f) ◦ ωX, where X and Y are the domain and codomain of f, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We have ω↑⊗↑ ◦ CGC( ) = ω↑⊗↑ ◦ flipV C,V C = flipW,W ◦ω↑⊗↑ = GR( ) ◦ ω↑⊗↑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For a ∈ D, v ∈ V , and y, z ∈ C, we have ω↑ ◦ CGC( aop ⊗ y): v ⊗ z �→ (−1)¯a¯vω↑(va ⊗ yz) and GR( aop ⊗ y) ◦ ω↑ = G( aop⊗y ) ◦ ω↑ : v ⊗ z �→ (−1)¯a¯vω↑(va ⊗ yz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For f ∈ V ∗, v ∈ V , and y, z ∈ C, we have ω 1 ◦ CGC( ): (f ⊗ y) ⊗ (v ⊗ z) �→ f(v)yz and GR( ) ◦ ω↓⊗↑ = G( ) ◦ ω↓⊗↑ : (f ⊗ y) ⊗ (v ⊗ z) �→ f(v)yz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Finally, we have ω↑⊗↓ ◦ CGC( ): 1 �→ � v∈BR V v ⊗ v∗ and GR( ) ◦ ω 1 : 1 �→ � v∈BC V C v ⊗ v∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' These are equal since any R-basis of V is also a C basis of V C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The verifications for the remaining generating morphisms and are similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' □ Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' When k = R and A is a central real division superalgebra, the oriented incarnation superfunctor Gm|n of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1 is full for all m, n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Suppose A = D is a central real division superalgebra and m, n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We wish to show that, for all objects X, Y in OBk(Dop), the R-linear map G: HomOBk(Dop;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='m−n)(X, Y ) → Homgl(m|n,D)(G(X), G(Y )) is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' As explained in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='7, this map is surjective if and only if the complexified map GC : HomOBk(Dop;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='m−n)(X, Y )C → Homgl(m|n,D)(G(X), G(Y ))C is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' This follows from Propositions 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='8 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='9 and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Consequences for real Lie groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We assume throughout this subsection that D ∈ {R, H}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Then OBR(D) is a monoidal category, and there is no need to work in the setting of supercategories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' In fact, OBR(D) is a spherical pivotal category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' (We refer the reader to [Sel11, §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='3] for the definition of spherical pivotal category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=') In any spherical pivotal category C, we have a trace map Tr: � X∈C EndC(X) → EndC(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' In terms of string diagrams, this corresponds to closing a diagram off to the right or left: Tr � f � = f = f , where the second equality follows from the axioms of a spherical category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We say that a morphism f ∈ HomC(X, Y ) is negligible if Tr(f ◦ g) = 0 for all g ∈ HomC(Y, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The negligible morphisms 26 SAIMA SAMCHUCK-SCHNARCH AND ALISTAIR SAVAGE form a two-sided tensor ideal N of C, and the quotient C/N is called the semisimplification of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For m ∈ N, let ON R(D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' m) denote the tensor ideal of negligible morphisms of Kar(OBR(D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' m)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For an associative k-algebra A and m ∈ N, let gl(m, A)-tmodk denote the monoidal category of tensor gl(m, A)-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' By definition, this is the full subcategory of gl(m, A)-modk whose objects are direct summands of V ⊗r ⊗ (V ∗)⊗s, r, s ∈ N, where V = Am is the natural module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For D ∈ {R, H} and m ∈ N, the oriented incarnation functor induces an equiva- lence of monoidal categories Kar(OBR(Dop;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' m))/ON R(Dop;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' m) → gl(m, D)-tmodR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Let D ∈ {R, H}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Since the category gl(m, D)-tmodR is idempotent complete, Gm = Gm|0 induces a monoidal functor Kar(Gm): Kar(OBR(Dop;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' m)) = gl(m, D)-tmodR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' By Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='10, this functor is full.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Every object in gl(m, D)-tmodR is completely reducible, since its complexification is a tensor module for gl(m, D)C ∼= gl(m, DC), hence completely reducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Thus, the category gl(m, D)-tmodR is semisimple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' In addition, by Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='5, EndOBR(Dop;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='m)(1) = R1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Thus, by [SW22, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='9], the kernel of Kar(Gm) is equal to ON R(Dop;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Therefore, Kar(Gm) induces a full and faithful functor OBR(Dop;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' m)/ON R(Dop;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' m) → gl(m, D)-tmodR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Finally, since the image of Kar(Gm) contains all summands of tensor powers of the natural module Dm and its dual (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' all tensor modules), it is essentially surjective, hence an equivalence of categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' □ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Superhermitian forms over involutive superalgebras In Section 9, we will introduce our second main diagrammatic supercategory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Then, in Section 10, we will define the corresponding incarnation superfunctor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' These constructions will depend on superhermitian forms over involutive superalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' In the current section, we cover the important properties of these forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Then, in Section 8, we further specialize to the case where the superalgebra is a real division superalgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Involutive superalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' An anti-involution of a superalgebra A is a homomorphism of associative superalgebras A → Aop squaring to the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Equivalently, it is a k-linear map ⋆: A → A, a �→ a⋆, such that (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1) (ab)⋆ = (−1)¯a¯bb⋆a⋆ and (a⋆)⋆ = a for all a, b ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' An involutive superalgebra is a pair (A, ⋆), where ⋆ is an anti-involution of an associative superalgebra A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We will typically use the notation ⋆ or ⋄ for anti-involutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' If (A, ⋆) is an involutive superalgebra and V is a right A-supermodule, then we let V ⋆ denote the left A-supermodule that is equal to V as a k-supermodule, and with A-action given by (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2) a · v := (−1)¯a¯vva⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Recall the definition of the Nakayama automorphism ζ of a Frobenius superalgebra from Sec- tion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' An involutive Frobenius superalgebra is a triple (A, ⋆, τ) such that (A, τ) is Frobenius superalgebra, (A, ⋆) is an involutive superalgebra, and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='3) ζ2(a) = a and τ(a⋆) = τ(a) for all a ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' DIAGRAMMATICS FOR REAL SUPERGROUPS 27 We will discuss our main examples of interest, the involutive real division superalgebras, in Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' However, let us mention here some other important examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Examples 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' (a) The identity map is an anti-involution of any supercommutative Frobenius superalgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' (b) As a special case of (a), if A = k[x]/(xn) for some n ≥ 1, we can take ⋆ to be the identity map and τ(�n−1 r=0 arxr) = an−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' (c) If G is a finite group, we can take A = kG, g⋆ = g−1 for all g ∈ G, and τ to be projection onto the identity element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' If (A, ⋆, τ) is an involutive Frobenius superalgebra, then (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='4) ζ(a⋆) = ζ(a)⋆ for all a ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For all a, b ∈ A, we have τ(ab) (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='3) = τ((ab)⋆) = (−1)¯a¯bτ(b⋆a⋆) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='5) = (−1)¯a¯bτ(a⋆ζ(b⋆)) and τ(ab) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='5) = (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='3) (−1)¯a¯bτ(ζ(b)a) (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='3) = (−1)¯a¯bτ((ζ(b)a)⋆) = (−1)¯a¯bτ(a⋆ζ(b)⋆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Then (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='4) follows from the nondegeneracy of the Frobenius form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' □ The following corollary will play an important role;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' see Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' If (A, ⋆, τ) is an involutive Frobenius algebra, then so is (A, ⋄, τ), where a⋄ = ζ(a)⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Suppose (A, ⋆, τ) is an involutive Frobenius superalgebra with Nakayama automor- phism ζ, and let BA be a homogeneous k-basis of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Then the left dual basis to B⋆ A := {b⋆ : b ∈ BA} is given by (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='5) (b⋆)∨ = ζ(b∨)⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For b, c ∈ BA, we have τ � ζ(c∨)⋆b⋆� (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1) = (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='3) (−1) ¯b¯cτ(bζ(c∨)) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='5) = τ(c∨b) = δbc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' □ Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' If (A, ⋆, τ) is an involutive Frobenius superalgebra, then (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='6) strA(a) = strA(a⋆) for all a ∈ A, where strA is the supertrace map defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='12), we compute strA(a) = � b∈BA (−1) ¯bτ(b∨ba) (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='3) = � b∈BA τ � a⋆b⋆(b∨)⋆� = � b∈BA (−1) ¯bτ(a⋆b∨b) = strA(a⋆), where, in the second-to-last equality we changed to a sum over {(b∨)⋆ : b ∈ BA} and used that, for b, c ∈ BA, (−1)¯cτ(c⋆(b∨)⋆) = τ(b∨c) = δbc, and so ((b∨)⋆)∨ = (−1)¯bb⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' □ 28 SAIMA SAMCHUCK-SCHNARCH AND ALISTAIR SAVAGE 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Supersymmetric forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Let (A, ⋄) denote an involutive superalgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For ν ∈ {±1}, a (ν, ⋄)-supersymmetric form on a right A-supermodule V is a homogeneous k-bilinear form Φ: V × V → k such that (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='7) Φ(v, w) = ν(−1)¯v ¯wΦ(w, v) and Φ(va, w) = (−1)¯a ¯wΦ(v, wa⋄) for all v, w ∈ V and a ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' If Φ is a nondegenerate (ν, ⋄)-supersymmetric form on V and BV is a k-basis of V , then the left dual basis B∨ V = {v∨ : v ∈ BV } of V is defined by Φ(v∨, w) = δvw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Note that v∨ = v + ¯Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Recall the left A-actions on V ∗ and V ⋄ given in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2) and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' A (ν, ⋄)-supersymmetric form Φ of parity σ induces a parity-preserving homomorphism of left A-supermodules (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='8) V ⋄ → ΠσV ∗, v �→ πσΦv, where Φv(w) = Φ(v, w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' This is an isomorphism if and only if Φ is nondegenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' If a right A-supermodule V admits a (ν, ⋄)-supersymmetric form, then (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='9) strV (a⋄) = strV (a) for all a ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Let Φ be a (ν, ⋄)-supersymmetric form on V , let BV be a k-basis of V , and let B∨ V = {v∨ : v ∈ BV } denote the left dual basis with respect to Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Then Φ(w, v∨) (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='7) = ν(−1) ¯w(¯v+¯Φ)Φ(v∨, w) = ν(−1)¯v+¯v ¯Φδvw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Thus (v∨)∨ = (−1)¯v+¯v ¯Φv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Then we compute strV (a⋄) = � v∈BV (−1)¯vv∗(va⋄) = � v∈BV (−1)¯vΦ(v∨, va⋄) (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='7) = � v∈BV (−1)¯v+¯a¯vΦ(v∨a, v) (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='7) = ν � v∈BV (−1)¯v ¯ΦΦ(v, v∨a) = � v∈BV (−1)¯vΦ(v∨, va) = � v∈BV strV (a), where, in the second-to-last equality, we changed to a sum over the basis B∨ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' □ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Superhermitian forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Our main source of examples of (ν, ⋄)-supersymmetric forms will come from superhermitian forms over involutive Frobenius superalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Let V be a finitely- generated right supermodule over an involutive superalgebra (A, ⋆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' A homogeneous map ϕ: V × V → A is a ⋆-sesquilinear form if it is k-bilinear and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='10) ϕ(va, wb) = (−1)¯a( ¯ϕ+¯v)a⋆ϕ(v, w)b, for all a, b ∈ A, v, w ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' (In our cases of interest, A will be purely even whenever ¯ϕ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=') If, in addition, there exists ν ∈ {±1} such that (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='11) ϕ(v, w) = ν(−1)¯v ¯wϕ(w, v)⋆ for all v, w ∈ V, then we say that ϕ is a (ν, ⋆)-superhermitian form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We say that ϕ is unimodular if the map V → HomA(V, A), v �→ ϕ(v, −), v ∈ V, is an isomorphism of k-supermodules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' If A is a division superalgebra, then ϕ is unimodular if and only if it is nondegenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' DIAGRAMMATICS FOR REAL SUPERGROUPS 29 We say that two (ν, ⋆)-superhermitian forms ϕ1 and ϕ2 are equivalent if there exists a homoge- neous f ∈ AutA(V ) such that ϕ2(v, w) = ϕ1(f(v), f(w)) for all v, w ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Note that, when A = k, a (ν, id)-superhermitian form is the same as a (ν, id)-supersymmetric form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Example 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' If A = C, ⋆ is complex conjugation, and V is purely even, then a (1, ⋆)-superhermitian form is the familiar notion of a hermitian form, while a (−1, ⋆)-superhermitian form is a skew- hermitian form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' On the other hand, a (1, id)-superhermitian form is a symmetric C-bilinear form, while a (−1, id)-superhermitian form is a skew-symmetric C-bilinear form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' An even (ν, ⋆)-superhermitian form on V is equivalent to an even (−ν, ⋆)-superhermitian form on the parity-shifted supermodule ΠV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Thus, for even forms, one can assume ν = 1 without losing any generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' However, this is not the case for odd forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' An odd (ν, ⋆)-superhermitian form on V is equivalent to an odd (ν, ⋆)-superhermitian form on the parity shift ΠV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Since odd forms are important for the periplectic Lie superalgebras we wish to include, we consider general ν in the current paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' If (A, ⋆, τ) is an involutive Frobenius superalgebra, and ϕ is a nondegenerate (ν, ⋆)- superhermitian form on V , then the composite (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='12) Φ = τ ◦ ϕ: V × V → k is a nondegenerate (ν, ⋄)-supersymmetric form on V , with a⋄ = ζ(a)⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For all v, w ∈ V , we have Φ(v, w) = τ(ϕ(v, w)) (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='11) = ν(−1)¯v ¯wτ(ϕ(w, v)⋆) (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='3) = ν(−1)¯v ¯wτ(ϕ(w, v)) = ν(−1)¯v ¯wΦ(w, v) and Φ(va, w) = τ(ϕ(va, w)) (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='10) = (−1)¯a( ¯ϕ+¯v)τ(a⋆ϕ(v, w)) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='5) = (−1)¯a ¯wτ(ϕ(v, w)ζ(a⋆)) (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='10) = (−1)¯a ¯wτ(ϕ(v, wζ(a⋆))) (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='4) = (−1)¯a ¯wΦ(v, wζ(a)⋆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Thus Φ is (ν, ⋄)-supersymmetric, with a⋄ = ζ(a)⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' It remains to show that Φ is nondegenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Suppose v ∈ V is nonzero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Then, since ϕ is nondegenerate, there exists w ∈ V such that ϕ(v, w) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Since τ is nondegenerate, there exists a ∈ A such that 0 ̸= τ(ϕ(v, w)a) = τ(ϕ(v, wa)) = Φ(v, wa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' □ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Adjoint operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Suppose that (A, ⋆) is an involutive superalgebra, and that ϕ is a uni- modular (ν, ⋆)-superhermitian form on a right A-supermodule V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For all X ∈ EndA(V ), there exists a unique X† ∈ EndA(V ) such that ϕ(v, Xw) = (−1) ¯ X¯vϕ(X†v, w) for all v, w ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Fix v ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The map w �→ (−1) ¯ X¯vϕ(v, Xw) is an element of HomA(V, A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Thus, since ϕ is unimodular, there exists a unique v′ ∈ V such that ϕ(v′, w) = (−1) ¯ X¯vϕ(v, Xw) for all v ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We define X†v = v′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' It is then straightforward to verify that X† ∈ EndA(V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' □ The element X† is called the adjoint to X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' 30 SAIMA SAMCHUCK-SCHNARCH AND ALISTAIR SAVAGE Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The map X �→ X† is an anti-involution on EndA(V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' In particular, (X†)† = X, and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='13) (XY )† = (−1) ¯ X ¯Y Y †X† for all X, Y ∈ EndA(V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' This is a straightforward verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' □ Now suppose that (A, ⋆, τ) is an involutive Frobenius superalgebra, and consider the nondegen- erate (ν, ⋄)-supersymmetric form Φ = τ ◦ ϕ as in Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Then, for all X ∈ EndA(V ), we have Φ(v, Xw) = (−1) ¯ X¯vΦ(X†v, w) for all v, w ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' It follows that the definition of X† is the same if we use the (ν, ⋆)-superhermitian form ϕ or the corresponding (ν, ⋄)-supersymmetric form Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Harish-Chandra superpairs associated to superhermitian forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For ϕ either a uni- modular (ν, ⋆)-superhermitian form or a nondegenerate (hence unimodular) (ν, ⋄)-supersymmetric form, define Gred(ϕ) = {X ∈ AutA(V )0 : ϕ(Xv, Xw) = ϕ(v, w) for all v, w ∈ V } g(ϕ) = {X ∈ EndA(V ) : ϕ(Xv, w) = −(−1) ¯ X¯vϕ(v, Xw) for all v, w ∈ V } = {X ∈ EndA(V ) : X† = −X}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We have that g(ϕ) is a Lie superalgebra with the usual bracket: [X, Y ] = XY − (−1) ¯ X ¯Y Y X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The pair G(ϕ) := (Gred(ϕ), g(ϕ)) is a Harish-Chandra superpair;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' If ϕ1 and ϕ2 are equivalent forms, then the groups Gred(ϕ1) and Gred(ϕ2) are isomorphic, as are the Lie superalgebras g(ϕ1) and g(ϕ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' If (A, ⋆, τ) is an involutive Frobenius superalgebra and ϕ is a unimodular (ν, ⋆)-superhermitian form, then we have the nondegenerate (ν, ⋄)-supersymmetric form Φ = τ ◦ ϕ from Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' It follows from the nondegeneracy of τ that g(ϕ) = g(Φ) and Gred(ϕ) = Gred(Φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Superhermitian forms over involutive real division superalgebras For our purposes, the most important examples of involutive superalgebras come from real divi- sion superalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' In this section, we examine some important properties of the Lie superalgebras associated to superhermitian forms over involutive real division superalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For the quaternions, we have the anti-involution ⋆: H → R, (a + bi + cj + dk)⋆ = a − bi − cj − dk, a, b, c, d ∈ R, of quaternionic conjugation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' This restricts to complex conjugation on C and the identity map on R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The complex Clifford superalgebra Cl(C) has anti-involution (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1) ⋆: Cl(C) → Cl(C), (a + εb)⋆ = a⋆ + εb⋆i, a, b ∈ C, where, on the right-hand side, ⋆ denotes complex conjugation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Note that this is an anti-involution of real superalgebras, but not of complex superalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' In fact, there are no C-linear anti-involutions of Cl(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The notation ⋆ will always refer to the above involutions when working with the real division superalgebras R, C, H, and Cl(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Note that (D, ⋆, Re) is an involutive Frobenius superalgebra for D ∈ {R, C, H, Cl(C)}, as is (C, id, Re), where Re(a) is the real part of the even part of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' None of the other real division superalgebras Clr(R), r ∈ {1, 2, 3, 5, 6, 7}, admit anti- involutions, since they are not isomorphic to their opposite superalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' DIAGRAMMATICS FOR REAL SUPERGROUPS 31 If (A, ⋆) is an involutive superalgebra, then the complexification AC is also an involutive super- algebra, with involution (which we continue to denote by the same symbol) (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2) ⋆: AC → AC, (a ⊗ z)⋆ = a⋆ ⊗ z, a ∈ A, z ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Similarly, if (A, ⋆, τ) is an involutive Frobenius superalgebra, then so is (AC, ⋆, τ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Real case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' In this subsection we work over the involutive real division superalgebra (R, id).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' If ϕ: V × V → R is a nondegenerate (ν, id)-superhermitian form on an R-supermodule V , then its complexification ϕC : V C × V C → C is a nondegenerate (ν, id)-superhermitian form on V C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We have an isomorphism of complex Lie superalgebras g(ϕ)C ∼= g(ϕC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' It is straightforward to verify that the map X ⊗ a �→ Xa, X ∈ g(ϕ), a ∈ C, gives the desired isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' □ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Complex cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' In this subsection we work over the involutive real division superalgebra (D, ⋆), where D ∈ {C, Cl(C)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Suppose ϕ is a nondegenerate (ν, ⋆)-superhermitian form on Dm|n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Then we have an isomorphism of complex Lie superalgebras g(ϕ)C ∼= gl(m|n, D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' It follows from (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='13) with Y = aI that (aX)† = a⋆X† for all a ∈ D and X ∈ g(ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Multiplication by i gives an isomorphism of C-supermodules g(ϕ) ∼ = −→ {X ∈ gl(m|n, D) : X† = X}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Therefore, for every X ∈ gl(m|n, D), we have X = 1 2(X† − X) + 1 2(X† + X), with 1 2(X† − X) ∈ g(ϕ) and 1 2(X† + X) ∈ g(ϕ)i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' □ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Quaternionic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' In this subjection we work over the real involutive division superalgebra (H, ⋆) and we fix a nondegenerate (ν, ⋆)-superhermitian form ϕ on Hm|n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We will often view the quaternions as (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='3) H = C[j], j2 = −1, jzj−1 = z⋆, z ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Choosing the C-basis {1, j} for H, we will identify Hm|n with C2m|2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Similarly, under the inclusion ı of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='4), we will identify Matm|n(H) with a subring of Mat2m|2n(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' In particular, we have (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='4) Matm|n(H) = {X ∈ Mat2m|2n(C) : X(vj) = (Xv)j for all v ∈ Hm|n = C2m|2n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Note that, for v ∈ Hm|n = C2m|2n, we have (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='5) vj = Jv⋆, where J = Jm+n = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed J1 0 · · 0 0 J1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' 0 0 · · 0 J1 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 , J1 = � 0 −1 1 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' 32 SAIMA SAMCHUCK-SCHNARCH AND ALISTAIR SAVAGE It follows that, for X ∈ Mat2m|2n(C), (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='6) X ∈ Matm|n(H) ⇐⇒ JX⋆J−1 = X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Consider the C-linear maps projH C, projH jC: H → C, projH C(a + jb) = a, projH jC(a + jb) = b, a, b ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' It is straightforward to verify that (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='7) projH jC(zj) = projH C(z)⋆ for all z ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Define ϕ1 := projH C ◦ϕ, ϕj := projH jC ◦ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' These are precisely the components of ϕ with respect to the C-basis {1, j} of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The following lemma gives a precise relationship between ϕ1 and ϕj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For all v, w ∈ Hm|n = C2m|2n, we have ϕj(vj, w) = −ϕ1(v, w), ϕ1(vj, w) = ϕj(v, w), (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='8) ϕj(v, wj) = ϕ1(v, w)⋆, ϕ1(v, wj) = −ϕj(v, w)⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='9) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We have ϕ1(vj, w) + jϕj(vj, w) = ϕ(vj, w) = −jϕ(v, w) = −jϕ1(v, w) + ϕj(v, w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Comparing components gives (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Similarly ϕ1(v, wj) + jϕj(v, wj) = ϕ(v, wj) = ϕ(v, w)j (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='3) = jϕ1(v, w)⋆ − ϕj(v, w)⋆ implies (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' □ Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The form ϕj : C2m|2n × C2m|2n → C is nondegenerate and (−ν, id)-supersymmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' It follows from (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='3) that ϕj(av, wb) = aϕj(v, w)b, v, w ∈ V, a, b ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Also, ϕ(v, w) = ν(−1)¯v ¯wϕ(w, v)⋆ = ν(−1)¯v ¯w� ϕ1(w, v) + jϕj(w, v) �⋆ = ν(−1)¯v ¯w� ϕ1(w, v)⋆ + ϕj(w, v)⋆j⋆� = ν(−1)¯v ¯wϕ1(w, v)⋆ − jν(−1)¯v ¯wϕj(w, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Thus, ϕj : C2m|2n × C2m|2n → C is a (−ν, id)-supersymmetric form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' It follows from (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='9) that, for all v, w ∈ C2m|2n, we have ϕ(v, w) = ϕj(v, wj)⋆ + jϕj(v, w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Thus, ϕj is nondegenerate, since ϕ is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' □ Corollary 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We have (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='10) ϕj(v, wj) − ϕj(vj, w) = 2 Re ϕ(v, w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Using (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='8) and (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='9), we have ϕj(v, wj) − ϕj(vj, w) = ϕ1(v, w)⋆ + ϕ1(v, w) = 2 Re ϕ(v, w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' □ Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We have an isomorphism of complex Lie superalgebras g(ϕ)C ∼= g(ϕj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' DIAGRAMMATICS FOR REAL SUPERGROUPS 33 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For all X ∈ Matm|n(H) satisfying ϕj(Xv, w) = −(−1) ¯ X¯vϕj(v, Xw) for all v, w ∈ C2m|2n, we have ϕ1(Xv, w) (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='9) = ϕj(Xv, wj)⋆ = −(−1) ¯ X¯vϕj(v, X(wj))⋆ (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='4) = −(−1) ¯ X¯vϕj(v, (Xw)j)⋆ (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='9) = −(−1) ¯ X¯vϕ1(v, Xw).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Thus g(ϕ) = {X ∈ gl(m|n, H) : ϕj(Xv, w) = −(−1) ¯ X¯vϕj(v, Xw)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Furthermore, using Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='4, for X ∈ g(ϕ), a ∈ C, and v, w ∈ C2m|2n, we have ϕj((Xa)v, w) = −(−1) ¯ X¯vaϕj(v, Xw) = −(−1) ¯ X¯vϕj(v, Xw)a = −(−1) ¯ X¯vϕj(v, (Xa)w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' It then follows from Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='11(b) that g(ϕ)C = {X ∈ gl(2m|2n, C) : ϕj(Xv, w) = −(−1) ¯ X¯vϕj(v, Xw)} = g(ϕj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' □ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The unoriented supercategory In this section, we introduce the second of our two main diagrammatic supercategories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' After defining the supercategory, we deduce some of its additional properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We then prove a basis theorem for morphism spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Throughout this section, k is an arbitrary field, and (A, ⋄) is an involutive superalgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Definition of the supercategory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Definition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For σ ∈ Z2, we define Bσ k (A, ⋄) to be the strict monoidal supercategory generated by one object I and morphisms : I⊗2 → I⊗2, : I⊗2 → 1, : 1 → I⊗2, a : I → I, a ∈ A, subject to the relations = , = , = = (−1)σ , = , = , (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1) 1 = , λ a + µ b = λa+µb , b a = ab , a = a , a = a⋄ , (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2) for all a, b ∈ A and λ, µ ∈ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The parity of a is ¯a, the morphisms and both have parity σ, and is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We refer to the morphisms a as tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For d ∈ k, we define Bσ k (A, ⋄;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' d) to be the quotient of Bσ k (A, ⋄) by the additional relations (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='3) a = d strA(a)1 1, a ∈ A, where strA is given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We call d the specialization parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Example 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' When A = k and ⋄ = id, we have a = a for all a ∈ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Thus, we can omit the generators a and all the relations involving them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Then, if σ = 0, the relations (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1) and (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='3) are the defining relations of the Brauer category, as given in [LZ15, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For general σ, Bσ k (k, id;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' d) is isomorphic to the marked Brauer category of [KT17], although the description there looks somewhat different since the authors do not use the concept of a monoidal supercategory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The following relations hold in Bσ k (A, ⋄) for all a ∈ A: = (−1)σ , = , a = a , a = a⋄ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='4) 34 SAIMA SAMCHUCK-SCHNARCH AND ALISTAIR SAVAGE Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For the second relation in (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='4), we have = (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2) = (−1)σ = (−1)σ = , where, for the unadorned equalities, we have used the third, sixth, and fourth equalities in (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1), in that order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Then, for the first relation in (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='4), we compute = (−1)σ = (−1)σ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2) = = = = (−1)σ , where, for the unadorned equalities, we use fourth relation in (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1), the sixth relation in (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1), the second relation in (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='4), the fifth relation in (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1), and finally the fourth relation in (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The third relation in (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='4) follows from the fourth relation in (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2) after composing on the top and bottom with the crossing and using the first relation in (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For the last relation in (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='4), we have a (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1) = (−1)σ a (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2) = (−1)σ+σ¯a a (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2) = (−1)σ+σ¯a a⋄ (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1) = (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2) a⋄ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' □ It follows from the defining relations that “bubbles” are central in Bσ k (A, ⋄): a = a for all a ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Note also that (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='5) ab (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2) = a b (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2) = a⋄ b (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2) = (−1)¯a¯b a⋄ b (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='4) = (−1)¯a¯b b a (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2) = (−1)¯a¯b ba and (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='6) a (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='4) = (−1)σ a (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2) = (−1)σ a (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1) = (−1)σ a (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2) = (−1)σ a⋄ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Remark 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Suppose that (A, ⋄, τ) is an involutive Frobenius superalgebra, σ = 1, and d ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' In B1 k(A, ⋄;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' d), we have d strA(a)1 1 = a (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='6) = − a⋄ = −d strA(a⋄)1 1 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='6) = −d strA(a)1 1 for all a ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' In applications to representation theory, we will assume that the characteristic of k is not two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' In this case, it follows that strA = 0 or 1 1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' In the former case, we have B1 k(A, ⋄;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' d) = B1 k(A, ⋄;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' In the latter case, B1 k(A, ⋄;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' d) is the zero supercategory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Thus, when σ = 1 and (A, ⋄) can be endowed with the structure of an involutive Frobenius superalgebra, we will usually assume that d = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For any d ∈ k, we have isomorphisms of monoidal supercategories (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='7) Ξ⋄ : Bσ k (A, ⋄) → Bσ k (Aop, ⋄) and Ξ⋄ : Bσ k (A, ⋄;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' d) → Bσ k (Aop, ⋄;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' d), given by applying the involution ⋄ to all tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The basis theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' This subsection is dedicated to the proof of a basis theorem giving bases for the morphisms spaces of the Bσ k (A, ⋄).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Our method involves embedding this supercategory into the superadditive envelope of the oriented supercategory OBk(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Even in the case A = k, when B0 k(k, id) is the usual Brauer category, this method of proof is new.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Recall, from Section 2, that, for a monoidal supercategory C, we let Add(Cπ) denote its super- additive envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The objects of Add(Cπ) are formal direct sums of objects of the Π-envelope Cπ, DIAGRAMMATICS FOR REAL SUPERGROUPS 35 and morphisms in Add(Cπ) are matrices of morphisms in Cπ, which we will write as sums of their components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For example, 0 0 + σ σ + σ σ + (−1)σ 0 0 : (↑ ⊕Πσ↓) ⊗ (↑ ⊕Πσ↓) → (↑ ⊕Πσ↓) ⊗ (↑ ⊕Πσ↓) is a morphism in Add(OBk(A)) with components 0 0 : ↑ ⊗ ↑ → ↑ ⊗ ↑, σ σ : Πσ↑ ⊗ ↓ → Πσ↓ ⊗ ↑, σ σ : Πσ↓ ⊗ ↑ → Πσ↑ ⊗ ↓, (−1)σ 0 0 : ↓ ⊗ ↓ → ↓ ⊗ ↓, and all other components equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Fix σ ∈ Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' There exists a unique monoidal superfunctor D: Bσ k (A, ⋄) → Add(OBk(A)π) such that D(I) = ↑ ⊕Πσ↓ and D � � = 0 0 + σ σ + σ σ + (−1)σ 0 0 , D ( ) = 0 σ + 0 σ , D ( ) = σ 0 + (−1)σ σ 0 , D � a � = a 0 0 + (−1)σ¯a a⋄ σ σ , a ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' If (A, ⋄) can be endowed with the structure of an involutive Frobenius superalgebra, then, for all d ∈ k, this induces a monoidal superfunctor D: Bσ k (A, ⋄;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' 2d) → Add(OBk(A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' d)π), where we assume that d = 0 if σ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' (See Remark 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=') Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We must verify that D respects the defining relations of Bk(A, ⋄) from Definition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For the first relation in (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1), we compute D � � = 0 0 + σ σ + σ σ + 0 0 = 0 0 + σ σ + σ σ + 0 0 = D � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The proof of the second relation in (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1) is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For the third relation in (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1), we have D � � = D( ⊗ ) ◦ D( ⊗ ) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='11) = � 0 σ + 0 σ + (−1)σ σ 0 + (−1)σ σ 0 � � σ 0 + (−1)σ σ 0 + 0 σ + (−1)σ 0 σ � = 0 0 + σ σ = 0 0 + σ σ = D � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Similarly, for the fourth relation in (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1), we have D � � = D( ⊗ ) ◦ D( ⊗ ) 36 SAIMA SAMCHUCK-SCHNARCH AND ALISTAIR SAVAGE (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='11) = � 0 σ + 0 σ + σ 0 + σ 0 � � σ 0 + (−1)σ σ 0 + (−1)σ 0 σ + 0 σ � = (−1)σ � 0 0 + σ σ � = (−1)σ � 0 0 + σ σ � = (−1)σD � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The fifth relation in (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1) is straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For the sixth relation in (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1), we compute D � � = D( ⊗ ) ◦ D( ⊗ ) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='11) = � 0 σ + 0 σ + (−1)σ σ 0 + (−1)σ σ 0 � � 0 0 + σ σ + σ σ + (−1)σ 0 0 + σ σ + 0 0 + 0 0 + (−1)σ σ σ � = 0 σ + 0 σ + σ 0 + (−1)σ σ 0 and D � � = D( ⊗ ) ◦ D( ⊗ ) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='11) = � 0 σ + 0 σ + σ 0 + σ 0 � � 0 0 + σ σ + σ σ + (−1)σ 0 0 + σ σ + 0 0 + 0 0 + (−1)σ σ σ � = 0 σ + 0 σ + σ 0 + (−1)σ σ 0 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='9) = 0 σ + 0 σ + σ 0 + (−1)σ σ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The first, second, and fourth relations in (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2) are straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For the third relation in (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2), we compute D � a b � = a b 0 0 + a⋄ b⋄ σ σ = ab 0 0 + (−1)¯a¯b b⋄a⋄ σ σ (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1) = ab 0 0 + (ab)⋄ σ σ = D � ab � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Finally, for the last relation in (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2), we have D � a � = D( ) ◦ D( a ⊗ ) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='11) = � 0 σ + 0 σ � � a 0 0 + (−1)σ¯a a σ σ + (−1)σ¯a a⋄ σ σ + a⋄ 0 0 � = (−1)σ¯a � a⋄ 0 σ + a 0 σ � = (−1)σ¯a � a⋄ 0 σ + a 0 σ � and D � a⋄ � = D( ) ◦ D( ⊗ a⋄ ) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='11) = � 0 σ + 0 σ � � a⋄0 0 + (−1)σ¯a a σ σ + (−1)σ¯a a⋄σ σ + a0 0 � DIAGRAMMATICS FOR REAL SUPERGROUPS 37 = (−1)σ¯a � a⋄ 0 σ + a 0 σ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' It remains to prove the final statement of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' This statement is clear if d = 0, and so it suffices to assume σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' In this case, we have D � a � = a + a⋄ = a+a⋄ where, in the last equality, we used [MS, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='24)] to convert the clockwise bubble to a counterclockwise one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Then the assertion follows from the fact that strA(a + a⋄) (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='6) = 2 strA(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' □ We are now ready to state and prove the basis theorem for Bσ k (A, ⋄;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For r, s ∈ N, an (r, s)- Brauer diagram is a string diagram representing a morphism in HomBσ k (A,⋄;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='d)(I⊗r, I⊗s) such that: there no tokens on any string and no closed strings (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' strings with no endpoints);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' no string has more than one critical point;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' there are no self-intersections of strings and no two strings cross each other more than once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' A perfect matching of a finite set is a partition of that set into subsets of size 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Numbering the bottom endpoints 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' , r from left to right and the top endpoints r +1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' , r +s from left to right, each (r, s)-Brauer diagram induces a perfect matching of {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' , r + s}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Let D(r, s) denote a set of (r, s)-Brauer diagrams, with each perfect matching of {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' , r + s} induced by exactly one element of D(r, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For r, s ∈ N, let D•(r, s) denote the set of all morphisms that can be obtained from elements of D(r, s) by adding exactly one token to each string according to Convention 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For example, is a possible element of D(5, 7) and b1 b2 b3 b4 b5 b6 , b1, b2, b3, b4, b5, b6 ∈ BA, are the corresponding elements of D•(5, 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We expect the following theorem to hold for an arbitrary involutive superalgebra (A, ⋄).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' However, since our proof relies on Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='5, it assumes that A also admits the structure of a Frobenius superalgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' As explained in Convention 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='3, this assumption holds whenever A is a real or complex division superalgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For all r, s ∈ N and d ∈ k, the set D•(r, s) is k-basis for the morphism space HomBσ k (A,⋄;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='d)(I⊗r, I⊗s) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We first prove that the elements of D•(r, s) span HomBσ k (A,⋄;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='d)(I⊗r, I⊗s) as k-supermodule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Using the last two relations in (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2) and the last two relations in (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='4), tokens on strings with endpoints can be moved near the appropriate endpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Then, using the third relation in (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2), we can reduce the number of tokens to precisely one on each string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' (Recall that a string with no token is the same as a string with a token labelled by the identity element of A, using the first relation in (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=') Next, using the second relation in (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2), we can write any diagram as a linear combination of ones where the tokens are labelled by elements of BA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Finally, using the relations (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1) and the first two relations in (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='4), we can move the strings so that they are positioned to agree with some 38 SAIMA SAMCHUCK-SCHNARCH AND ALISTAIR SAVAGE element of D(r, s), together with some bubbles to the right of this element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Then we can evaluate all bubbles using (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' It remains to prove linear independence of D•(r, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Under the functor D of Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='5, each element of D•(r, s) is sent, up to sign and parity shift, to a sum over all possible orientations of the strands, with the map a �→ ±a⋄ applied to labels of tokens on downward pointing strands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' It follows from Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='5 that these images are linearly independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Therefore, D•(r, s) is linearly independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' □ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The unoriented incarnation superfunctor In this section, we introduce the main application of the supercategory Bσ k (A, ⋄) to the rep- resentation theory of supergroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We begin by defining a very general unoriented incarnation superfunctor and proving an asymptotic faithfulness result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We then turn our attention to the special cases where k ∈ {R, C} and A is a division superalgebra over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' When k = C, fullness of the incarnation functor follows from known results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' When k = R, we state the fullness result, whose proof is split into three cases, proved in Sections 11 to 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Definition of the superfunctor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' In this subsection we work over an arbitrary field k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Let (A, ⋄) be an involutive superalgebra, let V be a right A-supermodule, and let Φ be a nondegenerate (ν, ⋄)-supersymmetric form of parity σ on V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Fix a homogeneous k-basis BV of V , and let B∨ V = {b∨ : b ∈ BV } be the left dual basis with respect to Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Recall the notation flip and ρa from Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' It follows from (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1) that (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1) ρa⋄ρb⋄ = ρ(ab)⋄, a, b ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' There exists a unique monoidal superfunctor, which we call the unoriented incar- nation superfunctor associated to Φ, FΦ : Bσ k (A, ⋄) → G(Φ)-smodk such that FΦ(I) = V and (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2) FΦ � � = ν flip, FΦ ( ) = Φ, FΦ( a ) = ρa⋄, a ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' This superfunctor also satisfies the following: FΦ ( ) = Φ′ : k → V ⊗ V, 1 �→ � v∈BV (−1)σ¯vv ⊗ v∨, and (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='3) FΦ � a � = strV (a)1 1 for all a ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='4) If V = Am|n for some m, n ∈ N, then FΦ induces a monoidal superfunctor FΦ : Bσ k (A, ⋄;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' ν(m − n)) → G(Φ)-smodk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We first show that (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2) and (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='3) indeed yield a superfunctor FΦ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We must show that it respects the relations (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1) and (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The first two relations in (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1) are clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For the third equality in (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1), we compute FΦ � � : v Φ′⊗1V �−−−−→ � w∈BV (−1)σ ¯ww ⊗ w∨ ⊗ v 1V ⊗Φ �−−−−→ � w∈BV Φ(w∨, v)v = v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For the fourth equality in (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1), we compute FΦ � � : v 1V ⊗Φ′ �−−−−→ � w∈BV (−1)σ(¯v+ ¯w)v ⊗ w ⊗ w∨ Φ⊗1V �−−−−→ (−1)σ � w∈BV Φ(v, w)w∨ = (−1)σv, where, to simplify the sign, we used the fact that Φ(v, w) = 0 unless ¯v + ¯w = σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' DIAGRAMMATICS FOR REAL SUPERGROUPS 39 The fifth equality in (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1) follows immediately from the fact that Φ is a (ν, ⋄)-supersymmetric k-bilinear form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For the sixth equality in (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1), we compute FΦ � � : u ⊗ v ⊗ w ν flip ⊗1V �−−−−−−→ ν(−1)¯u¯vv ⊗ u ⊗ w 1V ⊗Φ �−−−−→ ν(−1)(σ+¯u)¯vΦ(u, w)v, and FΦ � � u ⊗ v ⊗ w 1V ⊗ν flip �−−−−−−→ ν(−1)¯v ¯wu ⊗ w ⊗ v Φ⊗1V �−−−−→ ν(−1)¯v ¯wΦ(u, w)v = ν(−1)(σ+¯u)¯vΦ(u, w)v, where, in the final equality, we used the fact that Φ(u, w) = 0 unless ¯w = σ + ¯u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The first two relations in (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2) are straightforward, while the third follows from (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For the fourth relation in (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2), we compute FΦ � a � : u ⊗ v ρa⊗1V �−−−−→ (−1)¯a¯uua⋆ ⊗ v ν flip �−−−→ ν(−1)¯u¯v+¯a(¯u+¯v)v ⊗ ua⋆, and FΦ � a � : u ⊗ v ν flip �−−−→ ν(−1)¯u¯vv ⊗ u 1V ⊗ρa �−−−−→ ν(−1)¯u¯v+¯a(¯u+¯v)v ⊗ ua⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Finally, for the fifth relation in (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2), we compute FΦ � a � : u ⊗ v ρa⊗1V �−−−−→ (−1)¯a¯uua⋄ ⊗ v Φ �−→ (−1)¯a¯uΦ(ua⋄, v) (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='7) = (−1)¯a(¯u+¯v)Φ(u, va), and FΦ � a⋄ � : u ⊗ v 1V ⊗ρa⋄ �−−−−−→ (−1)¯a(¯u+¯v)u ⊗ va Φ �−→ (−1)¯a(¯u+¯v)Φ(u, va).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Next we prove (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Using Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='10, we have Φ(w, v∨) (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='7) = ν(−1) ¯w(¯v+σ)Φ(v∨, w) = ν(−1)¯v+σ¯vδvw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Thus, the k-basis of V left dual to B∨ V is given by (v∨)∨ = ν(−1)¯v+σ¯vv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Therefore, Fϕ � a � : 1 Φ′ �−→ ν � v∈BV (−1)¯vv∨ ⊗ v 1V ⊗ρa �−−−−→ ν � v∈BV (−1)¯vv∨ ⊗ va⋄ Φ �−→ ν � v∈BV (−1)¯vΦ(v∨, va⋄) = ν strV (a), where, in the final equality, we used Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The fact that FΦ factors through Bσ k (A, ⋄;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' ν(m−n)) when V = Am|n then follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' It remains to prove that, for any functor as in the first sentence of the theorem, we have Fϕ( ) = Φ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Suppose that Fϕ( ): 1 �→ � u,v∈BV auvu ⊗ v∨, auv ∈ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Then, for all v ∈ BV , we have v = Fϕ � � (v) = Fn � � (v) Fn � ⊗ � �−−−−−−−→ � u,w∈BV auwu ⊗ w∨ ⊗ v 1V ⊗Φ �−−−−→ � u∈BV (−1)σ¯uauvu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' It follows that auv = (−1)σ¯vδuv for all u, v ∈ BV , and so Fn( ) = Φ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' □ 40 SAIMA SAMCHUCK-SCHNARCH AND ALISTAIR SAVAGE 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Asymptotic faithfulness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For the remainder of this section, we assume that V = Am|n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Fix a k-basis BA of A with the property that b⋄ = ±b for all b ∈ BA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' If 2m + 2n ≥ r + s, then the elements FΦ(f), f ∈ D•(r, s), are linearly independent, over k, in HomG(Φ)(V ⊗r, V ⊗s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We have a commutative diagram HomBσ k (A,⋄)(I⊗r, I⊗s) HomBσ k (A,⋄)(I⊗(r+s), 1) HomG(Φ) � V ⊗r, V ⊗s� HomG(Φ) � V ⊗(r+s), k � FΦ ∼ = FΦ ∼ = where the horizontal maps are the usual isomorphisms that hold in any rigid monoidal supercategory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' In particular, the top horizontal map is the k-linear isomorphism given on diagrams by · · · · �→ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' · · · · with inverse · · · · �→ ± · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' · · , where the rectangles denotes some diagram, and the sign (which is needed only when σ = 1) is determined by the parity of this diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Applying the top horizontal map to an element of D•(r, s), then sliding tokens along strands to the correct position, yields an element of D•(r + s, 0) up to sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Therefore, it suffices to prove the theorem in the case where s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Suppose m, n ∈ N satisfy 2m + 2n ≥ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' If r is odd, then D•(r, 0) = ∅, and the proposition holds trivially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Therefore, we suppose that r is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' In what follows we number the strand endpoints 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' , r from left to right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Given f ∈ D•(r, 0), we enumerate the strands in f in order of the numbering of their right endpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Let bi, 1 ≤ i ≤ r 2, denote the label of the token on the i-th strand of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For 1 ≤ i ≤ r, define vi = � ej if the j-th strand in f has right endpoint in position i (ejbj)′ if the j-th strand in f has left endpoint in position i, where e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' , em+n denotes the standard A-basis of V = Am|n, and where {(eib)∨ : 1 ≤ i ≤ m + n, b ∈ BA} denotes the basis of V left dual to {eib : 1 ≤ i ≤ m + n, b ∈ BA} with respect to Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Now define vf = v1 ⊗ v2 ⊗ · · · ⊗ vr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For example, if f = b1 b2 b3 b4 , then vf = (e4b4)∨ ⊗ (e1b1)∨ ⊗ (e2b2)∨ ⊗ e1 ⊗ e2 ⊗ (e3b3)∨ ⊗ e3 ⊗ e4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' It is straightforward to verify that FΦ(f)(vg) = ±δf,g, for all f, g ∈ D•(r, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' It follows that the elements of FΦ(f), f ∈ D•(r, 0), are linearly independent, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' □ DIAGRAMMATICS FOR REAL SUPERGROUPS 41 Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' If 2m + 2n ≥ r + s, then the induced k-supermodule homomorphism FΦ : HomBσ k (A,⋄;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='ν(m−n))(I⊗r, I⊗s) → HomG(Φ)(V ⊗r, V ⊗s) is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' This follows immediately from Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='6 and Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' □ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Fullness over the real and complex numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We are now ready to state the last of our main results: the fullness of the unoriented incarnation functor in the case of a central real division superalgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We begin by stating the fullness result over the complex numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' If Φ is a nondegenerate (ν, id)-supersymmetric form on Cm|n of parity σ, then the unoriented incarnation superfunctor FΦ: Bσ C(C, id;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' ν(m − n)) → G(Φ)-smodC of Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1 is full.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' First consider the case σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' As explained in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1, we may assume that ν = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Then B0 C(C, id;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' m − n) is the usual Brauer category and the result was proved in [LZ17, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='6];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' see also [ES16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Next consider the case σ = 1, in which case we must have m = n, as explained in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' By (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='11), we may again assume that ν = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Then B0 C(C, id;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' 0) is the periplectic Brauer category, introduced in [KT17] as B(0, −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' In this case, the result was proved in [CE21, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1], with the key ingredient being [DLZ18, §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' □ In the remainder of the paper, we will be mostly concerned with the case where A is a real division superalgebra (see Convention 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='3), and where the (ν, ⋄)-supersymmetric form comes from a (ν, ⋆)- superhermitian form, as in Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' To simplify notation we define, for (D, ⋆) an involutive real division superalgebra, and d ∈ R, (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='5) Bσ R(D) := Bσ R(D, ⋄), Bσ R(D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' d) := Bσ R(D, ⋄;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' d), where a⋄ = (−1)¯aa⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Suppose (D, ⋆) is an involutive central real division superalgebra, and ϕ is a nonde- generate (ν, ⋆)-superhermitian form on Dm|n of parity σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Let Φ be the corresponding nondegenerate (ν, ⋄)-supersymmetric form on Dm|n, as in Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Then the unoriented incarnation super- functor FΦ : Bσ R(D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' ν(m − n)) → G(Φ)-smodR of Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1 is full.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The proof of Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='5 will be broken into three parts: we prove it holds for (D, ⋆) = (R, id) in Proposition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='4, we prove it holds for (D, ⋆) = (C, ⋆) and (D, ⋆) = (Cl(C), ⋆) in Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='4, and we prove it holds for (D, ⋆) = (H, ⋆) in Proposition 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Remark 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' As explained in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='6, the forgetful functor G(Φ)-smodR → g(Φ)-smodR is full and faithful when Gred(Φ) is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' In this case, we can replace the target supercate- gories G(Φ)-smodC and G(Φ)-smodR in Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='4 and Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='5 by g(Φ)-smodC and g(Φ)-smodR, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' It follows from the descriptions of Gred(Φ) in Appendix A that we can make this replacement whenever σ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' In addition, when σ = 0, we can make this replacement in Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='5 when (D, ⋆) ∈ {(C, ⋆), (Cl(C), ⋆)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' 42 SAIMA SAMCHUCK-SCHNARCH AND ALISTAIR SAVAGE 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Consequences for the semisimple cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For (D, ⋆) ∈ {(R, id), (C, ⋆), (H, ⋆)}, we see that Bk(D) is a monoidal category (as opposed to a monoidal supercategory).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' In fact, the category Bk(D) is a spherical pivotal category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We also have ⋄ = ⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For d ∈ Z, we let NR(D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' d) denote the tensor ideal of negligible morphisms of Kar(BR(D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' d));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' see Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For a (ν, ⋆)-supersymmetric form Φ on Dm|n, we let G(Φ)-tsmodR denote the monoidal supercat- egory of tensor G(Φ)-supermodules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' By definition, this is the full sub-supercategory of G(Φ)-smodR whose objects are direct summands of V ⊗r, r ∈ N, where V = Dm|n is the natural supermodule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We let G(Φ)-tsmod′ R denote the underlying category of G(Φ)-tsmodR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' By definition, this is the category with the same objects as G(Φ)-tsmodR, but whose morphisms are the even G-supermodule homo- morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' So G(Φ)-tsmod′ R is a monoidal category (as opposed to a monoidal supercategory).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' If n = 0, so that G(Φ) is a real group (as opposed to a supergroup), then we write G(Φ)-tmodR for the category of tensor modules, defined in the same way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For (D, ⋆) ∈ {(R, id), (C, ⋆), (H, ⋆)} and p, q, m, n ∈ N, p + q = m, the unoriented incarnation functor induces equivalences of monoidal categories Kar(BR(R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' m))/NR(R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' m) → O(p, q)-tmodR, Kar(BR(R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' −2n))/NR(R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' −2n) → OSp(0|2n, R)-tsmod′ R, Kar(BR(R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' 1 − 2n))/NR(R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' 1 − 2n) → OSp(1|2n, R)-tsmod′ R, Kar(BR(C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' m))/NR(C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' m) → U(p, q)-tmodR, Kar(BR(H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' m))/NR(H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' m) → Sp(p, q)-tmodR, Kar(BR(H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' −n))/NR(H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' −n) → O(m, H)-tsmod′ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Note that, while OSp(0|2n, R) is isomorphic to Sp(2n, R), we write OSp(0|2n, R)-tsmod′ R in Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='7, instead of Sp(2n, R)-tmodR, since we view the natural module R2n as purely odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Similarly, in O(n, H)-tsmod′ R, we view the natural module Hn as purely odd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' see Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The proof is similar to that of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' To simplify notation, we give the proof for O(p, q), since the other cases are analogous;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' see Remark 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='8(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Let Φ = ϕp,q|2n be the (ν, ⋆)-supersymmetric form on Rm defined in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='6), so that G(Φ) = O(p, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Since the category O(p, q)-tmodR is idempotent complete, FΦ induces a monoidal functor Kar(FΦ): Kar(BR(R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' m)) → O(p, q)-tmodR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' By Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='5, this functor is full.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Every object in O(p, q)-tmodR is completely reducible, since its complexification is a tensor module for O(m, C), hence completely reducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Thus, the category O(p, q)-tmodR is semisimple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' In addition, by Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='6, EndBR(R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='m)(1) = R1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Thus, by [SW22, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='9], the kernel of Kar(FΦ) is equal to NR(R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Therefore, Kar(FΦ) induces a full and faithful functor BR(R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' m)/NR(R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' m) → O(p, q)-tmodR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Finally, since the image of Kar(FΦ) contains all summands of tensor powers of the natural module Rm (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' all tensor modules), it is essentially surjective, hence an equivalence of categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' □ Remark 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' (a) Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='7 involves precisely the supergroups with Lie superalgebras that are real forms of complex Lie superalgebras whose finite-dimensional modules are all semisimple;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' see [DH76, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' (b) As explained in Remark 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='6, we can replace OSp(0|2n, R), U(p, q), and Sp(p, q) in The- orem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='7 by sp(0|2n, R), u(p, q), and sp(p, q), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' However, we cannot replace O(p, q), OSp(1|2n, R), or O(n, H) by their Lie superalgebras, since the orthogonal groups are not connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' DIAGRAMMATICS FOR REAL SUPERGROUPS 43 Corollary 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' If p, p′, q, q′ ∈ N satisfy p + q = p′ + q′, then we have equivalences of monoidal categories O(p, q)-tmodR ≃ O(p′, q′)-tmodR, U(p, q)-tmodR ≃ U(p′, q′)-tmodR, Sp(p, q)-tmodR ≃ Sp(p′, q′)-tmodR, sending the natural supermodule to the natural supermodule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We will extend Corollary 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='9 to equivalences of more general supergroups in Propositions 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='5, 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='5 and 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Remark 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' It is crucial that Corollary 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='9 involves O(p, q) and U(p, q), as opposed to SO(p, q) and SU(p, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For example, let V = C2 ∼= R4 denote the natural U(2)-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' By restriction, this is also the natural module for SU(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Direct computation shows that C ∼= EndU(2)(V ) ⊆ EndSU(2)(V ) ∼= H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' On the other hand, all irreducible modules of SU(1, 1) ∼= SL(2, R) have endomorphism algebra isomorphic to R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Thus, the categories SU(1, 1)-tmodR and SU(2)-tmodR are not equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' If W = C2 ∼= R4 denotes the natural module of U(1, 1), which is also the natural module for SU(1, 1) by restriction, we have C ∼= EndU(1,1)(W) ⊆ EndSU(1,1)(W) ∼= Mat2(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The equivalence U(2)-tmodR ≃ U(1, 1)-tmodR of Corollary 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='9 sends V to W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Both modules have endomorphism algebra isomorphic to C only if we use the full unitary groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Unoriented fullness: real case In this section, we prove Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='5 in the case (D, ⋆) = (R, id).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Note that, since R is purely even, we have ⋄ = ⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Recall, from Section 7, that, if (A, ⋆) is an involutive superalgebra, then so is (AC, ⋆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The proof of the following result is analogous to that of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Proposition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For any real involutive superalgebra (A, ⋆) and σ ∈ Z2, there is an isomorphism of monoidal supercategories Bσ R(A, ⋆)C ∼ = −→ Bσ C(AC, ⋆) given on objects by I �→ I and on morphisms by �→ , �→ , �→ , a �→ a⊗1 , a ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For all d ∈ k, this induces an isomorphism of monoidal supercategories Bσ R(A, ⋆;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' d)C ∼ = −→ Bσ C(AC, ⋆;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' d) Fix m, n ∈ N, set V = Rm|n, and let Φ be a nondegenerate (ν, id)-supersymmetric form on V of parity σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' (See Appendices A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='3 and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='7 for a classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=') We have a natural identification V C = Cm|n, and we extend Φ to a nondegenerate (ν, id)-supersymmetric form ΦC : V C × V C → C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Lemma 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For all G(Φ)-supermodules U and W, we have an isomorphism of C-supermodules HomG(Φ)(U, W)C ∼ = −→ HomG(ΦC)(U C, W C), f ⊗ a �→ f ⊗ a, f ∈ HomG(Φ)(U, W), a ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' 44 SAIMA SAMCHUCK-SCHNARCH AND ALISTAIR SAVAGE Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' First suppose that σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Then, as explained in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1, we may assume that ν = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' In this case, the classification of the nondegenerate (1, id)-supersymmetric forms is recalled in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We see that n must be even, and Gred(Φ) = O(p, q) × Sp(n, R) for some p, q ∈ N, p + q = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The group Sp(n, R) is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' On the other hand, if m ≥ 1, then O(p, q) has four connected components if p, q ≥ 1, and two connected components if p = 0 or q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Suppose p, q ≥ 1, the proof in the other case being analogous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Then choose elements X1, X2, X3 in O(p, q), one from each of its connected components not containing the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='23) and Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1, we have an isomorphism HomG(Φ)(U, W)C ∼= HomX1,X2,X3,g(ΦC)(U C, W C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Now, Gred(ΦC) ∼= O(m, C) × Sp(n, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The group Sp(n, C) is connected, while O(m, C) has two connected components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Reordering if necessary, we may assume that det(X1) = det(X2) = −1 = − det(X3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='18), we have HomX1,X2,X3,g(ΦC)(U C, W C) = HomX1,g(ΦC)(U C, W C) = HomG(ΦC)(U C, W C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The case σ = 1 is easier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' As explained in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='7, we have G(Φ) = GL(m, R) and G(ΦC) = GL(m, C), which are both connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Then, using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='22), we have HomG(Φ)(U, W)C = Homg(Φ)(U, W)C ∼= Homg(ΦC)(U C, W C) = HomG(ΦC)(U C, W C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' □ It follows from Lemma 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2 that we have a canonical full and faithful superfunctor (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1) ER : (G(Φ)-smodR)C → G(ΦC)-smodC sending V to V C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Since R is commutative, we can identify R and Rop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Let SR denote the isomor- phism of Proposition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1 when (A, ⋆) = (R, id).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Proposition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The diagram Bσ R(R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' ν(m − n))C Bσ C(C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' ν(m − n)) (G(Φ)-smodR)C G(ΦC)-smodC SR FC Φ FΦC ER commutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' To simplify notation, we set S = SR, E = ER, F = FΦC, and FC = FC Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' On objects, we have EFC(I) = V C = F(I) = FS(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For morphisms, we need to show that EFC(f) = FS(f) for f ∈ � , , � , where X and Y are the domain and codomain of f, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For f = , we have EFC( ) = ν flipV C,V C = FS( ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For f = , we have EFC( ) = FS( ): V C ⊗C V C → C, u ⊗ v �→ ΦC(u, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Finally, we consider f = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Let BV denote an R-basis of V , which we also view as a C-basis of V C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Then we have EFC( ) = FS( ): C → V C ⊗C V C, 1 �→ � v∈BV v ⊗ v∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' □ Proposition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='5 holds when (D, ⋆) = (R, id).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' DIAGRAMMATICS FOR REAL SUPERGROUPS 45 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We wish to show that the R-linear map FΦ : HomBσ R (R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='ν(m−n))(I⊗r, I⊗s) → HomG(Φ)(V ⊗r, V ⊗s) is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' This map is surjective if and only if the induced map (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2) FC Φ : HomBσ R (R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='ν(m−n))(I⊗r, I⊗s)C → HomG(Φ)(V ⊗r, V ⊗s)C is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' To show that (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2) is surjective, it suffices to show that the diagram (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='3) HomBσ R (R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='ν(m−n))(I⊗r, I⊗s)C HomG(Φ)(V ⊗r, V ⊗s)C HomBσ C (C);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='ν(m−n)(I⊗r, I⊗s) HomG(ΦC) � (V C)⊗r, (V C)⊗s� SR ∼ = FC Φ ∼ = ER FΦC commutes, where surjectivity of the bottom horizontal map follows from Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Com- mutativity of this diagram follows from Proposition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' □ If ϕp,q|2n is the form defined in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='6), then G(ϕp,q|2n) = OSp(p, q|2n, R) is the indefinite or- thosymplectic supergroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Recall the definition G(Φ)-tsmodR of the monoidal supercategory of tensor G(Φ)-supermodules from Section 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Proposition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' If p, p′, q, q′, n ∈ N satisfy p + q = p′ + q′, then we have an equivalence of monoidal supercategories, OSp(p, q|2n, R)-tsmodR ≃ OSp(p′, q′|2n, R)-tsmodR sending the natural supermodule to the natural supermodule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Viewing B0 R(R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' m − n) as a subcategory of B0 R(R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' m − n)C, it follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='19) and the commutativity of (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='3), with σ = 0 and ν = 1, that ker(FΦ) = S−1 R (ker(FΦC)) ∩ B0 R(R, m − n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' In particular, ker(FΦ) depends only on ΦC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' As noted in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2, ΦC depends only on p + q and n, up to equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Hence, both OSp(p, q|2n)-tsmodR and OSp(p′, q′|2n)-tsmodR are equivalent to the quotient of B0 R(R, m − n) by this common kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' □ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Unoriented fullness: complex cases In this section, we prove Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='5 in the case where (D, ⋆) is either (C, ⋆) or (Cl(C), ⋆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Throughout this section, we assume that (D, ⋆) is one of these two complex involutive superalge- bras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Recall, from Convention 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='3, that D is also a real Frobenius superalgebra, with Nakayama automorphism ζ given by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1), so that a⋄ = (−1)¯aa⋆, a ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='5, we can assume that the specialization parameter d is zero when D = Cl(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' If V is a complex vector superspace, we let V ⋆ denote the conjugate complex vector superspace, which is a special case of the construction described in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Precisely, V ⋆ is equal to V as an R-vector superspace, but the C-action is given by (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1) V ⋆ × C → V ⋆, (v, a) �→ va⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Above, and elsewhere, the juxtaposition vb, for b ∈ C and v ∈ V or V ⋆, will always denote the C-action on V (as opposed to the C-action on V ⋆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Recall the notion of complexification from Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We have an isomorphism of C-vector spaces (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2) V C ∼ = −→ V ⊕ V ⋆, v �→ 1 √ 2(v, v), v ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' 46 SAIMA SAMCHUCK-SCHNARCH AND ALISTAIR SAVAGE Note that C-linearity implies that v ⊗ a �→ 1 √ 2(va, va⋆), v ∈ V, a ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Recall, from Section 2, the superadditive envelope Add(Cπ) of a supercategory C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Fix d ∈ k and σ ∈ Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' There exists a unique C-linear monoidal superfunctor SD: Bσ R(D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' d)C → Add(OBC(D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' d)π) such that SD(I) = ↑ ⊕Πσ↓ and SD � � = 0 0 + σ σ + σ σ + (−1)σ 0 0 , (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='3) SD ( ) = 0 σ + 0 σ , SD ( ) = σ 0 + (−1)σ σ 0 , (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='4) SD � a � = a 0 0 + (−1)σ¯a a⋄ σ σ , a ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='5) The functor SD is full and faithful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The superfunctor SD is the complexification DC of the superfunctor of Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='5, with d replaced by d/2, followed by the superfunctor Add(OBR(D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' d/2))C → Add(OBC(D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' d)) given by imposing the relations a = a , a ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Note that the doubling of the specialization parameter comes from the fact that, by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='5, strR D(1) = sdimR D = 2(sdimC D) = 2 strC D(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' It remains to prove that SD is full and faithful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For r, s ∈ N, the functor SD induces a C-linear map (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='6) HomBσ R (D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='d)(I⊗r, I⊗s)C → HomAdd(OBσ C(D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='d)π)((↑ ⊕Πσ ↓)⊗r, (↑ ⊗ ↓)⊗s) ∼= � X1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=',Xr,Y1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=',Ys∈{↑,Πσ↓} HomOBσ C(D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='d)π(X1 ⊗ · · · ⊗ Xr, Y1 ⊗ · · · ⊗ Ys).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' By Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='6, HomBσ R (D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='d)(I⊗r, I⊗s)C has R-basis D•(r, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Thus, it has dimension (dimR D)(r+s)/2(r + s − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' if r + s is even and dimension zero if r + s is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' (We use here the fact that the number of perfect matchings of a set of size 2n is (2n − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' := (2n − 1)(2n − 3) · · · 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=') On the other hand, by Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='6, � X1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=',Xr,Y1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=',Ys∈{↑,Πσ↓} HomOBC(D)(X1 ⊗ · · · ⊗ Xr, Y1 ⊗ · · · ⊗ Ys) has the same dimension since, when r + s is even, there are (r + s − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' perfect matchings of the r + s endpoints, 2(r+s)/2 choices for the orientations of the strands, and then (dimC D)(r+s)/2 ways to put a token labelled b ∈ BC D on each strand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Thus, to prove that (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='6) is an isomorphism, it suffices to prove that it is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' To do this, it is enough to show that each generating morphism of OBσ C(D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' d), with appropriate parity shifts, is in the image of SD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Noting that SD � 1 2 − i 2 i � = 0 0 and SD � 1 2 + i 2 i � = σ σ , we have 1 4SD � − i i − i i − i i � = 0 0 , 1 2SD � a − i ia � = a 0 0 , a ∈ D, DIAGRAMMATICS FOR REAL SUPERGROUPS 47 1 2SD � − i i � = 0 σ , 1 2SD � + i i � = 0 σ , 1 2SD � + i i � = σ 0 , (−1)σ 1 2SD � − i i � = σ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' □ Fix m, n ∈ N, and set V = Dm|n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' If D = Cl(C), we assume that n = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' see Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Note that, if ϕ is a (ν, ⋆)-superhermitian form on V , then iϕ is a (−ν, ⋆)-superhermitian form on V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Therefore, without loss of generality, we let ϕ be a nondegenerate (1, ⋆)-superhermitian form on V of parity σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' (See Appendices A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='4 and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='6 for a classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=') Let Φ = τ ◦ ϕ be the corresponding nondegenerate (ν, ⋄)-supersymmetric form, with a⋄ = (−1)¯aa⋆;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' see Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='10 and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Recall, from Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='5, that G(Φ) = G(ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Lemma 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For all G(Φ)-supermodules U and W, we have an isomorphism of C-supermodules HomG(Φ)(U, W)C ∼ = −→ Homgl(m|n,D)(U C, W C), f ⊗ a �→ f ⊗ a, f ∈ HomG(Φ)(U, W), a ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' As explained in Appendices A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='4, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='6 and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='7, Gred(Φ) is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Thus HomG(Φ)(U, W)C ∼= Homg(Φ)(U, W)C (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='22) ∼= Homgl(m|n,D)(U C, W C), where we used Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2 in the final isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' □ It follows from Lemma 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2 that we have a canonical full and faithful superfunctor (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='7) ED : (G(Φ)-smodR)C → gl(m|n, D)-smodC sending V to V C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Since D is complex division superalgebra, V is naturally a complex vector superspace, and hence the g(Φ)-supermodule V = Dm|n is naturally a supermodule over g(Φ)C ∼= gl(m|n, D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Recall the isomorphism Ξ⋄ of (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The next result shows that the diagram Bσ R(D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' m − n)C Add(OBC(D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' m − n)π) Add(OBC(Dop;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' m − n)π) (G(Φ)-smodR)C gl(m|n, D)-smodC SD FC Φ Ξ⋄ Gm|n ED commutes up to supernatural isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Recall the notation Φv introduced in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='8), and the notation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='5) for elements of a parity shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' There is an monoidal supernatural isomorphism of superfunctors η: EDFC Φ ∼ = −→ Gm|nΞ⋄SD determined by (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='8) ηI : V C ∼ = −→ V ⊕ ΠσV ∗, ηI(v) = 1 √ 2 (v, πσΦv) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' To simplify notation, we set G = Gm|n, S = SD, E = Eg, FC = FC Φ, and Ξ = Ξ⋄.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' First note that ηI is the composition of the isomorphisms of C-supermodules V C ∼ = −−−→ (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2) V ⊕ V ⋆ ∼ = −→ V ⊕ ΠσV ∗, where the second isomorphism uses the restriction of (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='8) to C-supermodules, noting that V ⋄ = V ⋆ as C-supermodules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Thus ηI is a parity-preserving isomorphism of C-vector superspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' It is straightforward to verify that it is also a homomorphism of g(Φ)C-supermodules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' 48 SAIMA SAMCHUCK-SCHNARCH AND ALISTAIR SAVAGE On objects, we have EFC(I) = V C ηI−→ ∼ = V ⊕ ΠσV ∗ = G(↑ ⊕Πσ↓) = GΞS(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For morphisms, we need to show that ηY ◦ EFC(f) = GΞS(f) ◦ ηX for f ∈ � , , , a : a ∈ D � , where X and Y are the domain and codomain of f, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For f = , we have ηI⊗I ◦ EFC( ) = ηI⊗I ◦ flipV C,V C = (flipV ⊗V +Πσ flipV ⊗V ∗ +Πσ flipV ⊗V ∗ +(−1)σ flipV ∗⊗V ∗) ◦ ηI⊗I = GΞS( ) ◦ ηI⊗I, where the sign of (−1)σ arises from the isomorphism (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For f = , noting that η 1 is the identity map C → C, we have, for v, w ∈ V , η 1 ◦ EFC( ): V C ⊗C V C → C, v ⊗ w �→ Φ(v, w), and GΞS( ) ◦ ηI⊗I = G � 0 σ + 0 σ � ηI⊗I: V C ⊗ V C → C, v ⊗ w �→ 1 2(v, πσΦv) ⊗ (w, πσΦw) �→ 1 2 � Φ(v, w) + (−1)¯v ¯wΦ(w, v) � (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='7) = Φ(v, w), as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' (Above, we have used the fact that the maps are uniquely determined by their values on v ⊗ w, for v, w ∈ V ⊆ V C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=') Next we consider f = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Let BC V be a homogeneous C-basis for V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Then BC V ∪ BC V i is a homogeneous R-basis for V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' It is straightforward to verify that, for all w ∈ BC V , we have (wi)∨ = w∨i and Φw∨ = w∗, where ∨ denotes left duals with respect to Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Identifying v ∈ V and πσf ∈ ΠσV ∗ with (v, 0), (0, πσf) ∈ V ⊕ ΠσV ∗, we can write (v, πσf) as v + πσf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Using this convention,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='ηI⊗I ◦ EFC( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='): C → (V ⊕ ΠσV ∗) ⊗C (V ⊕ ΠσV ∗) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='∼ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='−→ (V ⊗C V ) ⊕ Πσ(V ⊗C V ∗) ⊕ Πσ(V ∗ ⊗C V ) ⊕ (V ∗ ⊗C V ∗) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='is the map given by ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1 �→ 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='w∈BC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='(−1)σ ¯w(w + πσΦw) ⊗ (w∨ + πσΦw∨) + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='w∈BC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='(−1)σ ¯w(iw + πσΦiw) ⊗ (w∨i + πσΦw∨i) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='= 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='w∈BC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='(−1)σ ¯w(w + πσΦw) ⊗ (w∨ + πσΦw∨) + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='w∈BC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='(−1)σ ¯w(wi − πσΦwi) ⊗ (w∨i − πσΦw∨i) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='w∈BC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='(−1)σ ¯ww ⊗ πσΦw∨ + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='w∈BC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='(−1)σ ¯wπσΦw ⊗ w∨ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='w∈BC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='(−1)σ ¯ww ⊗ πσw∗ + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='w∈BC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='(−1) ¯wπσw∗ ⊗ w ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='�→ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='w∈BC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='πσw ⊗ w∗ + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='w∈BC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='(−1) ¯wπσw∗ ⊗ w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' DIAGRAMMATICS FOR REAL SUPERGROUPS 49 where,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' in the final equality,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' we used the fact that the last sum is independent of the choice of basis to sum over the basis BC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='∨ V ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' and the fact that (w∨)∨ = (−1)σ ¯w+ ¯ww.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' On the other hand, we have GΞS( ) ◦ η 1 = G � σ 0 + (−1)σ σ 0 � : C → Πσ(V ⊗ V ∗) ⊕ Πσ(V ∗ ⊗C V ), 1 �→ � w∈BC V πσw ⊗ w∗ + � w∈BC V (−1) ¯wπσw∗ ⊗ w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Finally, for f = a , a ∈ D, we have ηI ◦ EFC( a ): V C → V ⊕ ΠσV ∗, v �→ (−1)¯a¯vva⋄ �→ (−1)¯a¯v 1 √ 2(va⋄, πσΦva⋄) and GΞS( a ) ◦ ηI = G � (a⋄)op 0 0 + (−1)σ¯a aop σ σ � : V C → V ⊕ ΠσV ∗, v �→ 1 √ 2(v, πσΦv) (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='4) �−−−→ 1 √ 2 � (−1)¯a¯vva⋄, (−1)σ¯aaπσΦv� = (−1)¯a¯v 1 √ 2(va⋄, πσΦva⋄).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' □ Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='5 holds when (D, ⋆) is equal to (C, ⋆) or (Cl(C), ⋆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' To show that the superfunctor FΦ is full, we must show that the R-linear map FΦ : HomBσ R (D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='m−n)(I⊗r, I⊗s) → HomG(Φ)(V ⊗r, V ⊗s) is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' This map is surjective if and only if the induced map (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='9) FC Φ : HomBσ R (D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='m−n)(I⊗r, I⊗s)C → HomG(Φ)(V ⊗r, V ⊗s)C is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' To show that (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='9) is surjective, it suffices to show that the diagram HomBσ R (D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='m−n)(I⊗r, I⊗s)C HomG(Φ)(V ⊗r, V ⊗s)C HomAdd(OBC(D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='m−n)) ((↑ ⊕Πσ↓)⊗r, (↑ ⊕Πσ↓)⊗s) Homgl(m|n,D) � (V C)⊗r, (V C)⊗s� HomAdd(OBC(Dop;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='m−n)) ((↑ ⊕Πσ↓)⊗r, (↑ ⊕Πσ↓)⊗s) Homgl(m|n,D) ((V ⊕ ΠσV ∗)⊗r, (V ⊕ ΠσV ∗)⊗s) SD ∼ = FC Φ ∼ = ED ∼ = Ξ⋄ Gm|n ∼ = commutes, where the bottom-right vertical isomorphism is induced by the isomorphism (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Commutativity of this diagram follows from Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' □ As a special case of G(Φ), we have the supergroups U(p, q|r, s) and UQ(p, q);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' see Appen- dices A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='4 and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Recall the definition G(Φ)-tsmodR of the monoidal supercategory of tensor G(Φ)-supermodules from Section 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' If p, p′, q, q′, r, r′, s, s′ ∈ N satisfy p + q = p′ + q′ and r + s = r′ + s′, then we have equivalences of monoidal supercategories U(p, q|r, s)-tsmodR ≃ U(p′, q′|r′, s′)-tsmodR and UQ(p, q)-tsmodR ≃ UQ(p′, q′)-tsmodR, sending the natural supermodule to the natural supermodule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The proof is analogous to that of Proposition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='5, using the commutative diagram appearing in the proof of Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' □ 50 SAIMA SAMCHUCK-SCHNARCH AND ALISTAIR SAVAGE 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Unoriented fullness: quaternionic case In this section, we prove Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='5 in the case where (D, ⋆) = (H, ⋆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We will naturally view C = R + Ri as a subalgebra of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Suppose that V is an H-vector superspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Then, using (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='3), we have an isomorphism of C-vector spaces (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1) V ∼ = −→ V ⋆, v �→ vj, where V ⋆ denotes the conjugate C-vector superspace, as in (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Combining with (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2), this yields an isomorphism of C-vector spaces (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2) V C ∼ = −→ V ⊕ V, v �→ 1 √ 2(v, vj), v ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Note that C-linearity implies that v ⊗ a �→ 1 √ 2(va, vja) = 1 √ 2(va, va⋆j), v ∈ V, a ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Recall that the additive envelope of a linear category C is the category Add(C) whose objects are formal direct sums of objects in C, and whose morphisms are identified with matrices of morphisms in C in the usual way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We write morphisms in additive envelopes as sums of their components, as in Section 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' In what follows, we will often be considering the object I ⊕ I ∈ Add(BC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' In order to distinguish the two copies of I, we will color the first blue and the second red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We will then color diagrams in such a way that the color of their endpoints indicate which copy of I is involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' In order to make the diagrams also readable without color, blue strands will be made thick and the blue copy of I will be written in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Thus, for example, = � 1I 0 0 0 � , = � 0 0 0 1I � , = � 0 0 1I 0 � , = � 0 1I 0 0 � ∈ HomAdd(BC)(I ⊕ I, I ⊕ I), And = \uf8eb \uf8ec \uf8ec \uf8ed 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 \uf8f6 \uf8f7 \uf8f7 \uf8f8 ∈ EndAdd(BC)((I ⊕ I)⊗2) = EndAdd(BC)((I ⊗ I) ⊕ (I ⊗ I) ⊕ (I ⊗ I) ⊕ (I ⊗ I)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Proposition 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' There exists a unique C-linear monoidal functor SH : Bσ R(H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' d)C → Add(Bσ C(C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' −2d)) such that SH(I) = I ⊕ I and SH � � = − − − − , SH ( ) = − , SH ( ) = − , (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='3) SH � i � = i − i , SH � j � = − , SH � k � = i + i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='4) The functor SH is full and faithful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' To show that SH is well defined, we must show that it respects the relations (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1) and (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For the first relation in (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1), we verify that SH � � = + + + = + + + = SH � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The proof of the second relation in (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1) is similar, since both sides are mapped by SH to the negative of the sum over all possible colorings of the strands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' DIAGRAMMATICS FOR REAL SUPERGROUPS 51 For the third relation in (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1), we have SH � � = + = + = SH � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The proof of the fourth relation in (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1) is analogous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' To verify the fifth relation in (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1), we compute SH � � = − = − = ( − ) = SH ( ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For the sixth relation in (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1), we have SH � � = − + − = − + − = SH � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The first two relations in (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2) are straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For the third relation in (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2), it suffices to consider the cases where a, b ∈ {1, i, j, k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' These are all straightforward computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For example, we have SH � i i � = − − = SH � −1 � , SH � j j � = − − = − − = SH � −1 � , SH � j i � = i + i = SH � k � , SH � i j � = −i − i = SH � −k � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The other cases are analogous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' It is also straightforward to verify the fourth relation in (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2) by considering the cases a ∈ {1, i, j, k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For the fifth relation in (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2), we again consider the cases a ∈ {1, i, j, k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' When a = i, we have SH � i � = −i − i = SH � −i � = SH � i⋆ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The other cases are analogous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Finally, we show that SH respects (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' First note that, by (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='6), any bubble with a purely imaginary token is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' By (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1), any bubble with a token labelled by a ∈ R is equal to a times a bubble with no token.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Then the fact that SH respects (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='3) follows from the computation SH � � = − − = −2 , and the fact that, by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='5, strR H(a) = 4a and strC C(a) = a for all a ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' It remains to prove that SH is full and faithful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For r, s ∈ N, the functor SH induces a C-linear map (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='5) HomBσ R (H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='d)(I⊗r, I⊗s)C → HomAdd(Bσ C (C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='−2d))((I ⊕ I)⊗r, (I ⊕ I)⊗s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' By Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='6, HomBσ R (H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='d)(I⊗r, I⊗s)C has C-basis D•(r, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Thus, it has dimension 4(r+s)/2(r + s − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' = 2r+s(r + s − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' if r + s is even and dimension zero if r + s is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' (We use here the fact that the number of perfect matchings of a set of size 2n is (2n)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' := (2n − 1)(2n − 3) · · · 1, and that dimR H = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=') On the other hand, HomAdd(Bσ C (C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='−2d))((I⊕I)⊗r, (I⊕I)⊗s) has the same dimension, since, when r +s is even, there are (r + s − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' perfect matchings of the r + s endpoints, and then 2r+s ways to choose one of the colors blue or red for the endpoints of the strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Thus, to prove that (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='5) is an isomorphism, it suffices to prove that it is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' To do this, it is enough to show that all possible colorings of the generating morphisms of Bσ C(C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' −2d) lie in the image of SH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' First note that SH � 1 2 + i 2 i � = , SH �1 2 − i 2 i � = , (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='6) 52 SAIMA SAMCHUCK-SCHNARCH AND ALISTAIR SAVAGE SH � − 1 2 j − i 2 k � = , SH � 1 2 j − i 2 k � = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='7) Next, we compute − 1 4SH � + i i + i i − i i � = , SH �1 2 a + i 2 ia � = a , a ∈ C, 1 2SH � j − i k � = = , 1 2SH � j + i k � = = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Then, composing with the morphisms in (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='7) to change colors of strands, we see that all possible colorings of the generating morphisms lie in the image of SH, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' □ Fix m, n ∈ N, set V = Hm|n, and let ϕ be a nondegenerate (ν, ⋆)-superhermitian form on V of parity σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' (See Appendices A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='5 and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='7 for a classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=') Recall, from Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='4, that the induced form ϕj : C2m|2n × C2m|2n → C is nondegenerate, (−ν, id)-supersymmetric, and of parity σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Let Φ denote the (ν, ⋆)-supersymmetric form defined as in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='12), where we recall that the Frobenius form τ on H is projection onto the real part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Recall, from Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='5, that G(Φ) = G(ϕ) and g(Φ) = g(ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Lemma 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For all G(Φ)-supermodules U and W, we have an isomorphism of C-supermodules HomG(Φ)(U, W)C ∼ = −→ HomG(ϕj)(U C, W C), f ⊗ a �→ f ⊗ a, f ∈ HomG(Φ)(U, W), a ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' First suppose that σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Then, as explained in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1, we may assume that ν = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' By the results of Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='5, we see that Gred(Φ) has two connected components when n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' (The case n = 0 is easier, and similar to the σ = 1 case discussed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=') Fix X ∈ Gred(Φ) with det(X) = −1, so that X is in the connected component of Gred(Φ) not containing the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Then, using Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='6 we have HomG(Φ)(U, W)C (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='23) = HomX,g(ϕ)C(U C, W C) ∼= HomX,g(ϕj)(U C, W C) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='18) = HomG(ϕj)(U C, W C), where, in the final equality, we used the fact that Gred(ϕj) has two connected components, and X lies in the connected component not containing the identity, as explained in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The case σ = 1 is easier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' As explained in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='7, G(Φ) and G(ϕj) are both connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Then, using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='22), we have HomG(Φ)(U, W)C = Homg(Φ)(U, W)C ∼= Homg(ϕj)(U C, W C) = HomG(ϕj)(U C, W C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' □ It follows from Lemma 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2 that we have a canonical full and faithful superfunctor (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='8) EH : (G(Φ)-smodR)C → G(ϕj)-smodC sending V to V C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The next result shows that the diagram Bσ R(H, ν(m − n))C Add(Bσ C(C, ν(2n − 2m))) (G(Φ)-smodR)C G(ϕj)-smodC SH FC Φ Fϕj EH commutes up to supernatural isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Proposition 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' There is a monoidal supernatural isomorphism of functors θ: EHFC Φ ∼ = −→ FϕjSH determined by (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='9) θI : V C ∼ = −→ V ⊕ V, θI(v) = 1 √ 2(v, vj), v ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' DIAGRAMMATICS FOR REAL SUPERGROUPS 53 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' To simplify notation, we set S = SH and E = EH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' First note that θI is the isomorphism (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' On objects, we have EFC Φ(I) = V C θI−→ ∼ = V ⊕ V = Fϕj(I ⊕ I) = FϕjSH(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Here, and it what follows, in the isomorphism V C ∼= V ⊕ V , the first copy of V and its elements are denoted by bold blue characters and the second copy of V and its elements are denoted by red non-bold characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' This matches our diagrammatic conventions introduced earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For morphisms, we need to show that θY ◦ EFC Φ(f) = FϕjS(f) ◦ θX for f ∈ � , , , i , j � , where X and Y are the domain and codomain of f, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' (Since k = i ◦ j , we do not need to check f = k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=') For f = , we have θI⊗I ◦ EFC Φ( ) = νθI⊗I ◦ flipV C,V C = ν flipV ⊕V ,V ⊕V ◦θI⊗I = FϕjS( ) ◦ θI⊗I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Note that the negative sign appearing in the definition of S( ) cancels with the negative sign arising from the fact that ϕj is (−ν, id)-supersymmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For f = , noting that θ 1 is the identity map C → C, we have, for v, w ∈ V , θ 1 ◦ EFC Φ( ): V C ⊗C V C → C, v ⊗ w �→ Re ϕ(v, w), and FϕjS( ) ◦ θI⊗I = Fϕj( − ) ◦ θI⊗I: V C ⊗C V C → C, v ⊗ w �→ 1 2(v, vj) ⊗ (w, wj) �→ 1 2(ϕj(v, wj) − ϕj(vj, w)) (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='10) = Re ϕ(v, w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' (Above, we have used the fact that the maps are uniquely determined by their values on v ⊗ w, for v, w ∈ V ⊆ V C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=') Next we consider f = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Choose a C-basis BC V of V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Then {v, vi : v ∈ BC V } is an R-basis of V , and it is straightforward to verify that (vi)∨ = v∨i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Identifying w ∈ V and v ∈ V and with (w, 0), (0, v) ∈ V ⊕ V , we can write (w, v) as w + v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Using this convention, we have that θI⊗I ◦ EFC Φ( ): C → (V ⊕ V ) ⊗C (V ⊕ V ) is the map given by 1 �→ � v∈BR V (−1)σ¯vv ⊗ v∨ = � v∈BC V (−1)σ¯v(v ⊗ v∨ + vi ⊗ v∨i) �→ 1 2 � v∈BC V (−1)σ¯v� (v + vj) ⊗ (v∨ + v∨j) + (vi + vij) ⊗ (v∨i + v∨ij) � = � v∈BC V (−1)σ¯vv ⊗ v∨j + � v∈BC V (−1)σ¯vvj ⊗ v∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' On the other hand, for all v, w ∈ BC V , we have ϕj(w∨j, v) (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='8) = −ϕ1(w∨, v) = −δvw and ϕj(w∨, vj) (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='9) = ϕ1(w∨, v)⋆ = δvw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Thus, the C-bases left dual to BC V and {vj : v ∈ BC V } with respect to ϕj are {−v∨j : v ∈ BC V } and {v∨ : v ∈ BC V }, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Therefore, FϕjS( ) ◦ η 1 = Fϕj( − ): C → (V ⊗C V ) ⊕ (V ⊗C V ), 54 SAIMA SAMCHUCK-SCHNARCH AND ALISTAIR SAVAGE 1 �→ � v∈BC V (−1)σ¯vvj ⊗ v∨ + � v∈BC V (−1)σ¯vv ⊗ v∨j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For f = i , we have θI ◦ EFC Φ � i � : V C → V ⊕ V , v �→ −vi �→ 1 √ 2(−vi, −vij) = 1 √ 2(−vi, vji) and FϕjS � i � θI = Fϕj � i − i � ηI : V C → V ⊕ V , v �→ 1 √ 2(v, vj) �→ 1 √ 2(−vi, vji).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For f = j , we have θI ◦ EFC Φ � j � : V C → V ⊕ V , v �→ −vj �→ 1 √ 2(−vj, v) and FϕjS � j � θI = Fϕj � − � ηI : V C → V ⊕ V , v �→ 1 √ 2(v, vj) �→ 1 √ 2(−vj, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' □ Proposition 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='5 holds when (D, ⋆) = (H, ⋆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We wish to show that, for all r, s ∈ N, the R-linear map FΦ : HomBσ R (H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='ν(m−n))(I⊗r, I⊗s) → HomG(Φ)(V ⊗r, V ⊗s) is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' This map is surjective if and only if the map (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='10) FC Φ : HomBσ R (H,⋆;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='ν(m−n))(I⊗r, I⊗s)C → HomG(Φ)(V ⊗r, V ⊗s)C is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' To show that (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='10) is surjective, it suffices to show that the diagram HomBσ R (H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='ν(m−n))(I⊗r, I⊗s)C HomG(Φ)(V ⊗r, V ⊗s) ⊗R C HomAdd(Bσ C (C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='ν(2n−2m))) ((I ⊕ I)⊗r, (I ⊕ I)⊗s) HomG(ϕj) � (V C)⊗r, (V C)⊗s� HomG(ϕj)((V ⊕ V )⊗r, (V ⊕ V )⊗s) SH ∼ = FC Φ ∼ = EH Fϕj ∼ = commutes, where the diagonal isomorphism is induced by the isomorphism (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='9), and surjectivity of the bottom-left vertical arrow follows from Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Commutativity of this diagram follows from Proposition 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' □ As a special case of G(Φ), we have the quaternionic orthosymplectic supergroups OSp∗(n|p, q);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' see Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Recall the definition G(Φ)-tsmodR of the monoidal supercategory of tensor G(Φ)-supermodules from Section 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Proposition 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' If p, p′, q, q′, n ∈ N satisfy p + q = p′ + q′, then we have an equivalence of monoidal supercategories OSp∗(n|p, q)-tsmodR ≃ OSp∗(n|p′, q′)-tsmodR sending the natural supermodule to the natural supermodule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The proof is analogous to that of Proposition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='5, using the commutative diagram appearing in the proof of Proposition 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' □ DIAGRAMMATICS FOR REAL SUPERGROUPS 55 Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Classification of superhermitian forms In this appendix, we give the classification of (ν, ⋆)-superhermitian forms for the various choices of involutive division superalgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' In each case, we also give the corresponding Harish-Chandra superpair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The explicit descriptions given in this appendix show that the Lie superalgebras of the form g(ϕ), together with the Lie superalgebras gl(r|s, D) for a real division superalgebra D, give all the real forms of gl(m|n, C), osp(m|2n, C), p(m, C), and q(m, C);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' see Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Preliminaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For any supermatrix X, let X♯ := (X⋆)st = (Xst)⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The isomorphism (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='15), together with the isomorphism of superalgebras Matm|n(Aop) ∼ = −→ Matm|n(A), Xop �→ X⋆, shows that (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1) (XY )♯ = (−1) ¯ X ¯Y Y ♯X♯ whenever the product XY is defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' It follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='14) that, for X ∈ Mat(m|n)×(r|s)(A), we have (X♯)♯ = Sm|nXSr|s where Sp|q = � Ip 0 0 −Iq � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We will often omit the subscripts on Sp,q when its size is clear from the context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Since we identify Am|n = Mat(m|n)×(1|0)(A), and S1|0 = I1, we have (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2) � v0 v1 �♯ = � v⋆,tr 0 −v⋆,tr 1 � and (v♯)♯ = Sv for v = � v0 v1 � ∈ Am|n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Every sesquilinear form on Am|n is of the form (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='3) ϕ(v, w) = v♯Mw where M ∈ Matm|n(A) is homogeneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The form ϕ given by (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='3) is (ν, ⋆)-superhermitian if and only if M♯ = ν(−1) ¯ MMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For v, w ∈ Am|n, we have (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='4) ϕ(w, v)⋆ = ϕ(w, v)♯ (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1) = (−1)¯v ¯w+ ¯ M(¯v+ ¯w)v♯M♯(w♯)♯ (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2) = (−1)¯v ¯w+ ¯ M(¯v+ ¯w)v♯M♯Sw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Now, if ¯ϕ = ¯ M = 0, then ϕ(w, v) = 0 = ϕ(v, w) unless ¯v + ¯w = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Then (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='4) implies that ϕ is (⋆, ν)-superhermitian if and only if M = νM♯S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' On the other hand, if ¯ϕ = ¯ M = 1, then ϕ(w, v) = 0 = ϕ(v, w) unless ¯v + ¯w = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Then (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='4) implies that ϕ is (⋆, ν)-superhermitian if and only if M = −νM♯S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Combining both cases, and using the fact that S2 = I, the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' □ Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For ϕ as in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='3), we have X† = (M♯S)−1X♯M♯S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For v, w ∈ Am|n and X ∈ gl(m|n, A), we have (−1) ¯ X¯vϕ(X†v, w) (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1) = v♯(X†)♯Mw and ϕ(v, Xw) = v♯MXw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Thus (X†)♯M = MX =⇒ M♯SX†S = X♯M♯ =⇒ X† = (M♯S)−1X♯M♯S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' □ For the remainder of this section (D, ⋆), will denote an involutive division superalgebra over k ∈ {R, C}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Our goal is to classify the (ν, ⋆)-superhermitian forms over such superalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' An even (ν, ⋆)-superhermitian form V is equivalent to an even (−ν, ⋆)-superhermitian form on ΠV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Thus, for even forms, we will only treat the (1, ⋆)-superhermitian case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For odd forms, we will need 56 SAIMA SAMCHUCK-SCHNARCH AND ALISTAIR SAVAGE to treat both the ν = 1 and ν = −1 cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' (However, these lead to isomorphic Lie algebras;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' see (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=') An even (1, ⋆)-superhermitian form on a D-supermodule V = V0 ⊕V1 corresponds, when viewing V as a (non-super) k-module, to a superhermitian form on V0 and an skew-superhermitian form on V1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' This allows us to use well-known results classifying such forms up to equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' (See, for example, [Lew82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=') They are typically classified by dimension or signature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Case (C, id), k = C, even form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' There are no even nondegenerate (1, id)-superhermitian forms on Cm|n when n ∈ 2Z + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Every even nondegenerate (1, id)-superhermitian form on Cm|2n is equivalent to (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='5) ϕm|2n : (v, w) �→ vt \uf8eb \uf8ed Im 0 0 0 0 In 0 −In 0 \uf8f6 \uf8f8 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Then G(ϕm|2n) is the complex orthosymplectic supergroup OSp(m|2n, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We have Gred(ϕm|2n) = O(m, C) × Sp(2n, C) and g(ϕm|2n) = osp(m|2n, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The group Sp(2n, C) is connected, whereas, when m ≥ 1, O(m, C) has two connected components: the identity component SO(n, C) and the elements of O(m, C) with determinant −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' It follows that, for any nondegenerate even (ν, id)-superhermitian form ϕ on Cm|n, the group Gred(ϕ) has two connected components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Any X ∈ Gred(ϕ) with det(X) = −1 is in the connected component not containing the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Case (R, id), k = R, even form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' There are no even nondegenerate (1, id)-superhermitian forms on Rm|n when n ∈ 2Z + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Every even nondegenerate (1, id)-superhermitian form on Rm|2n is equivalent to (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='6) ϕp,q|2n: (v, w) �→ vt \uf8eb \uf8ec \uf8ec \uf8ed Ip 0 0 0 0 −Iq 0 0 0 0 0 In 0 0 −In 0 \uf8f6 \uf8f7 \uf8f7 \uf8f8 w, v, w ∈ Rm|2n, for some p, q ∈ N, p+q = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Furthermore, ϕp,q|2n ∼ ϕp′,q′|2n if and only if (p′, q′) = (p, q) or (p′, q′) = (q, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The supergroup G(ϕp,q|2n) is the indefinite orthosymplectic supergroup OSp(p, q|2n, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We have Gred(ϕp,q|2n) = O(p, q) × Sp(2n, R) and g(ϕp,q|2n) = osp(p, q|2n, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The real symplectic group Sp(2n, R) is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' When p, q ≥ 1, the indefinite orthogonal group O(p, q) has four connected components, corresponding to the two choices ±1 of determinant for the restriction to the two subspaces Rp × 0q and 0p × Rq of Rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' When p = 0 or q = 0, but p + q ≥ 1, O(p, q) has two connected components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Since ϕC p,q|2n is equivalent to the form ϕm|2n of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='5), it follows from Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1 that osp(p, q|2n, R)C = g(ϕp,q|2n)C ∼= g(ϕC p,q|2n) ∼= osp(m|2n, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Case (C, ⋆), k = R, even form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Every even nondegenerate (1, ⋆)-superhermitian form on Cm|n is equivalent to (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='7) ϕp,q|r,s: (v, w) �→ v♯ \uf8eb \uf8ec \uf8ec \uf8ed Ip 0 0 0 0 −Iq 0 0 0 0 iIr 0 0 0 0 −iIs \uf8f6 \uf8f7 \uf8f7 \uf8f8 w DIAGRAMMATICS FOR REAL SUPERGROUPS 57 for some p, q, r, s ∈ N, p + q = m, r + s = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Furthermore, ϕp,q|r,s ∼ ϕp′,q′|r′,s′ if and only if (p′, q′, r′, s′) ∈ {(p, q, r, s), (q, p, s, r), (r, s, p, q), (s, r, q, p)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We call U(p, q|r, s) := G(ϕp,q|r,s) the indefinite unitary supergroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We have Gred(ϕp,q|r,s) = U(p, q) × U(r, s) and g(ϕp,q|r,s) = u(p, q|r, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Since the indefinite unitary group U(p, q) is connected for all p, q ∈ N, it follows that Gred(ϕ) is connected for any nondegenerate even (ν, ⋆)-superhermitian form ϕ on Cm|n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' It follows from Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2 that we have an isomorphism of complex Lie superalgebras u(p, q|r, s)C = g(ϕp,q|r,s)C ∼= gl(m|n, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Case (H, ⋆), k = R, even form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Every even nondegenerate (1, ⋆)-superhermitian form on Hm|n is equivalent to (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='8) ϕp,q|n: (v, w) �→ v♯ \uf8eb \uf8ed Ip 0 0 0 −Iq 0 0 0 jIn \uf8f6 \uf8f8 w for some p, q ∈ N, p + q = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' (See, for example, [Lew82, §5, §6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=') We have ϕp,q|n ∼ ϕp′,q′|n′ ⇐⇒ (p′, q′, n) ∈ {(p, q, n), (q, p, n)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We call OSp∗(n|p, q) := G(ϕp,q|n) the quaternionic orthosymplectic supergroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We have Gred(ϕp,q|n) = O(n, H) × U(p, q, H) and g(ϕp,q|n) = osp∗(n|p, q), where O(n, H) is the quaternionic orthogonal group, sometimes denoted O∗(2n) in the literature, and U(p, q, H) is the indefinite quaternionic unitary group, which is equal to the indefinite symplectic group Sp(p, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The notation for g(ϕp,q|n) is not consistent in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For example, it is denoted osp∗(n|2m, 2p) in [Ser83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The indefinite symplectic group Sp(p, q) is connected;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' see [Kna02, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='145].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The quaternionic orthogonal group O(n, H) has two connected components when n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Viewing elements of O(n, H) as 2n × 2n complex matrices, the identity component, SO(n, H), of O(n, H) consists of those elements with determinant 1, while the other component consists of those elements with determinant −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' It follows that, for any nondegenerate even (ν, ⋆)- superhermitian form ϕ on Hm|n, n ≥ 1, the group Gred(ϕ) has two connected components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Any X ∈ Gred(ϕ) with det(X) = −1 is in the connected component not containing the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We have an isomorphism of complex Lie superalgebras g(ϕp,q|n)C ∼= osp(2n|2m, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' (Note the reversal in the order of m and n on the right-hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=') Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Consider the form (ϕp,q|n)j in the notation of Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Since a skew-supersymmetric form on C2m|2n is equivalent to a supersymmetric form on C2n|2m (via the parity shift map C2m|2n → C2n|2m), we see that (ϕp,q|n)j is equivalent to the form ϕ2n|2m given by replacing m and n in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='5) by 2n and 2m, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Thus, the result follows from Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' □ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Case (Cl(C), ⋆), k = R, even form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' In light of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='9, we assume in this case that n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Every even nondegenerate (1, ⋆)-superhermitian form on Cl(C)m is equivalent to (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='9) ϕp,q : (v, w) �→ v♯ � Ip 0 0 −Iq � w for unique p, q ∈ N, p + q = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We call UQ(p, q) := G(ϕp,q) the indefinite isomeric unitary supergroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We have Gred(ϕp,q) = U(p, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' 58 SAIMA SAMCHUCK-SCHNARCH AND ALISTAIR SAVAGE The notation for g(ϕp,q) is not consistent in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' It is sometimes denoted by uq(p, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Its simple quotient is denoted upsq(n, p) in [Ser83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Since the indefinite unitary group U(p, q) is connected, it follows that, for any nondegenerate (ν, ⋆)-superhermitian form on Cl(C)m, the group Gred(ϕ) is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' It follows from Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2 that we have an isomorphism of complex Lie superalgebras g(ϕp,q)C ∼= gl(m, Cl(C)) = q(m, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Odd forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Let k ∈ {R, C} and let (D, ⋆) be an arbitrary involutive division k-superalgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' If k = R and ϕ is a nondegenerate (ν, ⋆)-superhermitian form on Cl(C)m|n, then ε(1 + i)ϕ is a nondegenerate (ν, ⋆)-superhermitian form on Cl(C) of parity ¯ϕ + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Since we already treated the even forms above, we assume in this subsection that D ∈ {R, C, H}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' An odd form on Dm|n can only be nondegenerate when m = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Any odd nondegenerate (ν, ⋆)- superhermitian form on Dm|m is equivalent to (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='10) ϕν m : (v, w) �→ v♯ � 0 Im −νIm 0 � w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' (Recall, from (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2), the sign appearing in v♯.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=') With this form, it follows from Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2 that � X00 X01 X10 X11 �† = � X♯ 11 −νX♯ 01 νX♯ 10 X♯ 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Thus, g(ϕν m) = ��X Y Z −X♯ � : X, Y, Z ∈ Matm(D), Y = νY ♯, Z = −νZ♯ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' In particular, we have an isomorphism of Lie superalgebras (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='11) g(ϕν m) ∼ = −→ g(ϕ−ν m ), X �→ X#.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We also have Gred(ϕν m) = ��X 0 0 (X♯)−1 � : A = GL(m, D) � ∼= GL(m, D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' It follows that Gred(ϕ) is connected for any nondegenerate odd (ν, ⋆)-superhermitian form ϕ on Dm|n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' When (D, ⋆) is equal to (R, id) or (C, id), the Lie superalgebras g(ϕν m) = p(m, R) and g(ϕν m) = p(m, C) are the real and complex periplectic Lie superalgebras, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The notation for the other cases is less standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' When (D, ⋆) = (C, ⋆), the Lie superalgebra g(ϕm) is sometimes denoted up(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' When (D, ⋆) = (H, ⋆), g(ϕν m) is sometimes denoted p∗(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Their simple quotients are denoted usπ(m) and sπ∗(m), respectively, in [Ser83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' It follows from Propositions 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='6 and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='11) that we have isomorphisms of complex Lie superalgebras g(ϕν m)C ∼= \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 p(m, C) if (D, ⋆) = (R, id), gl(m|m, C) if (D, ⋆) = (C, ⋆), p(2m, C) if (D, ⋆) = (H, ⋆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' DIAGRAMMATICS FOR REAL SUPERGROUPS 59 Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Classification of real forms A classification of the real forms of the classical Lie algebras can be found in [FH91, §26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The classification of the real simple Lie superalgebras was first given in [Ser83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' In particular, [Ser83, Table 3] lists all the real forms of the simple subquotients of gl(m|n, C), osp(m|2n, C), p(m, C), and q(m, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' However, because gl(m|n, C), p(m, C) and q(m, C) are not simple, they are not covered by this classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Since the real forms of these Lie superalgebras do not seem to have appeared in the literature, we give a classification here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Throughout this subsection g denotes one of the superalgebras gl(m|n, C), q(m, C), or p(m, C), m, n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Let g′ = [g, g] and g′′ = g′/Z(g′), where [g, g] denotes the ideal of g generated by [X, Y ], X, Y ∈ g, and Z(g′) = {X ∈ g′ : [X, g′] = 0} denotes the center of g′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' For m, n ∈ N, gl(m|n, C)′ = sl(m|n, C) = {X ∈ gl(m|n, C) : str(X) = 0}, q(m, C)′ = {X ∈ q(m, C) = gl(m, Cl(C)) : tr(X)1 = 0}, p(m, C)′ = �� X00 X01 X10 −Xt 00 � ∈ gl(m|m, C) : tr(X00) = 0, Xt 01 = X01, Xt 10 = −X10 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Since, for m ̸= n, Z(sl(m|n, C)) = 0, Z(sl(m|m, C)) = CI2m, Z(q(m, C)′) = CIm, Z(p(m, C)′) = 0, we have gl(m|n, C)′′ = sl(m|n, C), gl(m|m, C)′′ = sl(m|m, C)/CI2m = psl(m|m, C), q(m, C)′′ = q(m, C)′/CIm, p(m, C)′′ = p(m, C)′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' The adjoint action of g on itself is given by Ad: g → Endg(g), Ad(X)(Y ) = [X, Y ], X, Y ∈ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' This restricts to give an action Ad′ : g → Endg(g′) of g on g′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We have ker(Ad′) = Z(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We show this for the case g = gl(m|n, C), since the other cases are analogous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Suppose X = �m+n r,s=1 arsErs, ars ∈ C, is a homogeneous element of ker(Ad′), where Ers denotes the matrix with a 1 in position (r, s), and a 0 in all other positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Then, for all 1 ≤ t, u ≤ m + n, we have 0 = [X, Etu] = m+n � r=1 artEru ± m+n � s=1 ausEts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Thus, ars = 0 whenever r ̸= s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' So X is diagonal, hence even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Then, for all 1 ≤ t, u ≤ m + n, we have 0 = [X, Etu] = (att − auu)Etu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Thus att = auu, and so X ∈ CIm+n = Z(gl(m|n, C)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' □ A conjugate-linear involution of g (also called a real structure on g) is an automorphism κ of g, considered as a real Lie superalgebra, satisfying κ2 = id and κ(aX) = a⋆κ(X) for all a ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Every real form of g is isomorphic to gκ = {X ∈ g : κ(X) = X} 60 SAIMA SAMCHUCK-SCHNARCH AND ALISTAIR SAVAGE for some conjugate-linear involution κ of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' If κ is a conjugate-linear involution of g, then κ restricts to a conjugate-linear involution κ′ of g′, which, in turn, induces a conjugate-linear involution κ′′ of g′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Suppose κ and χ are conjugate-linear involutions of g such that κ′′ = χ′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Then κ′ = χ′ and κ(X) − χ(Z) ∈ Z(g) for all X ∈ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Since the odd part of g′′ is equal to the odd part of g, we have κ|g1 = χ|g1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We see by inspection that [g1, g1] = g′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Thus, κ′ = χ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Then, for all X ∈ g, Y ∈ g′, we have [κ(X), χ(Y )] = [κ(X), κ(Y )] = κ([X, Y ]) = χ([X, Y ]) = [χ(X), χ(Y )].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' It follows that Ad′(κ(X)) = Ad′(χ(X)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Hence, by Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='1, we have κ(X) − χ(X) ∈ Z(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' □ Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Every real form of gl(m|n, C), osp(m|2n, C), p(m, C), q(m, C), m, n ∈ N, is isomorphic to either gl(r|s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' D) for a real division superalgebra D, or g(ϕ) for a (ν, ⋆)-superhermitian form ϕ on Dr|s, where (D, ⋆) is an involutive real division superalgebra, for some r, s ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' In the case of osp(m|2n, C), which is simple, we see from [Ser83, Table 3] that the real forms are osp(p, q|2n, R), p + q = m and, when m is even, osp∗(m 2 |p, q), p + q = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Then the result follows from Appendices A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='3 and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Now assume that g is one of the Lie superalgebras gl(m|n, C), q(m, C), or p(m, C), m, n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' All of the real forms described in Appendices A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='3 to A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='7 induce real forms of g′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Comparing to the real forms given [Ser83, Table 3] shows that we obtain all real forms of g′′ in this way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Thus, it remains to show that, up to isomorphism, a real form of g is determined by the corresponding real form of g′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Equivalently, it suffices to show that, if κ and χ are conjugate-linear involutions of g such that κ′′ = χ′′, then κ = χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Suppose κ and χ are conjugate-linear involutions of g such that κ′′ = χ′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Then, by Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2, κ′ = χ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' If g = p(m, C), then Z(g) = 0, and it follows from Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2 that κ = χ, and we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Now suppose that g = g(m|n, C), m ̸= n, or g = q(m, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Then g = g′ ⊕ CI, where I denotes the identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Both κ and χ must leave Z(g) = CI invariant, hence the corresponding real forms must be isomorphic to (g′)κ ⊕ R = (g′)χ ⊕ R, where R denotes the one-dimensional abelian real Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Finally, suppose that g = gl(m|m, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' We have a short exact sequence of Lie superalgebras 0 → sl(m|m, C) → gl(m|m, C) str −→ Z(g) = C → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Since κ and χ agree on sl(m|m, C), it follows from Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='2 that there exists a ∈ C such that κ(X) = χ(X) + a str(X)I, X ∈ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Then, for all X ∈ g, we have X = κ2(X) = κ (χ(X) + a str(X)I) = X + a⋆ str(X)⋆χ(I) + a str(X)I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Thus, a⋆ str(X)⋆χ(I) + a str(X) = 0 for all X ∈ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Choosing X = E11 and X = iE11, implies that a⋆χ(I) + a = 0 and a⋆χ(I) − a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Hence a = 0, and so κ = χ, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' □ DIAGRAMMATICS FOR REAL SUPERGROUPS 61 References [Bae20] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content=' Baez.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} +page_content='ca' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAzT4oBgHgl3EQfc_zP/content/2301.01414v1.pdf'} diff --git a/ydE0T4oBgHgl3EQfcgDn/content/tmp_files/2301.02365v1.pdf.txt b/ydE0T4oBgHgl3EQfcgDn/content/tmp_files/2301.02365v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..73519aae0d7726bd11c63cc551bc85fcf1ab401f --- /dev/null +++ b/ydE0T4oBgHgl3EQfcgDn/content/tmp_files/2301.02365v1.pdf.txt @@ -0,0 +1,357 @@ +arXiv:2301.02365v1 [math.GR] 6 Jan 2023 +ON THE CHARACTERIZATION OF SPORADIC SIMPLE GROUPS BY CODEGREES +MALLORY DOLORFINO, LUKE MARTIN, ZACHARY SLONIM, YUXUAN SUN, AND YONG YANG +Abstract. Let G be a finite group and Irr(G) the set of all irreducible complex characters of G. Define the +codegree of χ ∈ Irr(G) as cod(χ) := |G:ker(χ)| +χ(1) +and denote by cod(G) := {cod(χ)|χ ∈ Irr(G)} the codegree +set of G. Let H be one of the 26 sporadic simple groups. In this paper, we show that H is determined up +to isomorphism by cod(H). +1. Introduction +Let G be a finite group and Irr(G) the set of all irreducible complex characters of G. For any χ ∈ Irr(G), +define the codegree of χ, denoted by cod(χ), as cod(χ) := +|G:ker(χ)| +χ(1) +. Then define the codegree set of G +as cod(G) := {cod(χ)|χ ∈ Irr(G)}. The concept of codegrees was originally considered in [7], where the +codegree was defined as cod(χ) := +|G| +χ(1), and it was later modified to its current definition by [19] so that +cod(χ) is the same for G and G/N when N ≤ ker(χ). Several properties of codegrees have been studied, +such as the relationship between the codegrees and element orders, codegrees of p-groups, and groups with +few codegrees. +The codegree set of a group is closely related to the character degree set of a group, defined as cd(G) := +{χ(1)|χ ∈ Irr(G)}. The relationship between the character degree set and the group structure is an active +area of research, and many properties of group structure are largely determined by the character degree set. +In 1990, Bertram Huppert studied the character degrees of simple groups and made the following conjecture +about the relationship between a simple group H and a finite group G that have equal character degree sets. +Huppert’s Conjecture: Let H be a finite nonabelian simple group and G a finite group such that +cd(H) = cd(G). Then G ∼= H × A, where A is an abelian group. +Huppert’s conjecture has since been verified for many cases such as the alternating groups, sporadic +groups, and simple groups of Lie type with low rank, but it has yet to be verified for simple groups of Lie +type with high rank. Recently, a similar conjecture related to codegrees has been posed. +Codegree Version of Huppert’s Conjecture: Let H be a finite nonabelian simple group and G a +finite group such that cod(H) = cod(G). Then G ∼= H. +This conjecture appears in the Kourovka Notebook of Unsolved Problems in Group Theory as question +20.79 [16]. It has been verified for PSL(2, q), PSL(3, 4), Alt7, J1, 2B2(22f+1) where f ≥ 1, M11, M12, M22, M23 +and PSL(3, 3) by [1, 4, 10]. The conjecture has also been verified for PSL(3, q) and PSU(3, q) in [17] and +2G2(q) in [11]. Most of these results concern simple groups with less than 21 character degrees [3]. However, +in this paper, we provide a general proof verifying this conjecture for all sporadic simple groups. The methods +used may be generalized to simple groups of Lie type, giving promising results for characterizing all simple +groups by their codegree sets. +Theorem 1.1. Let H be a sporadic simple group and G a finite group. If cod(G) = cod(H), then G ∼= H. +Throughout the paper, we follow the notation used in Isaac’s Book [13] and the ATLAS of Finite Groups +[8]. +2. Preliminary Results +We first reproduce some lemmas which will be used in later proofs. +Lemma 2.1. [18, Lemma 4.2] Let S be a finite nonabelian simple group. Then there exists 1S ̸= χ ∈ Irr(S) +that extends to Aut(S). +2000 Mathematics Subject Classification. 20C15, 20D08. +1 + +Lemma 2.2. [14, Theorem 4.3.34] Let N be a minimal normal subgroup of G such that N = S1 × · · · × St +where Si ∼= S is a nonabelian simple group for each i = 1, . . . , t. If χ ∈ Irr(S) extends to Aut(S), then +χ × · · · × χ ∈ Irr(N) extends to G. +Lemma 2.3. [10, Remark 2.6] Let G be a finite group and H a finite nonabelian simple group with cod(G) = +cod(H). Then G is a perfect group. +Lemma 2.4. [12] Let G be a finite group and S a finite nonabelian simple group such that cod(S) ⊆ cod(G). +Then |S| divides |G|. +Lemma 2.5. Let G be a finite group with N ⊴ G. Then cod(G/N) ⊆ cod(G). +Proof. From [13, Lemma 2.22], we can define Irr(G/N) = {ˆχ(gN) = χ(g) | χ ∈ Irr(G) and N ⊆ ker(χ)}. +Take any ˆχ ∈ Irr(G/N). By definition, we know that ˆχ(1) = χ(1), so the denominators of cod(ˆχ) and cod(χ) +are equal. In addition, ker(ˆχ) ∼= ker(χ)/N, so |ker(χ)| = |N| · |ker(ˆχ)|. Thus |G/N : ker(ˆχ)| = +|G|/|N| +| ker(χ)|/|N| = +|G| +| ker(χ)|, so cod(ˆχ) = cod(χ) and therefore cod(G/N) ⊆ cod(G). +□ +Lemma 2.6. Let G be a finite group with normal subgroups N and M such that N ≤ M. Then, cod(G/M) ⊆ +cod(G/N). +Proof. By the Third Isomorphism Theorem, we know that G/M ∼= (G/N)/(M/N) is a quotient of G/N, +and by Lemma 2.5, cod(G/M) ⊆ cod(G/N). +□ +Lemma 2.7. Let G and H be finite groups such that cod(G) ⊆ cod(H). Then there are at least |cod(G)| +elements in cod(H) which divide |G|. +Proof. Let x ∈ cod(G), it is clear that x divides |G|. Since this is true for each x, the lemma follows. +□ +3. Main Results +Theorem 3.1. Let H be a sporadic simple group and G a finite group with cod(G) = cod(H). If N is a +maximal normal subgroup of G, then G/N ∼= H. +Proof. By Lemma 2.3, G is perfect. Thus G/N is a nonabelian simple group. By Lemma 2.6, we have +cod(G/N) ⊆ cod(G) = cod(H). If cod(G/N) = cod(G) = cod(H), we have G/N ∼= H, since no two non- +isomorphic finite simple groups have equal codegree sets. Therefore, it is sufficient to show that cod(G/N) = +cod(G). We can easily check that cod(K) ̸⊆ cod(H) for any two non-isomorphic sporadic groups H and +K. Thus G/N must belong to one of the 17 infinite families of nonabelian simple groups. We compute the +orders of these simple groups using the formulas given in [6]. +For each sporadic group H, we check all of the possibilities for G/N and restrict only to those which satisfy +Lemma 2.4. For example, let H ∼= M (where M denotes the Monster group). Then, we check through all +17 families of nonabelian simple groups and return those whose orders divide |H|. The result is given in +Table 1. The number in the third column represents the maximal prime power q = pk which satisfies this +condition (all smaller prime powers except those specified as being excluded also satisfy the condition). +2 + +Table 1. Possibilities for G/N given H ∼= M after applying Lemma 2.4: |G/N| | |H|. +G/N +n +Max q = pk +An +5-32 +n/a +1 +34, excl. 24, 25, 43, 26 +2 +52, excl. 11, 13, 24, 19, 23 +Ln+1(q) +3 +32 +4 +22 +5 +22 +2 +32 +3 +5 +O2n+1(q) +4 +3 +5 +2 +6 +2 +3 +5 +S2n(q) +4 +3 +5 +2 +6 +2 +G/N +n +Max q = pk +4 +3 +O+ +2n(q) +5 +3 +6 +2 +2 +23 +Un+1(q) +3 +3 +4 +2 +5 +2 +O− +2n(q) +5 +2 +6 +2 +2E6(q) +n/a +2 +3D4(q) +n/a +2 +Sz(q) +n/a +q = 23, 25 +2F4(2)′ +n/a +n/a +Now, we check each group in this table to see if it satisfies Lemma 2.7, i.e. we check how many elements +in the codegree set of H divide the order of G/N. For H ∼= M, we find that none of the possible groups +G/N have order divisible by more than 3 of the codegrees of M. This contradicts [2], which shows that +for any nonabelian simple group, |cod(G/N)| > 3. +Thus, if H ∼= M and cod(G/N) ⊆ cod(H), then +cod(G/N) = cod(H) and G/N ∼= H. +We repeat this process for all of the other sporadic simple groups. For each sporadic group, H, we first +check which nonabelian simple groups, G/N, satisfy |G/N| divides |H|. Second we check which of these +possibilities order divisible by more than 3 codegrees of H. We find two groups H such that the number of +codegrees of H dividing |G/N| is more than 3. These are H ∼= Suz with G/N ∼= O+ +8 (2) and H ∼= Fi23 with +G/N ∼= O+ +8 (3). We find 5 and 4 such codegrees respectively. For each of these cases, however, [3] shows +that |cod(G/N)| > 20. Thus by Lemma 2.7, we cannot have cod(G/N) ⊆ cod(H) if G/N ̸∼= H and thus +G/N ∼= H. +□ +Now we present the proof of Theorem 1.1. +Proof. Let G be a minimal counterexample and N a maximal normal subgroup of G. By Lemma 2.3, G is +perfect, and by Theorem 3.1, G/N ∼= H. In particular, N ̸= 1 as G ̸∼= H. +Step 1: N is a minimal normal subgroup of G. +Suppose L is a nontrivial normal subgroup of G with L < N. Then by Lemma 2.6, we have cod(G/N) ⊆ +cod(G/L) ⊆ cod(G). +However, cod(G/N) = cod(H) = cod(G) so equality must be obtained in each +inclusion. Thus, cod(G/L) = cod(H) which implies that G/L ∼= H, since G is a minimal counterexample. +This is a contradiction since we also have G/N ∼= H, but L < N. +Step 2: N is the only nontrivial, proper normal subgroup of G. +Otherwise we assume U is another proper nontrivial normal subgroup of G. If N is included in U, then +U = N or U = G since G/N is simple, a contradiction. Then N ∩ U = 1 and G = N × U. Since U is also +a maximal normal subgroup of G, we have N ∼= U ∼= H. Choose ψ1 ∈ Irr(N) and ψ2 ∈ Irr(U) such that +cod(ψ1) = cod(ψ2) = max(cod(H)). Set χ = ψ1 · ψ2 ∈ Irr(G). Then cod(χ) = (max(cod(H)))2 /∈ cod(G), a +contradiction. +Step 3: For each nontrivial χ ∈ Irr(G|N) := Irr(G) − Irr(G/N), χ is faithful. +By [13, Lemma 2.22], we have that Irr(G/N) = {χ ∈ Irr(G)|N ≤ ker(χ)}. Then it follows by the definition +of Irr(G|N) that if χ ∈ Irr(G|N), N ̸≤ ker(χ). Thus since N is the unique nontrivial, proper, normal subgroup +of G, ker(χ) = G or ker(χ) = 1. Therefore, ker(χ) = 1 for all nontrivial χ ∈ Irr(G|N). +Step 4: N is an elementary abelian group. +3 + +Suppose that N is not abelian. +Since N is a minimal normal subgroup, by [9, Theorem 4.3A (iii)], +N = Sn where S is a nonabelian simple group and n ∈ Z+. By Lemmas 2.1 and 2.2, there is a non-trivial +character χ ∈ Irr(N) which extends to some ψ ∈ Irr(G). Now, ker(ψ) = 1 by Step 3, so cod(ψ) = |G|/ψ(1) = +|G/N| · |N|/χ(1). This contradicts the fact that |G/N| is divisible by cod(ψ), as χ(1) < |N|, so N must +be abelian. Now to show that N is elementary abelian, let a prime p divide |N|. Then N has a p-Sylow +subgroup K, and K is the unique p-Sylow subgroup of N since N is abelian, so K is characteristic in N. +Thus, K is a normal subgroup of G, so K = N as N is minimal, so |N| = pn. Now, take the subgroup +N p = {np : n ∈ N} of N, which is proper by Cauchy’s theorem. Then since N p is characteristic in N, it +must be normal in G, so N p is trivial by the uniqueness of N. Therefore, every element of N has order p, so +N is elementary abelian. +Step 5: CG(N) = N. +First note that since N is normal, CG(N) ⊴ G. Additionally, since N is abelian by Step 4, N ≤ CG(N), +so by the maximality of N, we must have CG(N) = N or CG(N) = G. If CG(N) = N, we are done. +If not, then CG(N) = G. Therefore N must be in the center of G. Then since N is the unique minimal +normal subgroup of G by Step 2, we must have that |N| is prime. If not, there always exists a proper +non-trivial subgroup K of N, and K is normal since it is contained in Z(G), contradicting the minimality of +N. Moreover, since G is perfect, we have that Z(G) = N, and N is isomorphic to a subgroup of the Schur +multiplier of G/N [13, Corollary 11.20]. +If H is isomorphic to any of M11, M23, M24, J1, J4, Co2, Co3, Fi22, Fi23, He, HN, Ly, Th, or M, then by [8], +the Schur multiplier of H is trivial, so N = 1, a contradiction. +If H is isomorphic to Co1, then G ∼= 2.Co1 by [8]. But 2.Co1 has a character degree of 24, which gives a +codegree of 219 ·38 ·54 ·72 ·11·13·23 ∈ cod(G), a contradiction, since 219 ·38 ·54 ·72 ·11·13·23 /∈ cod(H). If H +is isomorphic to Fi22, then G ∼= 2.Fi22 or G ∼= 3.Fi22 by [8]. If G ∼= 2.Fi22, then 213 · 39 · 52 · 7 · 13 ∈ cod(G), +a contradiction. If G ∼= 3.Fi22, then 217 · 37 · 52 · 6 · 11 ∈ cod(G), a contradiction. +Similarly, for any sporadic simple group H with non-trivial Schur multiplier, we use [8] to reach a contra- +diction as we did above, by finding an element of cod(G) that is not in cod(H). +Thus CG(N) = N. +Step 6: Let λ be a non-trivial character in Irr(N) and ϑ ∈ Irr(IG(λ)|λ), the set of irreducible constituents +of λIG(λ), where IG(λ) is the inertia group of λ ∈ G. Then |IG(λ)| +ϑ(1) +∈ cod(G). Also, ϑ(1) divides |IG(λ)/N|, +and |N| divides |G/N|. Lastly, IG(λ) < G, i.e. λ is not G-invariant. +Let λ be a non-trivial character in Irr(N) and ϑ ∈ Irr(IG(λ)|λ). Let χ be an irreducible constituent of +ϑG. By [13, Corollary 5.4], we know χ ∈ Irr(G), and by [13, Definition 5.1], we have χ(1) = +|G| +|IG(λ)| · ϑ(1). +Moreover, we know tat ker(χ) = 1 by Step 2, and thus cod(χ) = +|G| +χ(1) = |IG(λ)| +ϑ(1) , so |IG(λ)| +ϑ(1) +∈ cod(G). Now, +since N is abelian, λ(1) = 1, so we have ϑ(1) = ϑ(1)/λ(1) which divides |IG(λ)| +|N| , so |N| divides +|IG(λ)| +ϑ(1) . +Moreover, we know that cod(G) = cod(G/N), and all elements in cod(G/N) divide |G/N|, so |N| divides +|G/N|. +Next, we want to show IG(λ) is a proper subgroup of G. To reach a contradiction, assume IG(λ) = G. +Then ker(λ) ⊴ G. From Step 2, we know ker(λ) = 1, and from Step 4, we know N is a cyclic group of prime +order. Thus by the Normalizer-Centralizer theorem, we have G/N = NG(N)/CG(N) ≤ Aut(N) so G/N is +abelian, a contradiction. +Step 7: Final contradiction. +From Step 4, N is an elementary abelian group of order pn for some prime p and integer n ≥ 1. By +the Normalizer-Centralizer theorem, H ∼= G/N = NG(N)/CG(N) ≤ Aut(N) and n > 1. Note that in +general, Aut(N) = GL(n, p). By Step 6, |N| divides |G/N|, so we only need to consider primes p such that +p2 divides |H|. For instance, if H ∼= M11, |H| = 24 · 32 · 5 · 11 so |N| = 22, 23, 24 or 32. Then, we can +compute |Aut(N)| = |GL(n, p)| which is equal to 6, 168, 20160, and 48 respectively. In each of these four +cases, |H| ∤ |Aut(N)|. +For each sporadic group H, we follow a similar procedure to computationally check which possibilities of +(p, n) satisfy pn divides |H| and |H| divides |GL(n, p)|. We find only the following 7 possible H which satisfy +this condition, listed in Table 2. In other words, Table 2 gives all groups H and pairs (p, n) such that pn +divides |H| and |H| divides |GL(n, p)|. +4 + +Table 2. Minimum degree of faithful representations of sporadic groups over Fpk. +Group +p +n +Minimum Degree +He +2 +9 − 10 +51 +Suz +2 +12 − 13 +110 +Fi22 +2, 3 +14 − 17, 8 − 9 +78, 77 +Fi23 +2 +18 +782 +Co2 +2 +12 − 18 +22 +Co1 +2 +16 − 21 +24 +B +2 +23 − 41 +4370 +The final column in the table above gives the minimal degree of a faithful representation of the group H +over a finite field of characteristic p [15]. As this minimal degree is larger than the largest possible n in each +case, we deduce that H cannot be isomorphic to a subgroup of GL(n, p). Hence, we have a contradiction for +any sporadic simple group H. Thus, N = 1 and G ∼= H. +□ +4. Acknowledgements +This research was conducted under NSF-REU grant DMS-1757233, DMS-2150205 and NSA grant H98230- +21-1-0333, H98230-22-1-0022 by Dolorfino, Martin, Slonim, and Sun during the Summer of 2022 under the +supervision of Yang. The authors gratefully acknowledge the financial support of NSF and NSA, and also +thank Texas State University for providing a great working environment and support. Yang was also par- +tially supported by grants from the Simons Foundation (#499532, #918096, YY). +Competing interests The authors declare none. +Data availability Statement: Data sharing not applicable to this article as no datasets were generated +or analysed during the current study. +References +[1] N. Ahanjideh, Nondivisibility among irreducible character co-degrees. Bull. Aust. Math. Soc., 105 (2022), no. 1, 68-74. +[2] F. Alizadeh, H. Behravesh, M. Ghaffarzadeh, M. Ghasemi, S. Hekmatara, Groups with few codegrees of irreducible char- +acters. Comm. Algebra, 47 (2019), no. 3, 1147-1152. +[3] K. Aziziheris, F. Shafiei, F. Shirjian, Simple groups with few irreducible character degrees. J. Algebra Appl., 20 (2021), +no. 8, 2150139. +[4] A. Bahri, Z. Akhlaghi, B. Khosravi, An analogue of Huppert’s conjecture for character codegrees. Bull. Aust. Math. Soc., +104 (2021), no. 2, 278-286. +[5] J. Bezanson, S. Karpinski, V. B. Shah, A. Edelman, Julia: A fast dynamic language for technical computing. ArXiv +Preprint, ArXiv:1209.5145. +[6] R. W. Carter, Simple Groups of Lie Type. Wiley, 1989. +[7] D. Chillag and M. Herzog, On character degrees quotients. Arch. Math., 55 (1990), no. 2, 25-29. +[8] J. H. Conway et. al, Atlas of Finite Groups. Oxford Clarendon Press, 1985. +[9] J. D. Dixon and B. Mortimer, Permutation Groups. Spring, 1996. +[10] M. Gintz, M. Kortje, M. laurence, Y. Liu, Z. Wang, Y. Yang, On the characterization of some nonabelian simple groups +with few codegrees. Comm. Algebra, 50 (2022), no. 9, 3932-3939. +[11] H. Guan, X. Zhang, Y. Yang, Recognizing Ree groups +2G2(q) using the codegree set. Bull. Aust. Math. Soc., +https://www.doi.org/10.1017/S0004972722001022. +[12] N. N. Hung, Group pseudo-algebras of finite simple groups. In progress +[13] I. M. Isaacs, Character Theory of Finite Groups. New York Academic Press, 1976. +[14] G. James and A. Kerber, The Representation Theory of the Symmetric Group. Addison-Wesley Publishing Company, 1981. +[15] C. Jansen, The minimal degrees of faithful representations of the sporadic simple groups and their covering groups. LMS +J. Comput. Math., 8 (2005), 122-144. +[16] E. I. Khukrho and V. D. Mazurov, Unsolved Problems in Group Theory. The Kourovka Notebook. No. 20. Russian Academy +of Sciences, 2022. +[17] Y. Liu, Y. Yang, Huppert’s analogue conjecture for PSL(3, q) and PSU(3, q). Results Math., 78 (2023), Article number: 7. +[18] A. Moreto, Complex group algebra of finite groups: Brauer’s problem 1. Adv. Math., 208 (2007), 236-248. +[19] G. Qian, Y. Wang, H. Wei, Co-degrees of irreducible characters in finite groups. J. Algebra, 312 (2007), no. 2, 946-955. +5 + +Mallory Dolorfino, Kalamazoo College, Kalamazoo, Michigan, USA, mallory.dolorfino19@kzoo.edu +Luke Martin, Gonzaga University, Spokane, Washington, USA, lwmartin2019@gmail.com +Zachary Slonim, University of California, Berkeley, Berkeley, California, USA, zachslonim@berkeley.edu +Yuxuan Sun, Haverford College, Haverford, Pennsylvania, USA, ysun1@haverford.edu +Yong Yang, Texas State University, San Marcos, Texas, USA, yang@txstate.edu +6 + diff --git a/ydE0T4oBgHgl3EQfcgDn/content/tmp_files/load_file.txt b/ydE0T4oBgHgl3EQfcgDn/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5b02c0058fe184b95068b4f083d92a6cb62e06c2 --- /dev/null +++ b/ydE0T4oBgHgl3EQfcgDn/content/tmp_files/load_file.txt @@ -0,0 +1,376 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf,len=375 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content='02365v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content='GR] 6 Jan 2023 ON THE CHARACTERIZATION OF SPORADIC SIMPLE GROUPS BY CODEGREES MALLORY DOLORFINO, LUKE MARTIN, ZACHARY SLONIM, YUXUAN SUN, AND YONG YANG Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Let G be a finite group and Irr(G) the set of all irreducible complex characters of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Define the codegree of χ ∈ Irr(G) as cod(χ) := |G:ker(χ)| χ(1) and denote by cod(G) := {cod(χ)|χ ∈ Irr(G)} the codegree set of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Let H be one of the 26 sporadic simple groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' In this paper, we show that H is determined up to isomorphism by cod(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Introduction Let G be a finite group and Irr(G) the set of all irreducible complex characters of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' For any χ ∈ Irr(G), define the codegree of χ, denoted by cod(χ), as cod(χ) := |G:ker(χ)| χ(1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Then define the codegree set of G as cod(G) := {cod(χ)|χ ∈ Irr(G)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' The concept of codegrees was originally considered in [7], where the codegree was defined as cod(χ) := |G| χ(1), and it was later modified to its current definition by [19] so that cod(χ) is the same for G and G/N when N ≤ ker(χ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Several properties of codegrees have been studied, such as the relationship between the codegrees and element orders, codegrees of p-groups, and groups with few codegrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' The codegree set of a group is closely related to the character degree set of a group, defined as cd(G) := {χ(1)|χ ∈ Irr(G)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' The relationship between the character degree set and the group structure is an active area of research, and many properties of group structure are largely determined by the character degree set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' In 1990, Bertram Huppert studied the character degrees of simple groups and made the following conjecture about the relationship between a simple group H and a finite group G that have equal character degree sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Huppert’s Conjecture: Let H be a finite nonabelian simple group and G a finite group such that cd(H) = cd(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Then G ∼= H × A, where A is an abelian group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Huppert’s conjecture has since been verified for many cases such as the alternating groups, sporadic groups, and simple groups of Lie type with low rank, but it has yet to be verified for simple groups of Lie type with high rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Recently, a similar conjecture related to codegrees has been posed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Codegree Version of Huppert’s Conjecture: Let H be a finite nonabelian simple group and G a finite group such that cod(H) = cod(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Then G ∼= H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' This conjecture appears in the Kourovka Notebook of Unsolved Problems in Group Theory as question 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content='79 [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' It has been verified for PSL(2, q), PSL(3, 4), Alt7, J1, 2B2(22f+1) where f ≥ 1, M11, M12, M22, M23 and PSL(3, 3) by [1, 4, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' The conjecture has also been verified for PSL(3, q) and PSU(3, q) in [17] and 2G2(q) in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Most of these results concern simple groups with less than 21 character degrees [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' However, in this paper, we provide a general proof verifying this conjecture for all sporadic simple groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' The methods used may be generalized to simple groups of Lie type, giving promising results for characterizing all simple groups by their codegree sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Let H be a sporadic simple group and G a finite group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' If cod(G) = cod(H), then G ∼= H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Throughout the paper, we follow the notation used in Isaac’s Book [13] and the ATLAS of Finite Groups [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Preliminary Results We first reproduce some lemmas which will be used in later proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' [18, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content='2] Let S be a finite nonabelian simple group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Then there exists 1S ̸= χ ∈ Irr(S) that extends to Aut(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' 2000 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' 20C15, 20D08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' 1 Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' [14, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content='34] Let N be a minimal normal subgroup of G such that N = S1 × · · · × St where Si ∼= S is a nonabelian simple group for each i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' , t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' If χ ∈ Irr(S) extends to Aut(S), then χ × · · · × χ ∈ Irr(N) extends to G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' [10, Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content='6] Let G be a finite group and H a finite nonabelian simple group with cod(G) = cod(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Then G is a perfect group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' [12] Let G be a finite group and S a finite nonabelian simple group such that cod(S) ⊆ cod(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Then |S| divides |G|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Let G be a finite group with N ⊴ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Then cod(G/N) ⊆ cod(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' From [13, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content='22], we can define Irr(G/N) = {ˆχ(gN) = χ(g) | χ ∈ Irr(G) and N ⊆ ker(χ)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Take any ˆχ ∈ Irr(G/N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' By definition, we know that ˆχ(1) = χ(1), so the denominators of cod(ˆχ) and cod(χ) are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' In addition, ker(ˆχ) ∼= ker(χ)/N, so |ker(χ)| = |N| · |ker(ˆχ)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Thus |G/N : ker(ˆχ)| = |G|/|N| | ker(χ)|/|N| = |G| | ker(χ)|, so cod(ˆχ) = cod(χ) and therefore cod(G/N) ⊆ cod(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' □ Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Let G be a finite group with normal subgroups N and M such that N ≤ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Then, cod(G/M) ⊆ cod(G/N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' By the Third Isomorphism Theorem, we know that G/M ∼= (G/N)/(M/N) is a quotient of G/N, and by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content='5, cod(G/M) ⊆ cod(G/N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' □ Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Let G and H be finite groups such that cod(G) ⊆ cod(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Then there are at least |cod(G)| elements in cod(H) which divide |G|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Let x ∈ cod(G), it is clear that x divides |G|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Since this is true for each x, the lemma follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Main Results Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Let H be a sporadic simple group and G a finite group with cod(G) = cod(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' If N is a maximal normal subgroup of G, then G/N ∼= H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content='3, G is perfect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Thus G/N is a nonabelian simple group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content='6, we have cod(G/N) ⊆ cod(G) = cod(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' If cod(G/N) = cod(G) = cod(H), we have G/N ∼= H, since no two non- isomorphic finite simple groups have equal codegree sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Therefore, it is sufficient to show that cod(G/N) = cod(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' We can easily check that cod(K) ̸⊆ cod(H) for any two non-isomorphic sporadic groups H and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Thus G/N must belong to one of the 17 infinite families of nonabelian simple groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' We compute the orders of these simple groups using the formulas given in [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' For each sporadic group H, we check all of the possibilities for G/N and restrict only to those which satisfy Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' For example, let H ∼= M (where M denotes the Monster group).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Then, we check through all 17 families of nonabelian simple groups and return those whose orders divide |H|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' The result is given in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' The number in the third column represents the maximal prime power q = pk which satisfies this condition (all smaller prime powers except those specified as being excluded also satisfy the condition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' 2 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Possibilities for G/N given H ∼= M after applying Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content='4: |G/N| | |H|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' G/N n Max q = pk An 5-32 n/a 1 34, excl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' 24, 25, 43, 26 2 52, excl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' 11, 13, 24, 19, 23 Ln+1(q) 3 32 4 22 5 22 2 32 3 5 O2n+1(q) 4 3 5 2 6 2 3 5 S2n(q) 4 3 5 2 6 2 G/N n Max q = pk 4 3 O+ 2n(q) 5 3 6 2 2 23 Un+1(q) 3 3 4 2 5 2 O− 2n(q) 5 2 6 2 2E6(q) n/a 2 3D4(q) n/a 2 Sz(q) n/a q = 23, 25 2F4(2)′ n/a n/a Now, we check each group in this table to see if it satisfies Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content='7, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' we check how many elements in the codegree set of H divide the order of G/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' For H ∼= M, we find that none of the possible groups G/N have order divisible by more than 3 of the codegrees of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' This contradicts [2], which shows that for any nonabelian simple group, |cod(G/N)| > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Thus, if H ∼= M and cod(G/N) ⊆ cod(H), then cod(G/N) = cod(H) and G/N ∼= H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' We repeat this process for all of the other sporadic simple groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' For each sporadic group, H, we first check which nonabelian simple groups, G/N, satisfy |G/N| divides |H|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Second we check which of these possibilities order divisible by more than 3 codegrees of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' We find two groups H such that the number of codegrees of H dividing |G/N| is more than 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' These are H ∼= Suz with G/N ∼= O+ 8 (2) and H ∼= Fi23 with G/N ∼= O+ 8 (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' We find 5 and 4 such codegrees respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' For each of these cases, however, [3] shows that |cod(G/N)| > 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Thus by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content='7, we cannot have cod(G/N) ⊆ cod(H) if G/N ̸∼= H and thus G/N ∼= H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' □ Now we present the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Let G be a minimal counterexample and N a maximal normal subgroup of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content='3, G is perfect, and by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content='1, G/N ∼= H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' In particular, N ̸= 1 as G ̸∼= H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Step 1: N is a minimal normal subgroup of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Suppose L is a nontrivial normal subgroup of G with L < N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Then by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content='6, we have cod(G/N) ⊆ cod(G/L) ⊆ cod(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' However, cod(G/N) = cod(H) = cod(G) so equality must be obtained in each inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Thus, cod(G/L) = cod(H) which implies that G/L ∼= H, since G is a minimal counterexample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' This is a contradiction since we also have G/N ∼= H, but L < N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Step 2: N is the only nontrivial, proper normal subgroup of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Otherwise we assume U is another proper nontrivial normal subgroup of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' If N is included in U, then U = N or U = G since G/N is simple, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Then N ∩ U = 1 and G = N × U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Since U is also a maximal normal subgroup of G, we have N ∼= U ∼= H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Choose ψ1 ∈ Irr(N) and ψ2 ∈ Irr(U) such that cod(ψ1) = cod(ψ2) = max(cod(H)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Set χ = ψ1 · ψ2 ∈ Irr(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Then cod(χ) = (max(cod(H)))2 /∈ cod(G), a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Step 3: For each nontrivial χ ∈ Irr(G|N) := Irr(G) − Irr(G/N), χ is faithful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' By [13, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content='22], we have that Irr(G/N) = {χ ∈ Irr(G)|N ≤ ker(χ)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Then it follows by the definition of Irr(G|N) that if χ ∈ Irr(G|N), N ̸≤ ker(χ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Thus since N is the unique nontrivial, proper, normal subgroup of G, ker(χ) = G or ker(χ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Therefore, ker(χ) = 1 for all nontrivial χ ∈ Irr(G|N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Step 4: N is an elementary abelian group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' 3 Suppose that N is not abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Since N is a minimal normal subgroup, by [9, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content='3A (iii)], N = Sn where S is a nonabelian simple group and n ∈ Z+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' By Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content='1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content='2, there is a non-trivial character χ ∈ Irr(N) which extends to some ψ ∈ Irr(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Now, ker(ψ) = 1 by Step 3, so cod(ψ) = |G|/ψ(1) = |G/N| · |N|/χ(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' This contradicts the fact that |G/N| is divisible by cod(ψ), as χ(1) < |N|, so N must be abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Now to show that N is elementary abelian, let a prime p divide |N|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Then N has a p-Sylow subgroup K, and K is the unique p-Sylow subgroup of N since N is abelian, so K is characteristic in N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Thus, K is a normal subgroup of G, so K = N as N is minimal, so |N| = pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Now, take the subgroup N p = {np : n ∈ N} of N, which is proper by Cauchy’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Then since N p is characteristic in N, it must be normal in G, so N p is trivial by the uniqueness of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Therefore, every element of N has order p, so N is elementary abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Step 5: CG(N) = N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' First note that since N is normal, CG(N) ⊴ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Additionally, since N is abelian by Step 4, N ≤ CG(N), so by the maximality of N, we must have CG(N) = N or CG(N) = G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' If CG(N) = N, we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' If not, then CG(N) = G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Therefore N must be in the center of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Then since N is the unique minimal normal subgroup of G by Step 2, we must have that |N| is prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' If not, there always exists a proper non-trivial subgroup K of N, and K is normal since it is contained in Z(G), contradicting the minimality of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Moreover, since G is perfect, we have that Z(G) = N, and N is isomorphic to a subgroup of the Schur multiplier of G/N [13, Corollary 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content='20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' If H is isomorphic to any of M11, M23, M24, J1, J4, Co2, Co3, Fi22, Fi23, He, HN, Ly, Th, or M, then by [8], the Schur multiplier of H is trivial, so N = 1, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' If H is isomorphic to Co1, then G ∼= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content='Co1 by [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' But 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content='Co1 has a character degree of 24, which gives a codegree of 219 ·38 ·54 ·72 ·11·13·23 ∈ cod(G), a contradiction, since 219 ·38 ·54 ·72 ·11·13·23 /∈ cod(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' If H is isomorphic to Fi22, then G ∼= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content='Fi22 or G ∼= 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content='Fi22 by [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' If G ∼= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content='Fi22, then 213 · 39 · 52 · 7 · 13 ∈ cod(G), a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' If G ∼= 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content='Fi22, then 217 · 37 · 52 · 6 · 11 ∈ cod(G), a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Similarly, for any sporadic simple group H with non-trivial Schur multiplier, we use [8] to reach a contra- diction as we did above, by finding an element of cod(G) that is not in cod(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Thus CG(N) = N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Step 6: Let λ be a non-trivial character in Irr(N) and ϑ ∈ Irr(IG(λ)|λ), the set of irreducible constituents of λIG(λ), where IG(λ) is the inertia group of λ ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Then |IG(λ)| ϑ(1) ∈ cod(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Also, ϑ(1) divides |IG(λ)/N|, and |N| divides |G/N|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Lastly, IG(λ) < G, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' λ is not G-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Let λ be a non-trivial character in Irr(N) and ϑ ∈ Irr(IG(λ)|λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Let χ be an irreducible constituent of ϑG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' By [13, Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content='4], we know χ ∈ Irr(G), and by [13, Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content='1], we have χ(1) = |G| |IG(λ)| · ϑ(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Moreover, we know tat ker(χ) = 1 by Step 2, and thus cod(χ) = |G| χ(1) = |IG(λ)| ϑ(1) , so |IG(λ)| ϑ(1) ∈ cod(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Now, since N is abelian, λ(1) = 1, so we have ϑ(1) = ϑ(1)/λ(1) which divides |IG(λ)| |N| , so |N| divides |IG(λ)| ϑ(1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Moreover, we know that cod(G) = cod(G/N), and all elements in cod(G/N) divide |G/N|, so |N| divides |G/N|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Next, we want to show IG(λ) is a proper subgroup of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' To reach a contradiction, assume IG(λ) = G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Then ker(λ) ⊴ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' From Step 2, we know ker(λ) = 1, and from Step 4, we know N is a cyclic group of prime order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Thus by the Normalizer-Centralizer theorem, we have G/N = NG(N)/CG(N) ≤ Aut(N) so G/N is abelian, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Step 7: Final contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' From Step 4, N is an elementary abelian group of order pn for some prime p and integer n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' By the Normalizer-Centralizer theorem, H ∼= G/N = NG(N)/CG(N) ≤ Aut(N) and n > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Note that in general, Aut(N) = GL(n, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' By Step 6, |N| divides |G/N|, so we only need to consider primes p such that p2 divides |H|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' For instance, if H ∼= M11, |H| = 24 · 32 · 5 · 11 so |N| = 22, 23, 24 or 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Then, we can compute |Aut(N)| = |GL(n, p)| which is equal to 6, 168, 20160, and 48 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' In each of these four cases, |H| ∤ |Aut(N)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' For each sporadic group H, we follow a similar procedure to computationally check which possibilities of (p, n) satisfy pn divides |H| and |H| divides |GL(n, p)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' We find only the following 7 possible H which satisfy this condition, listed in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' In other words, Table 2 gives all groups H and pairs (p, n) such that pn divides |H| and |H| divides |GL(n, p)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' 4 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Minimum degree of faithful representations of sporadic groups over Fpk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Group p n Minimum Degree He 2 9 − 10 51 Suz 2 12 − 13 110 Fi22 2, 3 14 − 17, 8 − 9 78, 77 Fi23 2 18 782 Co2 2 12 − 18 22 Co1 2 16 − 21 24 B 2 23 − 41 4370 The final column in the table above gives the minimal degree of a faithful representation of the group H over a finite field of characteristic p [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' As this minimal degree is larger than the largest possible n in each case, we deduce that H cannot be isomorphic to a subgroup of GL(n, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Hence, we have a contradiction for any sporadic simple group H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Thus, N = 1 and G ∼= H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Acknowledgements This research was conducted under NSF-REU grant DMS-1757233, DMS-2150205 and NSA grant H98230- 21-1-0333, H98230-22-1-0022 by Dolorfino, Martin, Slonim, and Sun during the Summer of 2022 under the supervision of Yang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' The authors gratefully acknowledge the financial support of NSF and NSA, and also thank Texas State University for providing a great working environment and support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Yang was also par- tially supported by grants from the Simons Foundation (#499532, #918096, YY).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Competing interests The authors declare none.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Data availability Statement: Data sharing not applicable to this article as no datasets were generated or analysed during the current study.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' Algebra, 312 (2007), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' 2, 946-955.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content=' 5 Mallory Dolorfino, Kalamazoo College, Kalamazoo, Michigan, USA, mallory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content='dolorfino19@kzoo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content='edu Luke Martin, Gonzaga University, Spokane, Washington, USA, lwmartin2019@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content='com Zachary Slonim, University of California, Berkeley, Berkeley, California, USA, zachslonim@berkeley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content='edu Yuxuan Sun, Haverford College, Haverford, Pennsylvania, USA, ysun1@haverford.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content='edu Yong Yang, Texas State University, San Marcos, Texas, USA, yang@txstate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} +page_content='edu 6' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQfcgDn/content/2301.02365v1.pdf'} diff --git a/ydE2T4oBgHgl3EQf3wjf/content/tmp_files/2301.04175v1.pdf.txt b/ydE2T4oBgHgl3EQf3wjf/content/tmp_files/2301.04175v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..2c6a8056b8c7da74a092da49fb135b9313a6fb4d --- /dev/null +++ b/ydE2T4oBgHgl3EQf3wjf/content/tmp_files/2301.04175v1.pdf.txt @@ -0,0 +1,1138 @@ +The magnetic field driven superconductor–metal +transition in disordered hole-overdoped cuprates +Lina G. Johnsen +Center for Quantum Spintronics, Department of Physics, +Norwegian University of Science and Technology, NO-7491 Trondheim, Norway +Abstract. +By solving the Bogoliubov–de Gennes equations for a d-wave supercon- +ductor, we explore how the interplay between disorder and the orbital depairing of +an external magnetic field influences the superconductor–metal transition of the hole- +overdoped cuprates. For highly disordered systems, we find granular Cooper paring to +persist above the critical field where the superfluid stiffness goes to zero. We also show +that because the vortices are attracted to regions where the superconducting pairing +is already weak, the Caroli–de Gennes–Matricon zero-bias peak in the local density of +states at the vortex cores disappears already at moderate disorder. +Keywords: Superconducting phase transition, disorder, critical field, hole-overdoped +cuprates, Bogoliubov–de Gennes equations +arXiv:2301.04175v1 [cond-mat.supr-con] 10 Jan 2023 + +The magnetic field driven superconductor–metal transition ... +2 +1. Introduction +The rich phase diagram of the cuprates allow for the study of a variety of different +phases by adjusting the concentration of dopants [1, 2]. While undoped cuprates are +antiferromagnetic Mott insulators, a sufficient level of doping can cause the onset of +high-temperature superconductivity [3]. In the hole-underdoped and optimally doped +regimes, superconductivity can no longer be described by Bardeen-Cooper-Schrieffer +(BCS) theory [4]. In these regimes, the normal state is not a Fermi-liquid, but rather +a pseudogap phase or a strange metal [2, 5, 6, 7]. As the temperature is decreased, +the onset of superconductivity is determined by the superfluid stiffness rather than +the Cooper pairing [8]. +Moreover, the competition between superconducting and +antiferromagnetic order causes magnetic structures to arise around impurities [9, 10, 11] +and vortex cores [12, 13, 14, 15, 16, 17]. +In the less studied hole-overdoped regime, a large Fermi surface with well defined +quasi-particles makes Fermi-liquid theory a more suitable description of the normal- +state [18, 19]. In the superconducting state, experimental observations fit better with +BCS theory [2, 20]. However, some puzzling observations include Cooper pairs forming +above the superconducting critical temperature (Tc) in Bi2Sr2CaCu2O8+x (Bi2212) +[21, 22, 23], and a large fraction of uncondenced electrons below Tc in several of +the hole-overdoped cuprates [24, 25, 26, 27, 28]. +Recent experimental studies of +La2−xSrxCuO4 (LSCO) [27, 29, 30, 31] found that the superfluid stiffness decreases +linearly with increasing temperature, and that the critical temperature depends on the +zero-temperature superfluid stiffness. The claim that these findings go beyond BCS +theory has been challenged by theoretical works [32, 33, 34]. +No consensus has yet +been reached, but it is clear that the relation between the superconducting pairing and +superfluid stiffness shows interesting properties even when described within the dirty +BCS framework. +As a prime example, Li et al. [35] predicted that upon increasing the concentration +of non-magnetic impurities in hole-overdoped Bi2212, the superfluid stiffness is lost, +and the superconductor transitions into a state with granular Cooper pairing and +spontaneous supercurrent loops. +Since the hole-overdoped cuprates are d-wave +superconductors, they are not protected by Anderson’s theorem [36, 37, 38] and +superconductivity is strongly suppressed in the vicinity of non-magnetic impurities +[39, 40]. +In Refs. [23, 35] modelling Bi2212, the authors in particular highlight the +importance of flat bands in the anti-nodal regions causing increased pair-breaking and +the transition into the granular state. +In this work, we study how the interplay between disorder and the orbital depairing +of an external magnetic field influences the superconductor–metal transition of the +hole-overdoped cuprates. By solving the Bogoliubov–de Gennes (BdG) equations for a +disordered d-wave superconductor, we study the superfluid stiffness and superconducting +pairing close to the transition. Like Li et al. [35], we consider a band-structure fitting +experimental measurements of Bi2212 [23] with a band-filling that gives rise to flat + +The magnetic field driven superconductor–metal transition ... +3 +bands close to the Fermi level in the antinodal regions. We find that when the system +becomes sufficiently disordered, granular Cooper pairing persists beyond the magnetic +field driven superconductor to metal transition. This allows us to conveniently reach the +intermediate regime predicted for the disorder driven superconductor–metal transition +in Ref. [35] by tuning an external magnetic field. Moreover, we show that the Caroli– +de Gennes–Matricon (CdGM) zero-bias peak in the local density of states (LDOS) +at the vortex cores vanishes at moderate disorder, as the vortices start penetrating +regions where the superconductivity is already weak. This sensitivity to disorder could +contribute to the elusiveness of the CdGM zero-bias peak in experimental studies +[41, 42, 43, 44, 45]. +2. Model +We consider a disordered type-II d-wave superconductor under an applied magnetic +field. The lattice structure considered is a two-dimensional (2D) square lattice, which +to good approximation models the quasi-2D structure of the cuprates [1]. Moreover, +we assume the superconducting film to be thin enough that the orbital effect of the +perpendicular magnetic field dominates over the Zeeman splitting. This system can be +described by the Hamiltonian +H = − +� +i,j,σ +ti,jeiφi,jc† +i,σcj,σ − +� +i,σ +(µ − Vi) ni,σ + +� +⟨i,j⟩ +� +∆i,jc† +i,↑c† +j,↓ + h.c. +� +. +(1) +Here, c† +i,σ, ci,σ, and ni,σ = c† +i,σci,σ are the creation, annihilation, and number operators +associated with a spin-σ electron at lattice site i. Each of the above terms are explained +in the following. +The first term describes hopping between neighboring lattice sites. +We include +hopping between nearest, next nearest and third nearest neighbors. These three types +of hopping are associated with hopping parameters ti,j = t, t′, t′′, respectively. The +applied magnetic field introduces an accumulated Peierls phase +φi,j = − π +ΦSC +0 +� ri +rj +dr · A(r) +(2) +when an electron moves from position rj to position ri. Here, ΦSC +0 += hc/2e is the +superconducting flux quantum, and A(r) = B(0, x, 0) is the vector potential in the +Landau gauge resulting from a homogeneous external magnetic field B. +The second term in Eq. (1) introduces the chemical potential µ and the disorder +potential Vi. We consider a random disorder potential in the range Vi ∈ [−V, V ]. The +chemical potential is adjusted in order to fix the hole density x while considering different +disorder strengths. The hole density is given by +x = +1 +NxNy +� +i,σ +� +1 − ni,σ +� +. +(3) + +The magnetic field driven superconductor–metal transition ... +4 +We consider hole-doped superconductors (0 < x ≤ 1) far away from half-filling +(x = 0). In this regime, the cuprates are purely superconducting without competing +antiferromagnetic order. +Experimental observations suggest a more conventional +behavior, where BCS theory captures many aspects of the superconductivity well [1, 2]. +The last term in Eq. (1) introduces the superconducting pairing arising from +a nearest-neighbor interaction described within the mean field approximation [46]. +The pairing correlation ∆i,j = J ⟨ci,↑cj,↓⟩ is used to calculate the spin-singlet pairing +∆S +i,j = (∆i,j + ∆j,i)/2. The d-wave spin-singlet paring is defined as +∆d +i = 1 +4 +� +∆+x +i ++ ∆−x +i +− ∆+y +i +− ∆−y +i +� +, +(4) +where ∆±x(y) +i += ∆i,i±x(y) exp +� +iφi,i±x(y) +� +. In order to make the numerical calculations +feasible, we need to scale down the lattice size compared to a realistic system. The +parameter J should ideally be chosen large enough that the vortex diameter is much +smaller than the width of the system, but still small enough that the vortex spans at least +a few lattice sites. The parameters chosen for each plot is given in the corresponding +figure text. We consider the zero-temperature limit as our theoretical framework do not +capture the effect of thermal fluctuations. +We calculate the spin-singlet d-wave pairing self-consistently from the Bogoliubov- +de Gennes equations of the Hamiltonian in Eq. (1). We follow the method in Ref. [46]. +For details, see the Appendix. In order to reduce the system size without disturbance +from edge effects, we define a magnetic unit cell containing an even number of +superconducting flux quanta and apply periodic boundary conditions at its edges. Thus, +we can solve the BdG equations for a periodic array of Mx × My magnetic unit cells of +size Nx × Ny. +In highly disordered materials, the existence of superconducting pairing is no longer +a good measure for whether the material is superconducting. This is because paring +can exist locally without any global phase coherence. For defining the superconducting +phase transition, we therefore introduce the superfluid stiffness Ds. We calculate the +superfluid stiffness from the Kubo formula [47, 48, 46] +Ds +πe2 = ⟨−Kx⟩ − Λxx(qx = 0, qy → 0, ω = 0), +(5) +that describes the linear response to a vector potential Axei(q·ri−ωt) applied in the x +direction. Above, ⟨Kx⟩ is the expectation value of the kinetic energy and Λxx(q, ω) +is the current-current correlation function. The kinetic energy associated with the x +oriented bonds is given by +Kx = − +1 +NxNy +� +i,δ,σ +δ2 +x +� +ti+δ,ieiφi+δ,ic† +i+δ,σci,σ + h.c. +� +. +(6) +The current-current correlation function is given by +Λxx(q, ω) = +i +NxNy +� ∞ +0 +dt eiωt ⟨[Jx(q, t), Jx(−q, 0)]⟩ , +(7) + +The magnetic field driven superconductor–metal transition ... +5 +where Jx(q, t) = � +i exp(−iq · ri)Jx(ri, t) is the Fourier transform of the x oriented +particle current +J(ri, t) = i +� +δ,σ +δx +� +ti+δ,ieiφi+δ,ic† +i+δ,σci,σ − h.c. +� +. +(8) +With these physical quantities, we are able to study whether superconducting pairing +is present, whether the material is superconducting, and how currents flow inside the +material. Finally, we define the local density of states in terms of the retarded Green’s +function [46] +Ni = − 1 +π +� +σ +ℑm +� +GR +i,σ,i,σ(ω) +� +, +(9) +GR +i,α,j,β(ω) = −i +� ∞ +0 +dt eiωt�� +ci,α(t), c† +j,β(0) +�� +. +(10) +This furthermore allows us to study the local density of states inside the vortex cores. +3. Results +In order to study how the strong pair breaking associated with the d-wave pairing +symmetry affects the magnetic field driven superconducting transition, we choose +parameters modelling Bi2212 at 22% hole doping. As shown in Fig. 1, this band filling +gives rise to flat bands close to the Fermi surface in the antinodal regions, causing +increased scattering between regions where the superconducting pairing has opposite +signs [23]. For the case of zero external magnetic field, such flat bands have been shown +to cause a strong suppression of the superconducting pairing and superfluid stiffness +under increasing disorder [35]. +By choosing parameters giving a high sensitivity to +disorder, our parameters allow us to study the opposite limit compared to the robust +conventional superconductor studied in Ref. [49]. +We first consider the how the superconducting pairing evolves under increasing +disorder in the presence of a constant magnetic field. +Fig. 2(a)-(e) show the +superconducting pairing inside a magnetic unit cell penetrated by four superconducting +flux quanta for various disorder strengths. While the vortices in a clean system form a +regular lattice due to the mutual repulsion between vortices, increasing disorder causes +the vortices to shift towards highly disordered regions where the superconducting pairing +is already weak. In Fig. 2(a)-(e), we typically find one vortex where the superconducting +pairing is at its weakest, and the other three vortices in or close to local minima +sufficiently far away from other vortices. In the highly disordered systems, a vortex +can be located close to a grain boundary if the vortex repulsion makes this energetically +favorable. It is however very unlikely to find vortices in the middle of a superconducting +grain. The magnetic field therefore suppresses the Cooper pairing in the regions where +the pairing is already weak, and causes superconducting pairing to survive in grains +surrounded by regions where the pairing is absent. +We will later show that this +granularity is associated with a vanishing superfluid stiffness. + +The magnetic field driven superconductor–metal transition ... +6 +Figure +1. +We consider a normal state band structure fitting experimental +measurements of Bi2212 at 22 % hole doping [23] using next nearest and third nearest +neighbor hopping parameters t′/t = −0.05 and t′′/t = 0.2, respectively [35]. Panel a) +shows the Fermi surface (black lines), and panel b) the bandstructure along the red +line in panel a). The band structure is plotted from the Γ point to the M point and +towards the X point. The position of the Γ, M, and X points in the first Brillouin zone +are indicated by a white, yellow, and black dot, respectively. In panel b), the Fermi +level is marked by the black dotted line, and the M point is marked by the yellow +dotted line. Scattering between the antinodal regions by a wave vector q = (±π, ±′π), +as illustrated by the white arrow in panel a), is pair breaking due to the d-wave pairing +having opposite signs in the antinodal regions around (±π, 0) and (0, ±π) [50]. The +flat band shown in panel b) increases the scattering between the antinodal regions, +thus making the d-wave pairing more sensitive to impurities [51]. +In Fig. 2(f)-(j), we study how the local density of states at the vortex cores changes +as the disorder increases. When the disorder is low, the LDOS at the vortex core show +a clear Caroli–de Gennes–Matricon zero-bias peak [52]. However, the CdGM zero-bias +peak vanishes already at moderate disorder where the pairing is not yet granular in +the absence of magnetic fields. This is clearly seen from Fig. 2(c) and (h). The zero- +bias peak is nearly absent, although before applying the magnetic field there is a clear +superconducting gap in the average density of states and the superconducting paring +always remains above 40% of its value in the clean system. Since the vortices are being +attracted to regions of high disorder where the superconducting pairing is minimal, the +suppression of the zero-bias peak is determined by the disorder potential in the most +strongly disordered regions. In these regions, we see that the superconducting gap in +the LDOS in the absence of an external magnetic field is more filled up than when we +average over the whole system, see especially panel (h). The sensitivity to disorder could +be a contributing factor to the absence of the CdGM zero-bias peak in hole-overdoped +cuprates. The zero-bias peak has been observed in conventional superconductors [53] +and more recently also in the cuprate YBa2Cu3O7−δ (Y123) [54]. Observing the CdGM +zero-bias peak in cuprates have otherwise proved difficult, and experimental studies of + +(b) +(a) +X +1 +T +(元,元) +X = 0.22 +二 +q +0 +a +0 +M +E +-2 +-3 +- +I +0 +XM +-T +T +kxaThe magnetic field driven superconductor–metal transition ... +7 +Figure 2. Panel (a)-(e): The d-wave pairing ∆d +i in a magnetic unit cell containing +4 vortices for impurity potentials V/t = 0.4, 0.5, 0.6, 0.7, and 0.8, respectively. The +corresponding plots for zero magnetic field are shown below each panel. The pairing +is scaled by its value ∆d +0 in a clean system without vortices. The arrows represent the +net current through each lattice site, and the red dots mark the positions of the vortex +cores determined from the phase of ∆d +i . Panel (f)-(j): The local density of states at +the vortex cores in panel (a)-(e) (blue), and at the corresponding lattice sites for zero +magnetic field (green). The LDOS is averaged over all vortex sites and their nearest +and next nearest neighbors. The black curves show the LDOS averaged over all lattice +sites in the absence of the magnetic field for the given impurity potential (dotted) and +in a clean system (solid). (Parameters: J/t = 0.9, Nx(y) = 28, Mx(y) = 10.) + +(b) +(a) +0.8 +0.6 +0.4 +0.2 +0 +/t= 0.4 +- Nosc = 4, averaged over ○ sites +- Nosc= O, averaged over ○ sites +- Nosc = O, averaged over all sites +— No= O, and V/t= 0 +0. +0.5 +-0.5 +0. +0.5 +-0.5 +1/m +1/m +(e) +(C) +90=1/1 +V/t=0.7 +(h) +( +-0.5 +0.5 +-0.5 +-0.5 +0.5 +0.5 +1/ +1/mThe magnetic field driven superconductor–metal transition ... +8 +Bi2212 have not shown signatures of a robust zero-bias peak [41, 42, 43, 44, 45]. +In Fig. 3, we plot the superconducting pairing and superfluid stiffness as a function +of the disorder strength and the applied magnetic field. Each data point is calculated +by averaging over all lattice sites and 70 − 100 impurity configurations. +The error +bars represent the standard deviation. The standard error of the mean is 12% − 10% +of the standard deviation. Since we are considering relatively small lattice sizes, the +superconducting transition is sensitive not only to the magnetic field and the impurity +strength, but also the impurity configuration. Although the superconducting transition +is sharper for a specific impurity configuration, the superconducting transition is seen +in our plots as a gradual transition where an increasing fraction of the impurity +configurations result in zero superfluid stiffness. This is represented by the error bars +dropping down to zero and the data points gradually approaching zero. +Note that +when plotting the superconducting pairing and superfluid stiffness as a function of the +magnetic field, we will not get a purely monotonous decrease, particularly for weaker +disorder strengths. This is because not all of the field strengths can produce a square +lattice of vortices in a clean system. This error decreases with increasing system size +and disorder. +In Fig. 3(a), (b) and (c), we plot the average superconducting pairing and superfluid +stiffness as a function of disorder for a system under a constant applied magnetic field. In +panel (a) where the applied magnetic field is weak, the superconducting pairing remains +finite for all impurity strengths and configurations, while the superfluid stiffness goes to +zero for an increasing fraction of the impurity configurations as the disorder is increased. +For strong disorder, we thus find a regime of finite superconducting pairing beyond the +superconducting transition as was predicted in Ref. [35] for zero applied magnetic field. +The superconducting pairing survives in islands that grow smaller and fewer in number +as the disorder increases. Such islands survives for much higher disorder strengths than +what is presented in the figure. In panel (b), we consider an applied field that is closer +to the critical field of the superconductor. While the superfluid stiffness is still more +suppressed than the superconducting pairing, the difference is smaller than for weaker +field strengths. In panel (c), we show the average superconducting pairing as a function +of the disorder strength for a weaker pairing potential. This demonstrates that there +is a second transition where the superconducting pairing also goes to zero. However, +this transition happens for a higher disorder strength and magnetic field than what is +reasonable to consider for the band width and system size in panel (a) and (b). +In Fig. 3(d), (e) and (f) we study the average superconducting pairing and superfluid +stiffness as a function of the applied magnetic field for different disorder strengths. +Despite the non-monotonous behavior caused by the small system size, we see that the +average pairing always remain finite, while some fraction of the impurity configurations +result in zero superfluid stiffness for the higher field strengths, similar to the results +in panel (a) and (b). +These results differs qualitatively for what is expected for a +clean system, where we know that the superconducting pairing and superfluid stiffness +must go to zero simultaneously at the critical field. Instead, we find that in disordered + +The magnetic field driven superconductor–metal transition ... +9 +systems, the intermediate granular regime appears also beyond the magnetic field driven +superconducting transition. +In panel (f), where the disorder strength is high, the +superfluid stiffness starts its transition at lower field strengths than in panel (d) and (e). +This, together with the results from panel (a) and (b), indicates that the intermediate +granular regime appears at lower field strengths with increasing disorder. As shown in +Fig. 2, the vortices contribute to suppressing superconductivity in the already disordered +regions and thus makes the superconductivity granular. Once the system is granular, +the magnetic field does not punch additional holes in the superconducting condensate +and the pairing decreases very slowly as the field is increased. +It is interesting to +Figure 3. Panel (a) and (b): The superconducting pairing ∆d and superfluid stiffness +Ds under increasing disorder for a system penetrated by 4 and 24 vortices, respectively, +for pairing potential J/t = 0.9. Panel (c): The superconducting pairing for 4 vortices +for a weaker pairing potential J/t = 0.3. Panel (d), (e), and (f): The superconducting +pairing and superfluid stiffness as a function the number of vortices penetrating the +system for disorder strengths V/t = 0.4, V/t = 0.6, and V/t = 0.8 for pairing potential +J/t = 0.9. +In all panels, the superconducting pairing and superfluid stiffness are +averaged over all lattice sites and impurity configurations, and plotted with respect to +their values ∆d +0 and D0 +s in a clean system with zero external magnetic field. The insets +show a zoom in on the data in the main plots. The error bars represent the standard +deviation. (In this Figure, Nx = Ny = 28, and Mx = My = 1.) + +A'/ D, / Do +Ds / D.0 +(a) +(q) +0.8 +0.8 0.8 +0.8 +A / Ag +0.1 +0.1 +0.1 +0. +(c) +J /t= 0.3 +0.6 0.6 +0.6 +0.6 +0.06 +CH +0.04 +1.0 +1.5 +2.0 +1.0 +1.5 +0.4 0.4 +0.4 +0.4 +J /t= 0.9 +J/t= 0.9 +10.02 +Noc = 24 +0.2 +0.2 0.2 +0.2 +0.9 +1/1 +0 +0 +9 +V/t +V/t +Ad/A Ds / Do +△d /A Ds / D° +Ds/ D.0 +(d) +(e) +(f) +0.1 +0.1 +0.1 +0.1 +0.1 +0.1 +KH +0.6 +0.6 0.6 +0.6 0.6 +0.6 +OL +0.4 +0.4 0.4T +0.4 0.4 +0.4 +28 +44 +60 +12 +28 +44 +60 +12 +28 +44 +60 +J / t= 0.9 +J / t= 0.9 +J / t= 0.9 +0.2 0.2 +0.2 0.2 +0.2 +0.2 +V/t= 0.6 +V/t= 0.4 +V/t= 0.8 +0The magnetic field driven superconductor–metal transition ... +10 +note that the separation between the two transitions where the superfluid stiffness and +superconducting pairing vanishes, also found for a conventional s-wave superconductor +in Ref. [49], persists despite our conservative choice of parameters where the flat +bands makes the d-wave pairing very sensitive to impurity scattering. As a result, the +intermediate regime of remnant superconducting pairing in the superconductor–metal +transition can be conveniently studied by tuning the external magnetic field, provided +that the system is sufficiently disordered. +4. Concluding remarks +We have here provided a description of the magnetic field driven superconductor–metal +transition in the disordered hole-overdoped cuprates when described solely within the +dirty-BCS theory. We find that the CdGM zero-bias peak in the local density of states +at the vortex cores vanishes already at moderate disorder, due to the vortices being +attracted to the most disordered regions. We also show that there is an intermediate +regime with remnant superconducting pairing at the superconductor–metal transition, +which can be reached by tuning an external magnetic field. It still debated to what +extent the more unconventional nature of the cuprates needs to be taken into account +in the description of the hole-overdoped regime. While we have here studied the low- +temperature limit, it is likely that at temperatures closer to the critical temperature, +thermal fluctuations could be the dominant cause for the loss of phase coherence of +the Cooper pairs. Moreover, experiments predict a pseudogap to exist in the antinodal +regions of the Fermi surface above the superconducting critical temperature, particularly +in underdoped to weakly overdoped samples [55]. +It is unclear whether a second +pseudogap could enter the density of states also beyond the disorder and field driven +superconducting transition in the highly disordered overdoped samples considered here. +Another open question is what the exact nature of the material is at disorder and field +strengths where both the superfluid stiffness and superconducting pairing is absent. +Experimental studies suggest that the hole-overdoped cuprates are metallic rather than +insulating in the normal-state suggesting that the material could be conducting beyond +the two transitions [30]. +Although it far outside the scope of this work to resolve +this debate, we find that the superconductor–metal transition in the disordered hole- +overdoped cuprates show some interesting features even when described within the BdG +framework. +Acknowledgments +This work was supported by the Research Council of Norway through its Centres of +Excellence funding scheme, Project No. 262633 ”QuSpin”. + +The magnetic field driven superconductor–metal transition ... +11 +Appendix A. Theoretical framework +Appendix A.1. The full Bogoliubov–de Gennes equations +In order to diagonalize the Hamiltonian in Eq. (1), we solve the Bogoliubov-de Gennes +equations of the system following the approach in Refs. [46, 56]. We first define a basis +ψi = +� +ci,↑ ci,↓ c† +i,↑ c† +i,↓ +�T, +(A.1) +and write the Hamiltonian in the form +H = H0 + 1 +2 +� +i,j +ψ† +iHi,jψj. +(A.2) +Above, H0 is a constant and Hi,j is a 4 × 4 matrix. The Hamiltonian can be written in +a diagonal form +H = H0 + 1 +2 +� +n +Enγ† +nγn +(A.3) +by solving the full BdG equations +� +j +Hi,jφj,n = Enφi,n. +(A.4) +Here, En are the eigenenergies and φi,n the eigenvectors labeled by n ∈ [1, 4NxNy]. +There are seemingly twice as many fermionic operators γn compared to our original +operators ci,σ, which means pairs of the new operators must related to each other. It +can be shown that there are two equivalent solutions +En , φi,n = +� +ui,n↑ ui,n↓ vi,n↑ vi,n↓ +�T, +(A.5) +−En , φi,n = +� +v∗ +i,n↑ v∗ +i,n↓ u∗ +i,n↑ u∗ +i,n↓ +�T. +(A.6) +Since the eigenenergies of these equivalent solutions differ only by a sign, we can write +the the Hamiltonian in a diagonal form +H = H0 − 1 +2 +� +n for En>0 +En + +� +n for En>0 +Enγ† +nγn +(A.7) +including only positive eigenenergies. The old operators are related to the new ones by +ci,σ = +� +n for En>0 +� +ui,n,σγn + v∗ +i,n,σγ† +n +� +. +(A.8) +Since the operators in the diagonalized Hamiltonian are now independent, the +expectation values of the new operators can be evaluated as +� +γ† +nγm +� += fFD(En)δn,m, +(A.9) +� +γ† +nγ† +m +� += +� +γnγm +� += 0 +(A.10) +for En > 0, where fFD(En) is the Fermi-Dirac distribution. + +The magnetic field driven superconductor–metal transition ... +12 +Appendix A.2. The reduced Bogoliubov–de Gennes equations +We can simplify our calculation by realizing that in the absence of spin-orbit coupling +and spin-flip scattering, the Hamiltonian matrix contains two independent blocks [46]. +It turns out that the two independent sets of BdG equations can be written in exactly +the same form and that while one results in positive eigenenergies, the other results +in negative eigenenergies. It is therefore far more efficient to solve the reduced BdG +equations +� +j +� +ϵi,j +∆i,j +∆∗ +j,i +−ϵj,i +� � +uj,n +vj,n +� += En +� +ui,n +vi,n +� +(A.11) +for all positive and negative eigenenergies labeled by n ∈ [1, 2NxNy]. For simplicity of +notation, we have defined +ϵi,j = −ti,jeiφi,j − (µ − Vi)δi,j. +(A.12) +The diagonalized Hamiltonian can then be written in the form +H = H0 − 1 +2 +� +n +|En| + +� +n +|En|γ† +nγn, +(A.13) +where the old operators can be written in terms of new operators using the relations +ci,↑ = +� +n for En>0 +ui,nγn + +� +n for En<0 +ui,nγ† +n, +(A.14) +ci,↓ = +� +n for En>0 +v∗ +i,nγ† +n + +� +n for En<0 +v∗ +i,nγn. +(A.15) +The expectation values of the new operators are given by +� +γ† +nγm +� += fFD(|En|)δn,m, +(A.16) +� +γ† +nγ† +m +� += +� +γnγm +� += 0. +(A.17) +The reduced BdG equations in their current form is suitable for studying systems of a +finite size. However, when studying vortex formation, it is beneficial to consider a larger +systems. Therefore, we next introduce periodic boundary conditions to eliminate edge +effects. +Appendix A.3. Boundary conditions and self-consistent solution +When applying an external magnetic field perpendicular to the sample, the translational +invariance of the lattice is broken by the Peierls phase. +However, by introducing +magnetic unit cells containing an even number of superconducting flux quanta, we can +regain the translational invariance of the lattice under translation between equivalent +sites in different magnetic unit cells [46, 57, 58]. This allows us to use periodic boundary + +The magnetic field driven superconductor–metal transition ... +13 +conditions, and we can consider smaller lattice sizes without the disturbance of edge +effects. +We consider Mx × My magnetic unit cells of size Nx × Ny. A translation between +magnetic unit cells is described by a vector Rlx,ly = (lxNxa, lyNya, 0), where lx(y) ∈ +[0, Mx(y) − 1] and a is the lattice constant. By applying periodic boundary conditions +through the magnetic Bloch theorem [59], our eigenvectors and eigenenergies acquire an +index +k = +2πlx +MxNxax + +2πly +MyNyay. +(A.18) +This allows us to solve the BdG equations for a system size of Nx ×Ny for MxMy values +of k, rather than for a system of size NxMx×NyMy. We choose to absorb a k dependent +phase factor into the eigenvector so that +� +ui,n,k +vi,n,k +� += eik·ri +� +˜ui,n,k +˜vi,n,k +� +. +(A.19) +The BdG equations now take the form +� +j +eik·(rj−ri) +� +ϵi,j +∆i,j +∆∗ +j,i +−ϵj,i +� � +˜uj,n,k +˜vj,n,k +� += En,k +� +˜ui,n,k +˜vi,n,k +� +. +(A.20) +These are solved together with the self-consistency equation for the superconducting +pairing correlations +∆i,j = +U +MxMy +� +n,k +eik·(ri−rj)˜ui,n,k(˜vj,n,k)∗[1 − fFD(En,k)]. +(A.21) +Inside a magnetic unit cell, the only phase factors we need to consider are the Peierls +phases associated with electron hopping. The nonzero Peierls phases are φi±y,i = ∓πφix +for nearest neighbor hopping, φi±x±′y,i = ∓′πφ(ix ± 1/2) for next nearest neighbor +hopping, and φi±y,i = ∓2πφix for third nearest neighbor hopping. These depend on the +magnetic field through φ = NΦSC +0 /NxNy = Ba2/ΦSC +0 . When site i and j in Eq. (A.20) +and (A.21) lies in different magnetic unit cells, we need to apply the translation Rlx,ly to +one of the eigenvectors so that all eigenvalues lie in the same magnetic unit cell. Upon +such a translation, the eigenvalues pick up an additional phase through the boundary +condition +� +˜ui,n,k(ri + Rlx,ly) +˜vi,n,k(ri + Rlx,ly) +� += +� +e−iχ(ri,Rlx,ly )/2˜ui,n,k(ri) +e+iχ(ri,Rlx,ly )/2˜vi,n,k(ri) +� +. +(A.22) +The phase +χ(ri, Rlx,ly) = 2π +ΦSC +0 +A(Rlx,ly) · ri = 2πφlxNxiy. +(A.23) + +The magnetic field driven superconductor–metal transition ... +14 +is the total phase picked up by the superconducting pairing through the translation +∆d +i(ri − Rlx,ly) = ∆d +i(ri)eiχ(ri,Rlx,ly ). +(A.24) +By solving the reduced BdG equation in Eq. (A.20) together with the self-consistency +equation in Eq. (A.21) and the boundary condition in Eq. (A.22), we obtain +eigenenergies and eigenvalues that we can use to calculate physical observables. +Appendix A.4. Physical observables +We here give the expressions for the physical observables in terms of the eigenenergies +and eigenvalues. The hole concentration is given by +x = +1 +NxNyMxMy +� +i,n,k +� +1 − |˜ui,n,k|2fFD(En,k) − |˜vi,n,k|2� +1 − fFD(En,k) +�� +(A.25) +and determines the doping level. The d-wave superconducting pairing is calculated using +Eq. (4) and Eq. (A.21). The superfluid stiffness is calculated from the Kubo formula in +Eq. (5), where we insert the expectation value of the kinetic energy associated with the +x oriented bonds +⟨−Kx⟩ = +1 +NxNyMxMy +� +i,δ,n,k +δ2 +x +� +ti+δ,ieiφi+δ,i� +˜u∗ +i+δ,n,k˜ui,n,ke−ik·δfFD(En) ++ ˜v∗ +i,n,k˜vi+δ,n,keik·δ� +1 − fFD(En) +�� ++ c.c +� +(A.26) +and the current-current correlation function +Λxx(q, ω) = +1 +NxNy(MxMy)2 +� +n,k,m,k′ +fFD(En,k) − fFD(Em,k′) +ω + iδ + En,k − Em,k′ An,k,m,k′(−q)Am,k′,n,k(q). +(A.27) +Above, δ = {x, y, x ± y, 2x, 2y} and +Am,k′,n,k(q) = +� +i,δ +δxti+δ,iei(q−k′+k)·ri�� +˜u∗ +i+δ,m,k′˜ui,n,ke−ik′·δ − ˜v∗ +i,m,k′˜vi+δ,n,keik·δ� +eiφi+δ,i ++ +� +˜v∗ +i+δ,m,k′˜vi,n,ke−ik′·δ − ˜u∗ +i,m,k′˜ui+δ,n,keik·δ� +e−iφi+δ,i� +. +(A.28) +The bond currents can be obtained by multiplying Eq. (A.26) with i and reversing the +sign of the complex conjugate. In Fig. 2, we have included bond currents along all bonds +by removing the factor δ2 +x. 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Rev., 133:A1038–A1044, 1964. + diff --git a/ydE2T4oBgHgl3EQf3wjf/content/tmp_files/load_file.txt b/ydE2T4oBgHgl3EQf3wjf/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c3043d1f8ea963519a24255e23d7be05ef04e1ae --- /dev/null +++ b/ydE2T4oBgHgl3EQf3wjf/content/tmp_files/load_file.txt @@ -0,0 +1,1156 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf,len=1155 +page_content='The magnetic field driven superconductor–metal transition in disordered hole-overdoped cuprates Lina G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' Johnsen Center for Quantum Spintronics, Department of Physics, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' By solving the Bogoliubov–de Gennes equations for a d-wave supercon- ductor, we explore how the interplay between disorder and the orbital depairing of an external magnetic field influences the superconductor–metal transition of the hole- overdoped cuprates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' For highly disordered systems, we find granular Cooper paring to persist above the critical field where the superfluid stiffness goes to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' We also show that because the vortices are attracted to regions where the superconducting pairing is already weak, the Caroli–de Gennes–Matricon zero-bias peak in the local density of states at the vortex cores disappears already at moderate disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' Keywords: Superconducting phase transition, disorder, critical field, hole-overdoped cuprates, Bogoliubov–de Gennes equations arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='04175v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='supr-con] 10 Jan 2023 The magnetic field driven superconductor–metal transition .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' Introduction The rich phase diagram of the cuprates allow for the study of a variety of different phases by adjusting the concentration of dopants [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' While undoped cuprates are antiferromagnetic Mott insulators, a sufficient level of doping can cause the onset of high-temperature superconductivity [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' In the hole-underdoped and optimally doped regimes, superconductivity can no longer be described by Bardeen-Cooper-Schrieffer (BCS) theory [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' In these regimes, the normal state is not a Fermi-liquid, but rather a pseudogap phase or a strange metal [2, 5, 6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' As the temperature is decreased, the onset of superconductivity is determined by the superfluid stiffness rather than the Cooper pairing [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' Moreover, the competition between superconducting and antiferromagnetic order causes magnetic structures to arise around impurities [9, 10, 11] and vortex cores [12, 13, 14, 15, 16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' In the less studied hole-overdoped regime, a large Fermi surface with well defined quasi-particles makes Fermi-liquid theory a more suitable description of the normal- state [18, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' In the superconducting state, experimental observations fit better with BCS theory [2, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' However, some puzzling observations include Cooper pairs forming above the superconducting critical temperature (Tc) in Bi2Sr2CaCu2O8+x (Bi2212) [21, 22, 23], and a large fraction of uncondenced electrons below Tc in several of the hole-overdoped cuprates [24, 25, 26, 27, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' Recent experimental studies of La2−xSrxCuO4 (LSCO) [27, 29, 30, 31] found that the superfluid stiffness decreases linearly with increasing temperature, and that the critical temperature depends on the zero-temperature superfluid stiffness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' The claim that these findings go beyond BCS theory has been challenged by theoretical works [32, 33, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' No consensus has yet been reached, but it is clear that the relation between the superconducting pairing and superfluid stiffness shows interesting properties even when described within the dirty BCS framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' As a prime example, Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' [35] predicted that upon increasing the concentration of non-magnetic impurities in hole-overdoped Bi2212, the superfluid stiffness is lost, and the superconductor transitions into a state with granular Cooper pairing and spontaneous supercurrent loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' Since the hole-overdoped cuprates are d-wave superconductors, they are not protected by Anderson’s theorem [36, 37, 38] and superconductivity is strongly suppressed in the vicinity of non-magnetic impurities [39, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' In Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' [23, 35] modelling Bi2212, the authors in particular highlight the importance of flat bands in the anti-nodal regions causing increased pair-breaking and the transition into the granular state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' In this work, we study how the interplay between disorder and the orbital depairing of an external magnetic field influences the superconductor–metal transition of the hole-overdoped cuprates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' By solving the Bogoliubov–de Gennes (BdG) equations for a disordered d-wave superconductor, we study the superfluid stiffness and superconducting pairing close to the transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' Like Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' [35], we consider a band-structure fitting experimental measurements of Bi2212 [23] with a band-filling that gives rise to flat The magnetic field driven superconductor–metal transition .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' 3 bands close to the Fermi level in the antinodal regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' We find that when the system becomes sufficiently disordered, granular Cooper pairing persists beyond the magnetic field driven superconductor to metal transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' This allows us to conveniently reach the intermediate regime predicted for the disorder driven superconductor–metal transition in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' [35] by tuning an external magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' Moreover, we show that the Caroli– de Gennes–Matricon (CdGM) zero-bias peak in the local density of states (LDOS) at the vortex cores vanishes at moderate disorder, as the vortices start penetrating regions where the superconductivity is already weak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' This sensitivity to disorder could contribute to the elusiveness of the CdGM zero-bias peak in experimental studies [41, 42, 43, 44, 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' Model We consider a disordered type-II d-wave superconductor under an applied magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' The lattice structure considered is a two-dimensional (2D) square lattice, which to good approximation models the quasi-2D structure of the cuprates [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' Moreover, we assume the superconducting film to be thin enough that the orbital effect of the perpendicular magnetic field dominates over the Zeeman splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' This system can be described by the Hamiltonian H = − � i,j,σ ti,jeiφi,jc† i,σcj,σ − � i,σ (µ − Vi) ni,σ + � ⟨i,j⟩ � ∆i,jc† i,↑c† j,↓ + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' (1) Here, c† i,σ, ci,σ, and ni,σ = c† i,σci,σ are the creation, annihilation, and number operators associated with a spin-σ electron at lattice site i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' Each of the above terms are explained in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' The first term describes hopping between neighboring lattice sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' We include hopping between nearest, next nearest and third nearest neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' These three types of hopping are associated with hopping parameters ti,j = t, t′, t′′, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' The applied magnetic field introduces an accumulated Peierls phase φi,j = − π ΦSC 0 � ri rj dr · A(r) (2) when an electron moves from position rj to position ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' Here, ΦSC 0 = hc/2e is the superconducting flux quantum, and A(r) = B(0, x, 0) is the vector potential in the Landau gauge resulting from a homogeneous external magnetic field B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' The second term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' (1) introduces the chemical potential µ and the disorder potential Vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' We consider a random disorder potential in the range Vi ∈ [−V, V ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' The chemical potential is adjusted in order to fix the hole density x while considering different disorder strengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' The hole density is given by x = 1 NxNy � i,σ � 1 − ni,σ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' (3) The magnetic field driven superconductor–metal transition .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' 4 We consider hole-doped superconductors (0 < x ≤ 1) far away from half-filling (x = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' In this regime, the cuprates are purely superconducting without competing antiferromagnetic order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' Experimental observations suggest a more conventional behavior, where BCS theory captures many aspects of the superconductivity well [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' The last term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' (1) introduces the superconducting pairing arising from a nearest-neighbor interaction described within the mean field approximation [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' The pairing correlation ∆i,j = J ⟨ci,↑cj,↓⟩ is used to calculate the spin-singlet pairing ∆S i,j = (∆i,j + ∆j,i)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' The d-wave spin-singlet paring is defined as ∆d i = 1 4 � ∆+x i + ∆−x i − ∆+y i − ∆−y i � , (4) where ∆±x(y) i = ∆i,i±x(y) exp � iφi,i±x(y) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' In order to make the numerical calculations feasible, we need to scale down the lattice size compared to a realistic system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' The parameter J should ideally be chosen large enough that the vortex diameter is much smaller than the width of the system, but still small enough that the vortex spans at least a few lattice sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' The parameters chosen for each plot is given in the corresponding figure text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' We consider the zero-temperature limit as our theoretical framework do not capture the effect of thermal fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' We calculate the spin-singlet d-wave pairing self-consistently from the Bogoliubov- de Gennes equations of the Hamiltonian in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' We follow the method in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' For details, see the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' In order to reduce the system size without disturbance from edge effects, we define a magnetic unit cell containing an even number of superconducting flux quanta and apply periodic boundary conditions at its edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' Thus, we can solve the BdG equations for a periodic array of Mx × My magnetic unit cells of size Nx × Ny.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' In highly disordered materials, the existence of superconducting pairing is no longer a good measure for whether the material is superconducting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' This is because paring can exist locally without any global phase coherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' For defining the superconducting phase transition, we therefore introduce the superfluid stiffness Ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' We calculate the superfluid stiffness from the Kubo formula [47, 48, 46] Ds πe2 = ⟨−Kx⟩ − Λxx(qx = 0, qy → 0, ω = 0), (5) that describes the linear response to a vector potential Axei(q·ri−ωt) applied in the x direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' Above, ⟨Kx⟩ is the expectation value of the kinetic energy and Λxx(q, ω) is the current-current correlation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' The kinetic energy associated with the x oriented bonds is given by Kx = − 1 NxNy � i,δ,σ δ2 x � ti+δ,ieiφi+δ,ic† i+δ,σci,σ + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' (6) The current-current correlation function is given by Λxx(q, ω) = i NxNy � ∞ 0 dt eiωt ⟨[Jx(q, t), Jx(−q, 0)]⟩ , (7) The magnetic field driven superconductor–metal transition .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' 5 where Jx(q, t) = � i exp(−iq · ri)Jx(ri, t) is the Fourier transform of the x oriented particle current J(ri, t) = i � δ,σ δx � ti+δ,ieiφi+δ,ic† i+δ,σci,σ − h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' (8) With these physical quantities, we are able to study whether superconducting pairing is present, whether the material is superconducting, and how currents flow inside the material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' Finally, we define the local density of states in terms of the retarded Green’s function [46] Ni = − 1 π � σ ℑm � GR i,σ,i,σ(ω) � , (9) GR i,α,j,β(ω) = −i � ∞ 0 dt eiωt�� ci,α(t), c† j,β(0) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' (10) This furthermore allows us to study the local density of states inside the vortex cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' Results In order to study how the strong pair breaking associated with the d-wave pairing symmetry affects the magnetic field driven superconducting transition, we choose parameters modelling Bi2212 at 22% hole doping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' 1, this band filling gives rise to flat bands close to the Fermi surface in the antinodal regions, causing increased scattering between regions where the superconducting pairing has opposite signs [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' For the case of zero external magnetic field, such flat bands have been shown to cause a strong suppression of the superconducting pairing and superfluid stiffness under increasing disorder [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' By choosing parameters giving a high sensitivity to disorder, our parameters allow us to study the opposite limit compared to the robust conventional superconductor studied in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' We first consider the how the superconducting pairing evolves under increasing disorder in the presence of a constant magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' 2(a)-(e) show the superconducting pairing inside a magnetic unit cell penetrated by four superconducting flux quanta for various disorder strengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' While the vortices in a clean system form a regular lattice due to the mutual repulsion between vortices, increasing disorder causes the vortices to shift towards highly disordered regions where the superconducting pairing is already weak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' 2(a)-(e), we typically find one vortex where the superconducting pairing is at its weakest, and the other three vortices in or close to local minima sufficiently far away from other vortices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' In the highly disordered systems, a vortex can be located close to a grain boundary if the vortex repulsion makes this energetically favorable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' It is however very unlikely to find vortices in the middle of a superconducting grain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' The magnetic field therefore suppresses the Cooper pairing in the regions where the pairing is already weak, and causes superconducting pairing to survive in grains surrounded by regions where the pairing is absent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' We will later show that this granularity is associated with a vanishing superfluid stiffness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' The magnetic field driven superconductor–metal transition .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' 6 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' We consider a normal state band structure fitting experimental measurements of Bi2212 at 22 % hole doping [23] using next nearest and third nearest neighbor hopping parameters t′/t = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='05 and t′′/t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='2, respectively [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' Panel a) shows the Fermi surface (black lines), and panel b) the bandstructure along the red line in panel a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' The band structure is plotted from the Γ point to the M point and towards the X point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' The position of the Γ, M, and X points in the first Brillouin zone are indicated by a white, yellow, and black dot, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' In panel b), the Fermi level is marked by the black dotted line, and the M point is marked by the yellow dotted line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' Scattering between the antinodal regions by a wave vector q = (±π, ±′π), as illustrated by the white arrow in panel a), is pair breaking due to the d-wave pairing having opposite signs in the antinodal regions around (±π, 0) and (0, ±π) [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' The flat band shown in panel b) increases the scattering between the antinodal regions, thus making the d-wave pairing more sensitive to impurities [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' 2(f)-(j), we study how the local density of states at the vortex cores changes as the disorder increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' When the disorder is low, the LDOS at the vortex core show a clear Caroli–de Gennes–Matricon zero-bias peak [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' However, the CdGM zero-bias peak vanishes already at moderate disorder where the pairing is not yet granular in the absence of magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' This is clearly seen from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' 2(c) and (h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' The zero- bias peak is nearly absent, although before applying the magnetic field there is a clear superconducting gap in the average density of states and the superconducting paring always remains above 40% of its value in the clean system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' Since the vortices are being attracted to regions of high disorder where the superconducting pairing is minimal, the suppression of the zero-bias peak is determined by the disorder potential in the most strongly disordered regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' In these regions, we see that the superconducting gap in the LDOS in the absence of an external magnetic field is more filled up than when we average over the whole system, see especially panel (h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' The sensitivity to disorder could be a contributing factor to the absence of the CdGM zero-bias peak in hole-overdoped cuprates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' The zero-bias peak has been observed in conventional superconductors [53] and more recently also in the cuprate YBa2Cu3O7−δ (Y123) [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' Observing the CdGM zero-bias peak in cuprates have otherwise proved difficult, and experimental studies of (b) (a) X 1 T (元,元) X = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='22 二 q 0 a 0 M E 2 3 I 0 XM T T kxaThe magnetic field driven superconductor–metal transition .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' 7 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' Panel (a)-(e): The d-wave pairing ∆d i in a magnetic unit cell containing 4 vortices for impurity potentials V/t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='6, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='7, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='8, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' The corresponding plots for zero magnetic field are shown below each panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' The pairing is scaled by its value ∆d 0 in a clean system without vortices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' The arrows represent the net current through each lattice site, and the red dots mark the positions of the vortex cores determined from the phase of ∆d i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' Panel (f)-(j): The local density of states at the vortex cores in panel (a)-(e) (blue), and at the corresponding lattice sites for zero magnetic field (green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' The LDOS is averaged over all vortex sites and their nearest and next nearest neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' The black curves show the LDOS averaged over all lattice sites in the absence of the magnetic field for the given impurity potential (dotted) and in a clean system (solid).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' (Parameters: J/t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='9, Nx(y) = 28, Mx(y) = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=') (b) (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='2 0 /t= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='4 Nosc = 4, averaged over ○ sites Nosc= O, averaged over ○ sites Nosc = O, averaged over all sites — No= O, and V/t= 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='5 1/m 1/m (e) (C) 90=1/1 V/t=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='7 (h) ( 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='5 1/ 1/mThe magnetic field driven superconductor–metal transition .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' 8 Bi2212 have not shown signatures of a robust zero-bias peak [41, 42, 43, 44, 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' 3, we plot the superconducting pairing and superfluid stiffness as a function of the disorder strength and the applied magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' Each data point is calculated by averaging over all lattice sites and 70 − 100 impurity configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' The error bars represent the standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' The standard error of the mean is 12% − 10% of the standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' Since we are considering relatively small lattice sizes, the superconducting transition is sensitive not only to the magnetic field and the impurity strength, but also the impurity configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' Although the superconducting transition is sharper for a specific impurity configuration, the superconducting transition is seen in our plots as a gradual transition where an increasing fraction of the impurity configurations result in zero superfluid stiffness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' This is represented by the error bars dropping down to zero and the data points gradually approaching zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' Note that when plotting the superconducting pairing and superfluid stiffness as a function of the magnetic field, we will not get a purely monotonous decrease, particularly for weaker disorder strengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' This is because not all of the field strengths can produce a square lattice of vortices in a clean system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' This error decreases with increasing system size and disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' 3(a), (b) and (c), we plot the average superconducting pairing and superfluid stiffness as a function of disorder for a system under a constant applied magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' In panel (a) where the applied magnetic field is weak, the superconducting pairing remains finite for all impurity strengths and configurations, while the superfluid stiffness goes to zero for an increasing fraction of the impurity configurations as the disorder is increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' For strong disorder, we thus find a regime of finite superconducting pairing beyond the superconducting transition as was predicted in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' [35] for zero applied magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' The superconducting pairing survives in islands that grow smaller and fewer in number as the disorder increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' Such islands survives for much higher disorder strengths than what is presented in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' In panel (b), we consider an applied field that is closer to the critical field of the superconductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' While the superfluid stiffness is still more suppressed than the superconducting pairing, the difference is smaller than for weaker field strengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' In panel (c), we show the average superconducting pairing as a function of the disorder strength for a weaker pairing potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' This demonstrates that there is a second transition where the superconducting pairing also goes to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' However, this transition happens for a higher disorder strength and magnetic field than what is reasonable to consider for the band width and system size in panel (a) and (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' 3(d), (e) and (f) we study the average superconducting pairing and superfluid stiffness as a function of the applied magnetic field for different disorder strengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' Despite the non-monotonous behavior caused by the small system size, we see that the average pairing always remain finite, while some fraction of the impurity configurations result in zero superfluid stiffness for the higher field strengths, similar to the results in panel (a) and (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' These results differs qualitatively for what is expected for a clean system, where we know that the superconducting pairing and superfluid stiffness must go to zero simultaneously at the critical field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' Instead, we find that in disordered The magnetic field driven superconductor–metal transition .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' 9 systems, the intermediate granular regime appears also beyond the magnetic field driven superconducting transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' In panel (f), where the disorder strength is high, the superfluid stiffness starts its transition at lower field strengths than in panel (d) and (e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' This, together with the results from panel (a) and (b), indicates that the intermediate granular regime appears at lower field strengths with increasing disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' 2, the vortices contribute to suppressing superconductivity in the already disordered regions and thus makes the superconductivity granular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' Once the system is granular, the magnetic field does not punch additional holes in the superconducting condensate and the pairing decreases very slowly as the field is increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' It is interesting to Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' Panel (a) and (b): The superconducting pairing ∆d and superfluid stiffness Ds under increasing disorder for a system penetrated by 4 and 24 vortices, respectively, for pairing potential J/t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' Panel (c): The superconducting pairing for 4 vortices for a weaker pairing potential J/t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' Panel (d), (e), and (f): The superconducting pairing and superfluid stiffness as a function the number of vortices penetrating the system for disorder strengths V/t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='4, V/t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='6, and V/t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='8 for pairing potential J/t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' In all panels, the superconducting pairing and superfluid stiffness are averaged over all lattice sites and impurity configurations, and plotted with respect to their values ∆d 0 and D0 s in a clean system with zero external magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' The insets show a zoom in on the data in the main plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' The error bars represent the standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' (In this Figure, Nx = Ny = 28, and Mx = My = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=") A'/ D, / Do Ds / D." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='0 (a) (q) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='8 A / Ag 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' (c) J /t= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='06 CH 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='4 J /t= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='9 J/t= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='9 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='02 Noc = 24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='9 1/1 0 0 9 V/t V/t Ad/A Ds / Do △d /A Ds / D° Ds/ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='0 (d) (e) (f) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='6 OL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='4T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='4 28 44 60 12 28 44 60 12 28 44 60 J / t= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='9 J / t= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='9 J / t= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='2 V/t= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='6 V/t= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='4 V/t= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='8 0The magnetic field driven superconductor–metal transition .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' 10 note that the separation between the two transitions where the superfluid stiffness and superconducting pairing vanishes, also found for a conventional s-wave superconductor in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' [49], persists despite our conservative choice of parameters where the flat bands makes the d-wave pairing very sensitive to impurity scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' As a result, the intermediate regime of remnant superconducting pairing in the superconductor–metal transition can be conveniently studied by tuning the external magnetic field, provided that the system is sufficiently disordered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' Concluding remarks We have here provided a description of the magnetic field driven superconductor–metal transition in the disordered hole-overdoped cuprates when described solely within the dirty-BCS theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' We find that the CdGM zero-bias peak in the local density of states at the vortex cores vanishes already at moderate disorder, due to the vortices being attracted to the most disordered regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' We also show that there is an intermediate regime with remnant superconducting pairing at the superconductor–metal transition, which can be reached by tuning an external magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' It still debated to what extent the more unconventional nature of the cuprates needs to be taken into account in the description of the hole-overdoped regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' While we have here studied the low- temperature limit, it is likely that at temperatures closer to the critical temperature, thermal fluctuations could be the dominant cause for the loss of phase coherence of the Cooper pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' Moreover, experiments predict a pseudogap to exist in the antinodal regions of the Fermi surface above the superconducting critical temperature, particularly in underdoped to weakly overdoped samples [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' It is unclear whether a second pseudogap could enter the density of states also beyond the disorder and field driven superconducting transition in the highly disordered overdoped samples considered here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' Another open question is what the exact nature of the material is at disorder and field strengths where both the superfluid stiffness and superconducting pairing is absent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' Experimental studies suggest that the hole-overdoped cuprates are metallic rather than insulating in the normal-state suggesting that the material could be conducting beyond the two transitions [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' Although it far outside the scope of this work to resolve this debate, we find that the superconductor–metal transition in the disordered hole- overdoped cuprates show some interesting features even when described within the BdG framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' Acknowledgments This work was supported by the Research Council of Norway through its Centres of Excellence funding scheme, Project No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' 262633 ”QuSpin”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' The magnetic field driven superconductor–metal transition .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' 11 Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' Theoretical framework Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' The full Bogoliubov–de Gennes equations In order to diagonalize the Hamiltonian in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' (1), we solve the Bogoliubov-de Gennes equations of the system following the approach in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' [46, 56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' We first define a basis ψi = � ci,↑ ci,↓ c† i,↑ c† i,↓ �T, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='1) and write the Hamiltonian in the form H = H0 + 1 2 � i,j ψ† iHi,jψj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='2) Above, H0 is a constant and Hi,j is a 4 × 4 matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' The Hamiltonian can be written in a diagonal form H = H0 + 1 2 � n Enγ† nγn (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='3) by solving the full BdG equations � j Hi,jφj,n = Enφi,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='4) Here, En are the eigenenergies and φi,n the eigenvectors labeled by n ∈ [1, 4NxNy].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' There are seemingly twice as many fermionic operators γn compared to our original operators ci,σ, which means pairs of the new operators must related to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' It can be shown that there are two equivalent solutions En , φi,n = � ui,n↑ ui,n↓ vi,n↑ vi,n↓ �T, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='5) −En , φi,n = � v∗ i,n↑ v∗ i,n↓ u∗ i,n↑ u∗ i,n↓ �T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='6) Since the eigenenergies of these equivalent solutions differ only by a sign, we can write the the Hamiltonian in a diagonal form H = H0 − 1 2 � n for En>0 En + � n for En>0 Enγ† nγn (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='7) including only positive eigenenergies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' The old operators are related to the new ones by ci,σ = � n for En>0 � ui,n,σγn + v∗ i,n,σγ† n � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='8) Since the operators in the diagonalized Hamiltonian are now independent, the expectation values of the new operators can be evaluated as � γ† nγm � = fFD(En)δn,m, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='9) � γ† nγ† m � = � γnγm � = 0 (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='10) for En > 0, where fFD(En) is the Fermi-Dirac distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' The magnetic field driven superconductor–metal transition .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' 12 Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' The reduced Bogoliubov–de Gennes equations We can simplify our calculation by realizing that in the absence of spin-orbit coupling and spin-flip scattering, the Hamiltonian matrix contains two independent blocks [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' It turns out that the two independent sets of BdG equations can be written in exactly the same form and that while one results in positive eigenenergies, the other results in negative eigenenergies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' It is therefore far more efficient to solve the reduced BdG equations � j � ϵi,j ∆i,j ∆∗ j,i −ϵj,i � � uj,n vj,n � = En � ui,n vi,n � (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='11) for all positive and negative eigenenergies labeled by n ∈ [1, 2NxNy].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' For simplicity of notation, we have defined ϵi,j = −ti,jeiφi,j − (µ − Vi)δi,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='12) The diagonalized Hamiltonian can then be written in the form H = H0 − 1 2 � n |En| + � n |En|γ† nγn, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='13) where the old operators can be written in terms of new operators using the relations ci,↑ = � n for En>0 ui,nγn + � n for En<0 ui,nγ† n, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='14) ci,↓ = � n for En>0 v∗ i,nγ† n + � n for En<0 v∗ i,nγn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='15) The expectation values of the new operators are given by � γ† nγm � = fFD(|En|)δn,m, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='16) � γ† nγ† m � = � γnγm � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='17) The reduced BdG equations in their current form is suitable for studying systems of a finite size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' However, when studying vortex formation, it is beneficial to consider a larger systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' Therefore, we next introduce periodic boundary conditions to eliminate edge effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' Boundary conditions and self-consistent solution When applying an external magnetic field perpendicular to the sample, the translational invariance of the lattice is broken by the Peierls phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' However, by introducing magnetic unit cells containing an even number of superconducting flux quanta, we can regain the translational invariance of the lattice under translation between equivalent sites in different magnetic unit cells [46, 57, 58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' This allows us to use periodic boundary The magnetic field driven superconductor–metal transition .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' 13 conditions, and we can consider smaller lattice sizes without the disturbance of edge effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' We consider Mx × My magnetic unit cells of size Nx × Ny.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' A translation between magnetic unit cells is described by a vector Rlx,ly = (lxNxa, lyNya, 0), where lx(y) ∈ [0, Mx(y) − 1] and a is the lattice constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' By applying periodic boundary conditions through the magnetic Bloch theorem [59], our eigenvectors and eigenenergies acquire an index k = 2πlx MxNxax + 2πly MyNyay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='18) This allows us to solve the BdG equations for a system size of Nx ×Ny for MxMy values of k, rather than for a system of size NxMx×NyMy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' We choose to absorb a k dependent phase factor into the eigenvector so that � ui,n,k vi,n,k � = eik·ri � ˜ui,n,k ˜vi,n,k � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='19) The BdG equations now take the form � j eik·(rj−ri) � ϵi,j ∆i,j ∆∗ j,i −ϵj,i � � ˜uj,n,k ˜vj,n,k � = En,k � ˜ui,n,k ˜vi,n,k � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='20) These are solved together with the self-consistency equation for the superconducting pairing correlations ∆i,j = U MxMy � n,k eik·(ri−rj)˜ui,n,k(˜vj,n,k)∗[1 − fFD(En,k)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='21) Inside a magnetic unit cell, the only phase factors we need to consider are the Peierls phases associated with electron hopping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' The nonzero Peierls phases are φi±y,i = ∓πφix for nearest neighbor hopping, φi±x±′y,i = ∓′πφ(ix ± 1/2) for next nearest neighbor hopping, and φi±y,i = ∓2πφix for third nearest neighbor hopping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' These depend on the magnetic field through φ = NΦSC 0 /NxNy = Ba2/ΦSC 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' When site i and j in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='20) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='21) lies in different magnetic unit cells, we need to apply the translation Rlx,ly to one of the eigenvectors so that all eigenvalues lie in the same magnetic unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' Upon such a translation, the eigenvalues pick up an additional phase through the boundary condition � ˜ui,n,k(ri + Rlx,ly) ˜vi,n,k(ri + Rlx,ly) � = � e−iχ(ri,Rlx,ly )/2˜ui,n,k(ri) e+iχ(ri,Rlx,ly )/2˜vi,n,k(ri) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='22) The phase χ(ri, Rlx,ly) = 2π ΦSC 0 A(Rlx,ly) · ri = 2πφlxNxiy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='23) The magnetic field driven superconductor–metal transition .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' 14 is the total phase picked up by the superconducting pairing through the translation ∆d i(ri − Rlx,ly) = ∆d i(ri)eiχ(ri,Rlx,ly ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='24) By solving the reduced BdG equation in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='20) together with the self-consistency equation in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='21) and the boundary condition in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='22), we obtain eigenenergies and eigenvalues that we can use to calculate physical observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' Physical observables We here give the expressions for the physical observables in terms of the eigenenergies and eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' The hole concentration is given by x = 1 NxNyMxMy � i,n,k � 1 − |˜ui,n,k|2fFD(En,k) − |˜vi,n,k|2� 1 − fFD(En,k) �� (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='25) and determines the doping level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' The d-wave superconducting pairing is calculated using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' (4) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' The superfluid stiffness is calculated from the Kubo formula in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' (5), where we insert the expectation value of the kinetic energy associated with the x oriented bonds ⟨−Kx⟩ = 1 NxNyMxMy � i,δ,n,k δ2 x � ti+δ,ieiφi+δ,i� ˜u∗ i+δ,n,k˜ui,n,ke−ik·δfFD(En) + ˜v∗ i,n,k˜vi+δ,n,keik·δ� 1 − fFD(En) �� + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='c � (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='26) and the current-current correlation function Λxx(q, ω) = 1 NxNy(MxMy)2 � n,k,m,k′ fFD(En,k) − fFD(Em,k′) ω + iδ + En,k − Em,k′ An,k,m,k′(−q)Am,k′,n,k(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='27) Above, δ = {x, y, x ± y, 2x, 2y} and Am,k′,n,k(q) = � i,δ δxti+δ,iei(q−k′+k)·ri�� ˜u∗ i+δ,m,k′˜ui,n,ke−ik′·δ − ˜v∗ i,m,k′˜vi+δ,n,keik·δ� eiφi+δ,i + � ˜v∗ i+δ,m,k′˜vi,n,ke−ik′·δ − ˜u∗ i,m,k′˜ui+δ,n,keik·δ� e−iφi+δ,i� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='28) The bond currents can be obtained by multiplying Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='26) with i and reversing the sign of the complex conjugate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' 2, we have included bond currents along all bonds by removing the factor δ2 x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' Finally, the local density of states is given by Ni = 1 MxMy � n,k � |˜ui,n,k|2δ(ω − En,k) + |˜vi,n,k|2δ(ω + En,k) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='29) In our numerical calculation, the δ-function is approximated by δ(x) = (1/π)[Γ/(x2 + Γ2)], where Γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' The magnetic field driven superconductor–metal transition .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' 15 [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' Damascelli, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' Hussain, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content='-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' Shen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' Angle-resolved photoemission studies of the cuprate superconductors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf'} +page_content=' Mod.' metadata={'source': 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